In

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 or ...

. Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.
For example, take the following matrix: $\backslash mathbf\; =\; \backslash begin-1\; \&\; \backslash tfrac\; \backslash \backslash \; 1\; \&\; -1\backslash end.$
The first step to compute its inverse is to create the augmented matrix $\backslash left(\backslash begin\; -1\; \&\; \backslash tfrac\; \&\; 1\; \&\; 0\; \backslash \backslash \; 1\; \&\; -1\; \&\; 0\; \&\; 1\; \backslash end\backslash right)\; .$
Call the first row of this matrix $R\_1$ and the second row $R\_2$. Then, add row 1 to row 2 $(R\_1\; +\; R\_2\; \backslash to\; R\_2).$ This yields $\backslash left(\backslash begin\; -1\; \&\; \backslash tfrac\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash tfrac\; \&\; 1\; \&\; 1\; \backslash end\backslash right).$
Next, subtract row 2, multiplied by 3, from row 1 $(R\_1\; -\; 3\backslash ,\; R\_2\; \backslash to\; R\_1),$ which yields $\backslash left(\backslash begin\; -1\; \&\; 0\; \&\; -2\; \&\; -3\; \backslash \backslash \; 0\; \&\; \backslash tfrac\; \&\; 1\; \&\; 1\; \backslash end\backslash right).$
Finally, multiply row 1 by –1 $(-R\_1\; \backslash to\; R\_1)$ and row 2 by 2 $(2\backslash ,\; R\_2\; \backslash to\; R\_2).$ This yields the identity matrix on the left side and the inverse matrix on the right:$\backslash left(\backslash begin\; 1\; \&\; 0\; \&\; 2\; \&\; 3\; \backslash \backslash \; 0\; \&\; 1\; \&\; 2\; \&\; 2\; \backslash end\backslash right).$
Thus, $\backslash mathbf^\; =\; \backslash begin\; 2\; \&\; 3\; \backslash \backslash \; 2\; \&\; 2\; \backslash end.$
The reason it works is that the process of Gaussian Elimination can be viewed as a sequence of applying left matrix mutliplication using elementary row operations using Elementary matrix ($\backslash mathbf\; E\_n$), such as $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; A\; =\; \backslash mathbf\; I.$
Applying right-multiplication using $\backslash mathbf\; A^,$ we get $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; I\; =\; \backslash mathbf\; I\; \backslash mathbf\; A^.$ And the right side $\backslash mathbf\; I\; \backslash mathbf\; A^\; =\; \backslash mathbf\; A^,$ which is the inverse we want.
To obtain $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; I,$ we create the augumented matrix by combining with and applying

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 a ...

of , is the matrix of cofactors, and represents the matrix

Moore-Penrose Inverse Matrix

{{DEFAULTSORT:Invertible Matrix Linear algebra Matrices Determinants Matrix theory

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 matrice ...

, an -by- 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 often ...

is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:$\backslash mathbf\; =\; \backslash mathbf\; =\; \backslash mathbf\_n\; \backslash $
where denotes the -by- 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 or ...

and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix .
A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular 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 bic ...

its 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 a ...

is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is -by- and the rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...

of is equal to (), then has a left inverse, an -by- matrix such that . If has rank (), then it has a right inverse, an -by- matrix such that .
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. However, in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of invertible matrices together with the operation of matrix multiplication (and entries from ring ) form a group, the general linear group of degree , denoted .
Properties

The invertible matrix theorem

Let be a square -by- matrix over a field (e.g., the field of real numbers). The following statements are equivalent (i.e., they are either all true or all false for any given matrix): * There is an -by- matrix such that . * The matrix has a left inverse (that is, there exists a such that ) ''or'' a right inverse (that is, there exists a such that ), in which case both left and right inverses exist and . * is invertible, that is, has an inverse, is nonsingular, and is nondegenerate. * is row-equivalent to the -by-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 or ...

.
* is column-equivalent to the -by- 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 or ...

.
* has pivot positions.
* has full rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...

