TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a matrix (plural matrices) is a
rectangular In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
or table of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s,
symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different con ...
s, or expressions, arranged in rows and columns, which is used to represent a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
or a property of such an object. For example, $\begin1 & 9 & -13 \\20 & 5 & -6 \end$ is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
. Therefore, the study of matrices is a large part of linear algebra, and most properties and
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
of abstract linear algebra can be expressed in terms of matrices. For example,
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
represents
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.'' Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a
noncommutative ring In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
, which is one of the most common examples of a noncommutative ring. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of a square matrix is a number associated to the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
if and only if it has a nonzero determinant, and the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of a square matrix are the roots of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
determinant. In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, matrices are widely used for specifying and representing
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s (for example
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s) and
coordinate change In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consid ...
s. In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis.

# Definition

A ''matrix'' is a rectangular array of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s (or other mathematical objects), called the ''entries'' of the matrix. Matrices are subject to standard
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
such as
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
. Most commonly, a matrix over a field ''F'' is a rectangular array of elements of ''F''. A real matrix and a complex matrix are matrices whose entries are respectively
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. More general types of entries are discussed below. For instance, this is a real matrix: :$\mathbf = \begin -1.3 & 0.6 \\ 20.4 & 5.5 \\ 9.7 & -6.2 \end.$ The numbers, symbols, or expressions in the matrix are called its ''entries'' or its ''elements''. The horizontal and vertical lines of entries in a matrix are called ''rows'' and ''columns'', respectively.

## Size

The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the numbers of rows and columns a matrix (in the usual sense) can have as long as they are positive integers. A matrix with ''m'' rows and ''n'' columns is called an ''m''×''n'' matrix, or ''m''-by-''n'' matrix, while ''m'' and ''n'' are called its ''dimensions''. For example, the matrix A above is a 3×2 matrix. Matrices with a single row are called '' row vectors'', and those with a single column are called '' column vectors''. A matrix with the same number of rows and columns is called a ''
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
''. A matrix with an infinite number of rows or columns (or both) is called an ''infinite matrix''. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an ''empty matrix''.

# Notation

The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in box brackets or
parentheses A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
, so that an $m \times n$ matrix A represented as $\mathbf = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \end = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \end.$ This may be abbreviated by writing only a single generic term, possibly along with indices, as in or $\mathbf=\left(a_\right)_$ in the case that $n=m$. Matrices are usually symbolized using
upper-case Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
letters (such as A in the examples above), while the corresponding
lower-case Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
letters, with two subscript indices (e.g., ''a'', or ''a''), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, as in $\underline$. The entry in the ''i''-th row and ''j''-th column of a matrix A is sometimes referred to as the ''i'',''j'' or (''i'', ''j'') entry of the matrix, and commonly denoted by ''a'' or ''a''. Alternative notations for that entry are ''A'' 'i,j''and ''A''. For example, the (1, 3) entry of the following matrix A is 5 (also denoted ''a'', ''a'', ''A'' ,3or ''A''): :$\mathbf=\begin 4 & -7 & \color & 0 \\ -2 & 0 & 11 & 8 \\ 19 & 1 & -3 & 12 \end$ Sometimes, the entries of a matrix can be defined by a formula such as ''a'' = ''f''(''i'', ''j''). For example, each of the entries of the following matrix A is determined by the formula ''a'' = ''i'' − ''j''. :$\mathbf A = \begin 0 & -1 & -2 & -3\\ 1 & 0 & -1 & -2\\ 2 & 1 & 0 & -1 \end$ In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as A = 'i''−''j'' or A = ((''i''−''j'')). If matrix size is ''m'' × ''n'', the above-mentioned formula ''f''(''i'', ''j'') is valid for any ''i'' = 1, ..., ''m'' and any ''j'' = 1, ..., ''n''. This can be either specified separately, or indicated using ''m'' × ''n'' as a subscript. For instance, the matrix A above is 3 × 4, and can be defined as A = 'i'' − ''j''(''i'' = 1, 2, 3; ''j'' = 1, ..., 4), or A = 'i'' − ''j'' Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an ''m''-by-''n'' matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an ''m''-by-''n'' matrix are indexed by and . This article follows the more common convention in mathematical writing where enumeration starts from 1. An asterisk is occasionally used to refer to whole rows or columns in a matrix. For example, ''a'' refers to the ''i'' row of A, and ''a'' refers to the ''j'' column of A. The
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all ''m''-by-''n'' real matrices is often denoted $\mathcal\left(m, n\right),$ or $\mathcal_\R.$ The set of all ''m''-by-''n'' matrices over another field or over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, is similarly denoted $\mathcal\left(m, n, R\right),$ or $\mathcal_\left(R\right).$ If , that is, in the case of square matrices, one does not repeat the dimension: $\mathcal\left(n, R\right),$ or Often, $M$ is used in place of $\mathcal M.$

# Basic operations

There are a number of basic operations that can be applied to modify matrices, called ''matrix addition'', ''scalar multiplication'', ''transposition'', ''matrix multiplication'', ''row operations'', and ''submatrix''.

## Addition, scalar multiplication, and transposition

Familiar properties of numbers extend to these operations of matrices: for example, addition is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, that is, the matrix sum does not depend on the order of the summands: A+B=B+A. The transpose is compatible with addition and scalar multiplication, as expressed by (''c''A) = ''c''(A) and (A+B)=A+B. Finally, (A)=A.

## Matrix multiplication

''Multiplication'' of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an ''m''-by-''n'' matrix and B is an ''n''-by-''p'' matrix, then their ''matrix product'' AB is the ''m''-by-''p'' matrix whose entries are given by
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of the corresponding row of A and the corresponding column of B: : where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''p''. For example, the underlined entry 2340 in the product is calculated as :$\begin \begin \underline & \underline 3 & \underline 4 \\ 1 & 0 & 0 \\ \end \begin 0 & \underline \\ 1 & \underline \\ 0 & \underline \\ \end &= \begin 3 & \underline \\ 0 & 1000 \\ \end. \end$ Matrix multiplication satisfies the rules (AB)C = A(BC) (
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
), and (A + B)C = AC + BC as well as C(A + B) = CA + CB (left and right
distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are ''m''-by-''n'' and ''n''-by-''k'' matrices, respectively, and Even if both products are defined, they generally need not be equal, that is: :AB ≠ BA, In other words, matrix multiplication is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
,
in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: :$\begin 1 & 2\\ 3 & 4\\ \end \begin 0 & 1\\ 0 & 0\\ \end= \begin 0 & 1\\ 0 & 3\\ \end,$ whereas :$\begin 0 & 1\\ 0 & 0\\ \end \begin 1 & 2\\ 3 & 4\\ \end= \begin 3 & 4\\ 0 & 0\\ \end.$ Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the Hadamard product and the
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors ...
. They arise in solving matrix equations such as the Sylvester equation.

## Row operations

There are three types of row operations: # row addition, that is adding a row to another. # row multiplication, that is multiplying all entries of a row by a non-zero constant; # row switching, that is interchanging two rows of a matrix; These operations are used in several ways, including solving
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...
s and finding matrix inverses.

## Submatrix

A submatrix of a matrix is obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2: :$\mathbf=\begin 1 & \color & 3 & 4 \\ 5 & \color & 7 & 8 \\ \color & \color & \color & \color \end \rightarrow \begin 1 & 3 & 4 \\ 5 & 7 & 8 \end.$ The minors and cofactors of a matrix are found by computing the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first ''k'' rows and columns, for some number ''k'', are the ones that remain; this type of submatrix has also been called a leading principal submatrix.

# Linear equations

Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an ''m''-by-''n'' matrix, x designates a column vector (that is, ''n''×1-matrix) of ''n'' variables ''x'', ''x'', ..., ''x'', and b is an ''m''×1-column vector, then the matrix equation :$\mathbf = \mathbf$ is equivalent to the system of linear equations :$\begin a_x_1 + a_x_2 + &\cdots + a_x_n = b_1 \\ &\ \ \vdots \\ a_x_1 + a_x_2 + &\cdots + a_x_n = b_m \end$ Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If ''n'' = ''m'' and the equations are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
, then this can be done by writing :$\mathbf = \mathbf^ \mathbf$ where A is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its
generalized inverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ...
.