; that is, .
* Based on the rank , the equation has only the trivial solution and the equation has exactly one solution for each in .
* The kernel of is trivial, that is, it contains only the null vector as an element,
* The columns of are linearly independent.
* The columns of span .
* .
* The columns of form a basis of .
* The linear transformation mapping to is a bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

from to .
* 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 a ...

of is nonzero: .In general, a square matrix over a commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

is invertible if and only if its determinant is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...

in that ring.
* The number 0 is not an 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 ...

of .
* The 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 tr ...

is an invertible matrix (hence rows of are linearly independent, span , and form a basis of ).
* The matrix can be expressed as a finite product of elementary matrices.
Other properties

Furthermore, the following properties hold for an invertible matrix : * $(\backslash mathbf\; A^)^\; =\; \backslash mathbf\; A$ * $(k\; \backslash mathbf\; A)^\; =\; k^\; \backslash mathbf\; A^$ for nonzero scalar * $(\backslash mathbf)^+\; =\; \backslash mathbf\; x^+\; \backslash mathbf\; A^$ if has orthonormal columns, where denotes theMoore–Penrose inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Ro ...

and is a vector
* $(\backslash mathbf\; A^\backslash top)^\; =\; (\backslash mathbf\; A^)^\backslash top$
* For any invertible -by- matrices and , $(\backslash mathbf)^\; =\; \backslash mathbf\; B^\; \backslash mathbf\; A^.$ More generally, if $\backslash mathbf\; A\_1,\; \backslash dots,\; \backslash mathbf\; A\_k$ are invertible -by- matrices, then $(\backslash mathbf\; A\_1\; \backslash mathbf\; A\_2\; \backslash cdots\; \backslash mathbf\; A\_\; \backslash mathbf\; A\_k)^\; =\; \backslash mathbf\; A\_k^\; \backslash mathbf\; A\_^\; \backslash cdots\; \backslash mathbf\; A\_2^\; \backslash mathbf\; A\_1^.$
*$\backslash det\; \backslash mathbf\; A^\; =\; (\backslash det\; \backslash mathbf\; A)^.$
The rows of the inverse matrix of a matrix are 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 ...

to the columns of (and vice versa interchanging rows for columns). To see this, suppose that where the rows of are denoted as $v\_i^$ and the columns of as $u\_j$ for $1\; \backslash leq\; i,j\; \backslash leq\; n.$ Then clearly, the Euclidean inner product of any two $v\_i^\; u\_j\; =\; \backslash delta\_.$ This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of 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 ...

vectors (but not necessarily orthonormal vectors) to the columns of are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse .
A matrix that is its own inverse (i.e., a matrix such that and ), is called an involutory matrix.
In relation to its adjugate

The adjugate of a matrix can be used to find the inverse of as follows: If is an invertible matrix, then : $A^\; =\; \backslash frac\; \backslash operatorname(A).$In relation to the identity matrix

It follows from the associativity of matrix multiplication that if : $\backslash mathbf\; =\; \backslash mathbf\; \backslash $ for ''finite square'' matrices and , then also : $\backslash mathbf\; =\; \backslash mathbf\backslash $Density

Over the field of real numbers, the set of singular -by- matrices, considered as a subset of is anull set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...

, that is, has Lebesgue measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...

. This is true because singular matrices are the roots of 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 a ...

function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...

, almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...

-by- matrices are invertible.
Furthermore, the -by- invertible matrices are a dense open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

in the topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

of all -by- matrices. Equivalently, the set of singular matrices is closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...

and nowhere dense
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywher ...

in the space of -by- matrices.
In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be 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 input ...