# Linear transformations

Matrices and matrix multiplication reveal their essential features when related to ''linear transformations'', also known as ''linear maps''. A real ''m''-by-''n'' matrix A gives rise to a linear transformation R → R mapping each vector x in R to the (matrix) product Ax, which is a vector in R. Conversely, each linear transformation ''f'': R → R arises from a unique ''m''-by-''n'' matrix A: explicitly, the of A is the ''i'' coordinate of ''f''(e), where e = (0,...,0,1,0,...,0) is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
with 1 in the ''j'' position and 0 elsewhere.
The matrix A is said to represent the linear map ''f'', and A is called the ''transformation matrix'' of ''f''. For example, the 2×2 matrix :$\mathbf = \begin a & c\\b & d \end$ can be viewed as the transform of the
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
into a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of ...
with vertices at , , , and . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors $\begin 0 \\ 0 \end, \begin 1 \\ 0 \end, \begin 1 \\ 1 \end$, and $\begin0 \\ 1\end$ in turn. These vectors define the vertices of the unit square. The following table shows several 2×2 real matrices with the associated linear maps of R. The original is mapped to the grid and shapes. The origin (0,0) is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of maps: if a ''k''-by-''m'' matrix B represents another linear map ''g'': R → R, then the composition is represented by BA since :(''g'' ∘ ''f'')(x) = ''g''(''f''(x)) = ''g''(Ax) = B(Ax) = (BA)x. The last equality follows from the above-mentioned associativity of matrix multiplication. The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensi ...
of the linear map represented by A. The
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
states that the dimension of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of a matrix plus the rank equals the number of columns of the matrix.

# Square matrix

A
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order ''n.'' Any two square matrices of the same order can be added and multiplied. The entries ''a'' form the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.

## Main types

:

### Diagonal and triangular matrix

If all entries of A below the main diagonal are zero, A is called an ''upper
triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal a ...
''. Similarly if all entries of ''A'' above the main diagonal are zero, A is called a ''lower triangular matrix''. If all entries outside the main diagonal are zero, A is called a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
.

### Identity matrix

The ''identity matrix'' I of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
are equal to 1 and all other elements are equal to 0, for example, :$\mathbf_1 = \begin 1 \end, \ \mathbf_2 = \begin 1 & 0 \\ 0 & 1 \end, \ \ldots , \ \mathbf_n = \begin 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end$ It is a square matrix of order ''n'', and also a special kind of
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
. It is called an identity matrix because multiplication with it leaves a matrix unchanged: :AI = IA = A for any ''m''-by-''n'' matrix A. A nonzero scalar multiple of an identity matrix is called a ''scalar'' matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.

### Symmetric or skew-symmetric matrix

A square matrix A that is equal to its transpose, that is, A = A, is a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with r ...
. If instead, A is equal to the negative of its transpose, that is, A = −A, then A is a
skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a ...
. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A = A, where the star or
asterisk The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star. Computer scientists and mathematicians often v ...
denotes the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex ...
of the matrix, that is, the transpose of the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a ...
of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.

### Invertible matrix and its inverse

A square matrix A is called ''
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
'' or ''non-singular'' if there exists a matrix B such that :AB = BA = I , where I is the ''n''×''n''
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
with 1s on the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
and 0s elsewhere. If B exists, it is unique and is called the '' inverse matrix'' of A, denoted A.

### Definite matrix

A symmetric real matrix is called ''positive-definite'' if the associated
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
: has a positive value for every nonzero vector in . If only yields negative values then is ''negative-definite''; if does produce both negative and positive values then is ''indefinite''. If the quadratic form yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. The table at the right shows two possibilities for 2-by-2 matrices. Allowing as input two different vectors instead yields the
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is line ...
associated to : :. In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' replaced respectively by
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
, Hermitian form, sesquilinear form, and
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex ...
.