.
Examples

An example with rank of n-1 to be a non-invertible matrix : $\backslash mathbf\; =\; \backslash begin\; 2\; \&\; 4\backslash \backslash \; 2\; \&\; 4\; \backslash end\; .$ We can easily see the rank of this 2*2 matrix is one, which is n-1≠n, so it is a non-invertible matrix. Consider the following 2-by-2 matrix: : $\backslash mathbf\; =\; \backslash begin-1\; \&\; \backslash tfrac\; \backslash \backslash \; 1\; \&\; -1\backslash end\; .$ The matrix $\backslash mathbf$ is invertible. To check this, one can compute that $\backslash det\; \backslash mathbf\; =\; -\backslash frac$, which is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix : $\backslash mathbf\; =\; \backslash begin\; -1\; \&\; \backslash tfrac\; \backslash \backslash \; \backslash tfrac\; \&\; -1\; \backslash end\; .$ The determinant of $\backslash mathbf$ is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.Methods of matrix inversion

Gaussian elimination

Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes , the right portion applied the same elementary row operation sequence will become .
Newton's method

A generalization ofNewton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...

as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed:
:$X\_\; =\; 2X\_k\; -\; X\_k\; A\; X\_k.$
Victor Pan and John Reif
John H. Reif (born 1951) is an American academic, and Professor of Computer Science at Duke University, who has made contributions to large number of fields in computer science: ranging from algorithms and computational complexity theory to robot ...

have done work that includes ways of generating a starting seed. Byte magazine
''Byte'' (stylized as ''BYTE'') was a microcomputer magazine, influential in the late 1970s and throughout the 1980s because of its wide-ranging editorial coverage. "''Byte'' magazine, the leading publication serving the homebrew market ..."
'' ...

summarised one of their approaches.
Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic.
Cayley–Hamilton method

The Cayley–Hamilton theorem allows the inverse of to be expressed in terms of , traces and powers of : : $\backslash mathbf^\; =\; \backslash frac\; \backslash sum\_^\; \backslash mathbf^s\; \backslash sum\_\; \backslash prod\_^\; \backslash frac\; \backslash operatorname\backslash left(\backslash mathbf^l\backslash right)^,$ where is dimension of , and is the trace of matrix given by the sum of the main diagonal. The sum is taken over and the sets of all $k\_l\; \backslash geq\; 0$ satisfying the linearDiophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...

: $s\; +\; \backslash sum\_^\; lk\_l\; =\; n\; -\; 1.$
The formula can be rewritten in terms of complete Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno ...

of arguments $t\_l\; =\; -\; (l\; -\; 1)!\; \backslash operatorname\backslash left(A^l\backslash right)$ as
: $\backslash mathbf^\; =\; \backslash frac\; \backslash sum\_^n\; \backslash mathbf^\; \backslash frac\; B\_(t\_1,\; t\_2,\; \backslash ldots,\; t\_).$
Eigendecomposition

If matrix can be eigendecomposed, and if none of its eigenvalues are zero, then is invertible and its inverse is given by : $\backslash mathbf^\; =\; \backslash mathbf\backslash mathbf^\backslash mathbf^,$ where is the square matrix whose -th column is the eigenvector $q\_i$ of , and is thediagonal 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 m ...

whose diagonal elements are the corresponding eigenvalues, that is, $\backslash Lambda\_\; =\; \backslash lambda\_i.$ If
is symmetric, is guaranteed to be an orthogonal matrix, therefore $\backslash mathbf^\; =\; \backslash mathbf^\backslash top\; .$ Furthermore, because is a diagonal matrix, its inverse is easy to calculate:
: $\backslash left;\; href="/html/ALL/l/Lambda^\backslash right.html"\; ;"title="Lambda^\backslash right">Lambda^\backslash right$
Cholesky decomposition

If matrix ispositive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...