### Orthogonal matrix

An ''orthogonal matrix'' is a square matrix with real entries whose columns and rows are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
s (that is,
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
vectors). Equivalently, a matrix A is orthogonal if its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
is equal to its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ...
: :$\mathbf^\mathrm=\mathbf^, \,$ which entails :$\mathbf^\mathrm \mathbf = \mathbf \mathbf^\mathrm = \mathbf_n,$ where I is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
of size ''n''. An orthogonal matrix A is necessarily
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
(with inverse ), unitary (), and normal (). The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of any orthogonal matrix is either or . A ''special orthogonal matrix'' is an orthogonal matrix with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
+1. As a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, every orthogonal matrix with determinant is a pure
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant reverses the orientation, i.e., is a composition of a pure reflection and a (possibly null) rotation. The identity matrices have determinant , and are pure rotations by an angle zero. The
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
analogue of an orthogonal matrix is a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose ...
.

## Main operations

### Trace

The
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace' ...
, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors: : tr(AB) = tr(BA). This is immediate from the definition of matrix multiplication: :$\operatorname\left(\mathbf\right) = \sum_^m \sum_^n a_ b_ = \operatorname\left(\mathbf\right).$ It follows that the trace of the product of more than two matrices is independent of
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, m ...
s of the matrices, however this does not in general apply for arbitrary permutations (for example, tr(ABC) ≠ tr(BAC), in general). Also, the trace of a matrix is equal to that of its transpose, that is, :tr(A) = tr(A).

### Determinant

The ''determinant'' of a square matrix A (denoted det(A) or , A, ) is a number encoding certain properties of the matrix. A matrix is invertible
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
its determinant is nonzero. Its
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
equals the area (in R) or volume (in R) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved. The determinant of 2-by-2 matrices is given by :$\det \begina&b\\c&d\end = ad-bc.$ The determinant of 3-by-3 matrices involves 6 terms ( rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions. The determinant of a product of square matrices equals the product of their determinants: :det(AB) = det(A) · det(B), or using alternate notation: :, AB, = , A, · , B, . Adding a multiple of any row to another row, or a multiple of any column to another column does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or ...
s using
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.

### Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying :$\mathbf \mathbf = \lambda \mathbf$ are called an ''eigenvalue'' and an ''eigenvector'' of A, respectively. The number λ is an eigenvalue of an ''n''×''n''-matrix A if and only if A−λI is not invertible, which is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equiv ...
to :$\det\left(\mathbf-\lambda \mathbf\right) = 0.$ The polynomial ''p'' in an indeterminate ''X'' given by evaluation of the determinant det(''X''I−A) is called the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ch ...
of A. It is a
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+ ...
of degree ''n''. Therefore the polynomial equation ''p''(λ)=0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of A are real. According to the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfie ...
, ''p''(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the
zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followe ...
.

# Computational aspects

Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of vectors x converging to an eigenvector when ''n'' tends to
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...
. To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
. As with other numerical situations, two main aspects are the
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to c ...
of algorithms and their numerical stability. Determining the complexity of an algorithm means finding
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an elem ...
s or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices. Calculating the matrix product of two ''n''-by-''n'' matrices using the definition given above needs ''n'' multiplications, since for any of the ''n'' entries of the product, ''n'' multiplications are necessary. The
Strassen algorithm In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although ...
outperforms this "naive" algorithm; it needs only ''n'' multiplications. A refined approach also incorporates specific features of the computing devices. In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the
conjugate gradient method In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterat ...
. An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion (adj(A) denotes the
adjugate matrix In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a diff ...
of A) :A = adj(A) / det(A) may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the
conditioning Conditioning may refer to: Science, computing, and technology * Air conditioning, the removal of heat from indoor air for thermal comfort ** Automobile air conditioning, air conditioning in a vehicle ** Ice storage air conditioning, air conditi ...
of linear algebraic problems, such as computing a matrix's inverse. Most computer
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programmin ...
s support arrays but are not designed with built-in commands for matrices. Instead, available external libraries provide matrix operations on arrays, in nearly all currently used programming languages. Matrix manipulation was among the earliest numerical applications of computers. The original
Dartmouth BASIC Dartmouth BASIC is the original version of the BASIC programming language. It was designed by two professors at Dartmouth College, John G. Kemeny and Thomas E. Kurtz. With the underlying Dartmouth Time Sharing System (DTSS), it offered an inte ...
had built-in commands for matrix arithmetic on arrays from its second edition implementation in 1964. As early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.