, then its inverse can be obtained as
: $\backslash mathbf^\; =\; \backslash left(\backslash mathbf^*\backslash right)^\; \backslash mathbf^\; ,$
where is the lower triangular Cholesky decomposition of , and denotes the conjugate transpose of .
Analytic solution

Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of ''small'' matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors: : $\backslash mathbf^\; =\; \backslash mathbf^\backslash mathrm\; =\; \backslash begin\; \backslash mathbf\_\; \&\; \backslash mathbf\_\; \&\; \backslash cdots\; \&\; \backslash mathbf\_\; \backslash \backslash \; \backslash mathbf\_\; \&\; \backslash mathbf\_\; \&\; \backslash cdots\; \&\; \backslash mathbf\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; \backslash mathbf\_\; \&\; \backslash mathbf\_\; \&\; \backslash cdots\; \&\; \backslash mathbf\_\; \backslash \backslash \; \backslash end$ so that : $\backslash left(\backslash mathbf^\backslash right)\_\; =\; \backslash left(\backslash mathbf^\backslash right)\_\; =\; \backslash left(\backslash mathbf\_\backslash right)$ where is thetranspose
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 tr ...

.
Inversion of 2 × 2 matrices

The ''cofactor equation'' listed above yields the following result for matrices. Inversion of these matrices can be done as follows: : $\backslash mathbf^\; =\; \backslash begin\; a\; \&\; b\; \backslash \backslash \; c\; \&\; d\; \backslash \backslash \; \backslash end^\; =\; \backslash frac\; \backslash begin\; \backslash ,\backslash ,\backslash ,d\; \&\; \backslash !\backslash !-b\; \backslash \backslash \; -c\; \&\; \backslash ,a\; \backslash \backslash \; \backslash end\; =\; \backslash frac\; \backslash begin\; \backslash ,\backslash ,\backslash ,d\; \&\; \backslash !\backslash !-b\; \backslash \backslash \; -c\; \&\; \backslash ,a\; \backslash \backslash \; \backslash end.$ This is possible because is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes. The Cayley–Hamilton method gives : $\backslash mathbf^\; =\; \backslash frac\; \backslash left;\; href="/html/ALL/l/\backslash left(\_\backslash operatorname\backslash mathbf\_\backslash right)\_\backslash mathbf\_-\_\backslash mathbf\_\backslash right.html"\; ;"title="\backslash left(\; \backslash operatorname\backslash mathbf\; \backslash right)\; \backslash mathbf\; -\; \backslash mathbf\; \backslash right">\backslash left(\; \backslash operatorname\backslash mathbf\; \backslash right)\; \backslash mathbf\; -\; \backslash mathbf\; \backslash right$Inversion of 3 × 3 matrices

A computationally efficient matrix inversion is given by : $\backslash mathbf^\; =\; \backslash begin\; a\; \&\; b\; \&\; c\backslash \backslash \; d\; \&\; e\; \&\; f\; \backslash \backslash \; g\; \&\; h\; \&\; i\backslash \backslash \; \backslash end^\; =\; \backslash frac\; \backslash begin\; \backslash ,\; A\; \&\; \backslash ,\; B\; \&\; \backslash ,C\; \backslash \backslash \; \backslash ,\; D\; \&\; \backslash ,\; E\; \&\; \backslash ,\; F\; \backslash \backslash \; \backslash ,\; G\; \&\; \backslash ,\; H\; \&\; \backslash ,\; I\backslash \backslash \; \backslash end^\backslash mathrm\; =\; \backslash frac\; \backslash begin\; \backslash ,\; A\; \&\; \backslash ,\; D\; \&\; \backslash ,G\; \backslash \backslash \; \backslash ,\; B\; \&\; \backslash ,\; E\; \&\; \backslash ,H\; \backslash \backslash \; \backslash ,\; C\; \&\; \backslash ,F\; \&\; \backslash ,\; I\backslash \backslash \; \backslash end$ (where the scalar is not to be confused with the matrix ). If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by : $\backslash begin\; A\; \&=\&\; (ei\; -\; fh),\; \&\backslash quad\&\; D\; \&=\&\; -(bi\; -\; ch),\; \&\backslash quad\&\; G\; \&=\&\; (bf\; -\; ce),\; \backslash \backslash \; B\; \&=\&\; -(di\; -\; fg),\; \&\backslash quad\&\; E\; \&=\&\; (ai\; -\; cg),\; \&\backslash quad\&\; H\; \&=\&\; -(af\; -\; cd),\; \backslash \backslash \; C\; \&=\&\; (dh\; -\; eg),\; \&\backslash quad\&\; F\; \&=\&\; -(ah\; -\; bg),\; \&\backslash quad\&\; I\; \&=\&\; (ae\; -\; bd).\; \backslash \backslash \; \backslash end$ The determinant of can be computed by applying the rule of Sarrus as follows: : $\backslash det(\backslash mathbf)\; =\; aA\; +\; bB\; +\; cC.$ The Cayley–Hamilton decomposition gives : $\backslash mathbf^\; =\; \backslash frac\backslash left(\; \backslash frac\backslash left;\; href="/html/ALL/l/(\backslash operatorname\backslash mathbf)^\_-\_\backslash operatorname(\backslash mathbf^)\backslash right.html"\; ;"title="(\backslash operatorname\backslash mathbf)^\; -\; \backslash operatorname(\backslash mathbf^)\backslash right">(\backslash operatorname\backslash mathbf)^\; -\; \backslash operatorname(\backslash mathbf^)\backslash right$ The general inverse can be expressed concisely in terms of thecross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

and triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...

. If a matrix $\backslash mathbf\; =\; \backslash begin\; \backslash mathbf\_0\; \&\; \backslash mathbf\_1\; \&\; \backslash mathbf\_2\backslash end$ (consisting of three column vectors, $\backslash mathbf\_0$, $\backslash mathbf\_1$, and $\backslash mathbf\_2$) is invertible, its inverse is given by
: $\backslash mathbf^\; =\; \backslash frac\backslash begin\; ^\backslash mathrm\; \backslash \backslash \; ^\backslash mathrm\; \backslash \backslash \; ^\backslash mathrm\; \backslash end.$
The determinant of , , is equal to the triple product of , , and —the volume of the parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...

formed by the rows or columns:
: $\backslash det(\backslash mathbf)\; =\; \backslash mathbf\_0\backslash cdot(\backslash mathbf\_1\backslash times\backslash mathbf\_2).$
The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of is orthogonal to the non-corresponding two columns of (causing the off-diagonal terms of $\backslash mathbf\; =\; \backslash mathbf^\backslash mathbf$ be zero). Dividing by
: $\backslash det(\backslash mathbf)\; =\; \backslash mathbf\_0\backslash cdot(\backslash mathbf\_1\backslash times\backslash mathbf\_2)$
causes the diagonal elements of to be unity. For example, the first diagonal is:
: $1\; =\; \backslash frac\; \backslash mathbf\backslash cdot(\backslash mathbf\_1\backslash times\backslash mathbf\_2).$
Inversion of 4 × 4 matrices

With increasing dimension, expressions for the inverse of get complicated. For , the Cayley–Hamilton method leads to an expression that is still tractable: : $\backslash mathbf^\; =\; \backslash frac\backslash left(\; \backslash frac\backslash left;\; href="/html/ALL/l/(\backslash operatorname\backslash mathbf)^\_-\_3\backslash operatorname\backslash mathbf\backslash operatorname(\backslash mathbf^)\_+\_2\backslash operatorname(\backslash mathbf^)\backslash right.html"\; ;"title="(\backslash operatorname\backslash mathbf)^\; -\; 3\backslash operatorname\backslash mathbf\backslash operatorname(\backslash mathbf^)\; +\; 2\backslash operatorname(\backslash mathbf^)\backslash right">(\backslash operatorname\backslash mathbf)^\; -\; 3\backslash operatorname\backslash mathbf\backslash operatorname(\backslash mathbf^)\; +\; 2\backslash operatorname(\backslash mathbf^)\backslash right$Blockwise inversion

Matrices can also be ''inverted blockwise'' by using the following analytic inversion formula: where , , and are matrix sub-blocks of arbitrary size. ( must be square, so that it can be inverted. Furthermore, and must be nonsingular.) This strategy is particularly advantageous if is diagonal and (theSchur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.
Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...

of ) is a small matrix, since they are the only matrices requiring inversion.
This technique was reinvented several times and is due to Hans Boltz (1923), who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz
Tadeusz Julian Banachiewicz (13 February 1882, Warsaw – 17 November 1954, Kraków) was a Polish astronomer, mathematician and geodesist.
Scientific career
He was educated at University of Warsaw and his thesis was on "reduction co ...