# Decomposition

There are several methods to render matrices into a more easily accessible form. They are generally referred to as ''matrix decomposition'' or ''matrix factorization'' techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank, or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices. The
LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a ...
factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The ''Gaussian elimination'' is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.
Singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
expresses any matrix A as a product UDV, where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or ''diagonalization'' expresses A as a product VDV, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique. ...
. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into
Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to ...
, that is to say matrices whose only nonzero entries are the eigenvalues λ to λ of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the ''n'' power of A (that is, ''n''-fold iterated matrix multiplication) can be calculated via :A = (VDV) = VDVVDV...VDV = VDV and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
''e'', a need frequently arising in solving
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
s, matrix logarithms and square roots of matrices. To avoid numerically
ill-conditioned In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
situations, further algorithms such as the Schur decomposition can be employed.

# Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension is
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realized as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers. Matrices, subject to certain requirements tend to form
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
known as matrix groups. Similarly under certain conditions matrices form rings known as
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
s. Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields.

## Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, that is, a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
where
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
operations are defined and well-behaved, may be used instead of R or C, for example
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s or
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, sub ...
s. For example,
coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studi ...
makes use of matrices over finite fields. Wherever
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance, they may be complex in the case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, such as C, from the outset. More generally, matrices with entries in a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
''R'' are widely used in mathematics. Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(''n'', ''R'') (also denoted M''n''(R)) of all square ''n''-by-''n'' matrices over ''R'' is a ring called
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
, isomorphic to the
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of the left ''R''- module ''R''. If the ring ''R'' is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, that is, its multiplication is commutative, then the ring M(''n'', ''R'') is also an associative algebra over ''R''. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of square matrices over a commutative ring ''R'' can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
in ''R'', generalising the situation over a field ''F'', where every nonzero element is invertible. Matrices over superrings are called supermatrices. Matrices do not always have all their entries in the same ring– or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
; but their sizes must fulfill certain compatibility conditions.

## Relationship to linear maps

Linear maps R → R are equivalent to ''m''-by-''n'' matrices, as described above. More generally, any linear map between finite-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s can be described by a matrix A = (''a''), after choosing bases v, ..., v of ''V'', and w, ..., w of ''W'' (so ''n'' is the dimension of ''V'' and ''m'' is the dimension of ''W''), which is such that :$f\left(\mathbf_j\right) = \sum_^m a_ \mathbf_i\qquad\mbox\ j=1,\ldots,n.$ In other words, column ''j'' of ''A'' expresses the image of v in terms of the basis vectors w of ''W''; thus this relation uniquely determines the entries of the matrix A. The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices. Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix A describes the transpose of the linear map given by A, with respect to the dual bases. These properties can be restated more naturally: the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
of all matrices with entries in a field $k$ with multiplication as composition is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equiv ...
to the category of finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s and linear maps over this field. More generally, the set of ''m''×''n'' matrices can be used to represent the ''R''-linear maps between the free modules ''R'' and ''R'' for an arbitrary ring ''R'' with unity. When ''n''=''m'' composition of these maps is possible, and this gives rise to the
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
of ''n''×''n'' matrices representing the
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of ''R''.

## Matrix groups

A
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
is a mathematical structure consisting of a set of objects together with a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary ...
, that is, an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a ''matrix group''. Since a group every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...
s. Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
of (that is, a smaller group contained in) their general linear group, called a special linear group. Orthogonal matrices, determined by the condition :MM = I, form the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. Every orthogonal matrix has
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
1 or −1. Orthogonal matrices with determinant 1 form a subgroup called ''special orthogonal group''. Every
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a matrix group, as one can see by considering the regular representation of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
. General groups can be studied using matrix groups, which are comparatively well understood, by means of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In ess ...
.

## Infinite matrices

It is also possible to consider matrices with infinitely many rows and/or columns even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication, and transposition can still be defined without problem; however, matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general. If ''R'' is any ring with unity, then the ring of endomorphisms of $M=\bigoplus_R$ as a right ''R'' module is isomorphic to the ring of column finite matrices $\mathrm_I\left(R\right)$ whose entries are indexed by $I\times I$, and whose columns each contain only finitely many nonzero entries. The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the row finite matrices $\mathrm_I\left(R\right)$ whose rows each only have finitely many nonzero entries. If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map ''f'': ''V''→''W'', bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector''v'' of
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s, only finitely many entries ''v'' are nonzero. Now the columns of A describe the images by ''f'' of individual basis vectors of ''V'' in the basis of ''W'', which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of ''A'' however: in the product A·''v'' there are only finitely many nonzero coefficients of ''v'' involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries because each of those columns does. Products of two matrices of the given type are well defined (provided that the column-index and row-index sets match), are of the same type, and correspond to the composition of linear maps. If ''R'' is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously, the matrices whose row sums are absolutely convergent series also form a ring. Infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that must be imposed. However, the explicit point of view of matrices tends to obfuscate the matter, and the abstract and more powerful tools of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

## Empty matrix

An ''empty matrix'' is a matrix in which the number of rows or columns (or both) is zero. Empty matrices help dealing with maps involving the zero vector space. For example, if ''A'' is a 3-by-0 matrix and ''B'' is a 0-by-3 matrix, then ''AB'' is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space ''V'' to itself, while ''BA'' is a 0-by-0 matrix. There is no common notation for empty matrices, but most
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
s allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows regarding the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...
occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite-dimensional space to itself has determinant1, a fact that is often used as a part of the characterization of determinants.

# Applications

There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
and
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics ana ...
, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.
Text mining Text mining, also referred to as ''text data mining'', similar to text analytics, is the process of deriving high-quality information from text. It involves "the discovery by computer of new, previously unknown information, by automatically extra ...
and automated
thesaurus A thesaurus (plural ''thesauri'' or ''thesauruses'') or synonym dictionary is a reference work for finding synonyms and sometimes antonyms of words. They are often used by writers to help find the best word to express an idea: Synonym dictiona ...
compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents. Complex numbers can be represented by particular real 2-by-2 matrices via :$a + ib \leftrightarrow \begin a & -b \\ b & a \end,$ under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
1, as above. A similar interpretation is possible for
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s and
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyper ...
s in general. Early
encryption In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can de ...
techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.
Computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation. Matrices over a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
are important in the study of
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to ...
.
Chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
makes use of matrices in various ways, particularly since the use of
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ch ...
to discuss molecular bonding and
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
. Examples are the
overlap matrix In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond ang ...
and the Fock matrix used in solving the Roothaan equations to obtain the
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
s of the
Hartree–Fock method In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often ...
.

## Graph theory

The
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite sim ...
of a finite graph is a basic notion of
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example "yes" and "no", respectively) are called logical matrices. The distance (or cost) matrix contains information about distances of the edges. These concepts can be applied to
website A website (also written as a web site) is a collection of web pages and related content that is identified by a common domain name and published on at least one web server. Examples of notable websites are Google, Facebook, Amazon, and ...
s connected by
hyperlink In computing, a hyperlink, or simply a link, is a digital reference to data that the user can follow or be guided by clicking or tapping. A hyperlink points to a whole document or to a specific element within a document. Hypertext is text wi ...
s or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be
sparse Sparse is a computer software tool designed to find possible coding faults in the Linux kernel. Unlike other such tools, this static analysis tool was initially designed to only flag constructs that were likely to be of interest to kernel dev ...
, that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in
network theory Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be def ...
.

## Analysis and geometry

The
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was devel ...
of a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point ...
''ƒ'': R → R consists of the
second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
s of ''ƒ'' with respect to the several coordinate directions, that is, : It encodes information about the local growth behaviour of the function: given a critical point x=(''x'',...,''x''), that is, a point where the first
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s $\partial f / \partial x_i$ of ''ƒ'' vanish, the function has a local minimum if the Hessian matrix is positive definite.
Quadratic programming Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear const ...
can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above). Another matrix frequently used in geometrical situations is the of a differentiable map ''f'': R → R. If ''f'', ..., ''f'' denote the components of ''f'', then the Jacobi matrix is defined as : If ''n'' > ''m'', and if the rank of the Jacobi matrix attains its maximal value ''m'', ''f'' is locally invertible at that point, by the implicit function theorem.
Partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For
elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or ex ...
s this matrix is positive definite, which has a decisive influence on the set of possible solutions of the equation in question. The
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen concerning a sufficiently fine grid, which in turn can be recast as a matrix equation.

## Probability theory and statistics

Stochastic matrices are square matrices whose rows are probability vectors, that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happ ...
s with finitely many states. A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain-like absorbing states, that is, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices. Statistics also makes use of matrices in many different forms.
Descriptive statistics A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and a ...
is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to
dimensionality reduction Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
techniques. The
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
encodes the mutual
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of num ...
of several
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. Another technique using matrices are linear least squares, a method that approximates a finite set of pairs (''x'', ''y''), (''x'', ''y''), ..., (''x'', ''y''), by a linear function :''y'' ≈ ''ax'' + ''b'', ''i'' = 1, ..., ''N'' which can be formulated in terms of matrices, related to the
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
of matrices. Random matrices are matrices whose entries are random numbers, subject to suitable
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s, such as
matrix normal distribution In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density ...
. Beyond probability theory, they are applied in domains ranging from
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
to
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
.

## Symmetries and transformations in physics

Linear transformations and the associated
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
play a key role in modern physics. For example,
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, an ...
s in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particl ...
are classified as representations of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...
of special relativity and, more specifically, by their behavior under the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...
. Concrete representations involving the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when ...
and more general
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\m ...
are an integral part of the physical description of
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, which behave as
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slig ...
s. For the three lightest
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All comm ...
s, there is a group-theoretical representation involving the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
SU(3); for their calculations, physicists use a convenient matrix representation known as the
Gell-Mann matrices The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group i ...
, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions,
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ...
. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
s are not the same as, but linearly related to the basic quark states that define particles with specific and distinct
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
es.

## Linear combinations of quantum states

The first model of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
(
Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states. This is also referred to as
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quant ...
. One particular example is the
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure"
eigenstates In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
. Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the
S-matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
, which encodes all information about the possible interactions between particles.

## Normal modes

A general application of matrices in physics is the description of linearly coupled harmonic systems. The
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ve ...
of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s, its
normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencie ...
s, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of
molecules A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioc ...
: the internal vibrations of systems consisting of mutually bound component atoms. They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.

## Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix analysis: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a ''refraction matrix'' describing the refraction at a lens surface, and a ''translation matrix'', describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.

## Electronics

Traditional mesh analysis and nodal analysis in electronics lead to a system of linear equations that can be described with a matrix. The behaviour of many
electronic component An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are ...
s can be described using matrices. Let ''A'' be a 2-dimensional vector with the component's input voltage ''v'' and input current ''i'' as its elements, and let ''B'' be a 2-dimensional vector with the component's output voltage ''v'' and output current ''i'' as its elements. Then the behaviour of the electronic component can be described by ''B'' = ''H'' · ''A'', where ''H'' is a 2 x 2 matrix containing one impedance element (''h''), one
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
element (''h''), and two
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
elements (''h'' and ''h''). Calculating a circuit now reduces to multiplying matrices.

# History

Matrices have a long history of application in solving
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...
s but they were known as arrays until the 1800s. The Chinese text ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'' written in 10th–2nd century BCE is the first example of the use of array methods to solve simultaneous equations, including the concept of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
s. In 1545 Italian mathematician
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
introduced the method to Europe when he published ''Ars Magna''.''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 , p. 564-565 The Japanese mathematician
Seki Seki may refer to: Places * Seki, Gifu, a city in Japan * Seki River, a river in Japan * Şəki, a city and provincial capital in Azerbaijan * Şəki (village), a village and municipality in Azerbaijan * Šeki, a small town in Slovenia * Se ...
used the same array methods to solve simultaneous equations in 1683. The Dutch mathematician'' '' Jan de Witt represented transformations using arrays in his 1659 book ''Elements of Curves'' (1659).''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 , p. 564 Between 1700 and 1710
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays. Cramer presented his rule in 1750. The term "matrix" (Latin for "womb","dam" (non-human female animal kept for breeding),"source", "origin", "list", "register", derived from '' mater''—mother) was coined by
James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...
in 1850, who understood a matrix as an object giving rise to several determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead, he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition. Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 Cayley published his ''A memoir on the theory of matrices'' in which he proposed and demonstrated the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfie ...
. The English mathematician Cuthbert Edmund Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = 'a''to represent a matrix where ''a'' refers to the'' i''th row and the ''j''th column. The modern study of determinants sprang from several sources. Number-theoretical problems led
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
to relate coefficients of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s, that is, expressions such as and
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are
non-commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
.
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = 'a''the following: replace the powers ''a'' by ''a'' in the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
:$a_1 a_2 \cdots a_n \prod_ \left(a_j - a_i\right)\;$, where Π denotes the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
of the indicated terms. He also showed, in 1829, that the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of symmetric matrices are real. Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
) level, see above; Kronecker's ''Vorlesungen über die Theorie der Determinanten'' and Weierstrass' ''Zur Determinantentheorie'', both published in 1903, first treated determinants
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
atically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established. Many theorems were first established for small matrices only, for example, the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfie ...
was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices. Frobenius, working on
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is line ...
s, generalized the theorem to all dimensions (1898). Also at the end of the 19th century, the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Wilhelm Jordan. In the early 20th century, matrices attained a central role in linear algebra, partially due to their use in classification of the hypercomplex number systems of the previous century. The inception of
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quant ...
by
Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
, Born and
Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...
led to studying matrices with infinitely many rows and columns. Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s on
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s, which, very roughly speaking, correspond to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, but with an infinity of independent directions.

## Other historical usages of the word "matrix" in mathematics

The word has been used in unusual ways by at least two authors of historical importance.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
and
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applic ...
in their '' Principia Mathematica'' (1910–1913) use the word "matrix" in the context of their axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function is identical to its extension: For example, a function Φ(''x, y'') of two variables ''x'' and ''y'' can be reduced to a ''collection'' of functions of a single variable, for example, ''y'', by "considering" the function for all possible values of "individuals" ''a'' substituted in place of variable ''x''. And then the resulting collection of functions of the single variable ''y'', that is, , can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" ''b'' substituted in place of variable ''y'': :
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
in his 1946 ''Introduction to Logic'' used the word "matrix" synonymously with the notion of
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
as used in mathematical logic.Tarski, Alfred; (1946) ''Introduction to Logic and the Methodology of Deductive Sciences'', Dover Publications, Inc, New York NY, .

* List of named matrices * * * * Irregular matrix * * * Matrix multiplication algorithm *
Tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
— A generalization of matrices with any number of indices

# References

* * * * * * * * * * * * * * * * * * * * * * * . * * * * * * * * * * * * * * * * * * * * * * *

## Physics references

* * * * * * * * *

## Historical references

* A. Cayley ''A memoir on the theory of matrices''. Phil. Trans. 148 1858 17-37; Math. Papers II 475-496 * , reprint of the 1907 original edition * * * * * * * *

* * * *