(1937), who generalized it and proved its correctness.
The nullity theorem says that the nullity of equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of equals the nullity of the sub-block in the upper right of the inverse matrix.
The inversion procedure that led to Equation () performed matrix block operations that operated on and first. Instead, if and are operated on first, and provided and are nonsingular, the result is
Equating Equations () and () leads to
where Equation () is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem.
If and are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization
By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
Since a blockwise inversion of an matrix requires inversion of two half-sized matrices and 6 multiplications between two half-sized matrices, it can be shown that a divide and conquer algorithm that uses blockwise inversion to invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally. Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of operations, while the best proven lower bound is .
This formula simplifies significantly when the upper right block matrix is the zero matrix. This formulation is useful when the matrices and have relatively simple inverse formulas (or pseudo inverses in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes
:$\backslash begin\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \&\; \backslash mathbf\; \backslash end^\; =\; \backslash begin\; \backslash mathbf^\; \&\; \backslash mathbf\; \backslash \backslash \; -\backslash mathbf^\backslash mathbf^\; \&\; \backslash mathbf^\; \backslash end.$
By Neumann series

If a matrix has the property that : $\backslash lim\_\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^n\; =\; 0$ then is nonsingular and its inverse may be expressed by aNeumann series A Neumann series is a mathematical series of the form
: \sum_^\infty T^k
where T is an operator and T^k := T^\circ its k times repeated application.
This generalizes the geometric series.
The series is named after the mathematician Carl Neuman ...

:
: $\backslash mathbf\; A^\; =\; \backslash sum\_^\backslash infty\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^n.$
Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducin ...

. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies
: $\backslash sum\_^\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^n\; =\; \backslash prod\_^\backslash left(\backslash mathbf\; I\; +\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^\backslash right)$.
Therefore, only matrix multiplications are needed to compute terms of the sum.
More generally, if is "near" the invertible matrix in the sense that
: $\backslash lim\_\; \backslash left(\backslash mathbf\; I\; -\; \backslash mathbf\; X^\; \backslash mathbf\; A\backslash right)^n\; =\; 0\; \backslash mathrm\; \backslash lim\_\; \backslash left(\backslash mathbf\; I\; -\; \backslash mathbf\; A\; \backslash mathbf\; X^\backslash right)^n\; =\; 0$
then is nonsingular and its inverse is
: $\backslash mathbf\; A^\; =\; \backslash sum\_^\backslash infty\; \backslash left(\backslash mathbf\; X^\; (\backslash mathbf\; X\; -\; \backslash mathbf\; A)\backslash right)^n\; \backslash mathbf\; X^~.$
If it is also the case that has rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...

1 then this simplifies to
: $\backslash mathbf\; A^\; =\; \backslash mathbf\; X^\; -\; \backslash frac~.$
''p''-adic approximation

If is a matrix withinteger
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

or rational coefficients and we seek a solution in arbitrary-precision rationals, then a -adic approximation method converges to an exact solution in , assuming standard matrix multiplication is used. The method relies on solving linear systems via Dixon's method of -adic approximation (each in ) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.
Reciprocal basis vectors method

Given an square matrix $\backslash mathbf\; =\; \backslash left;\; href="/html/ALL/l/x^\_\backslash right.html"\; ;"title="x^\; \backslash right">x^\; \backslash right$, $1\; \backslash leq\; i,j\; \backslash leq\; n$, with rows interpreted as vectors $\backslash mathbf\_\; =\; x^\; \backslash mathbf\_$ ( Einstein summation assumed) where the $\backslash mathbf\_$ are a standardorthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...

of 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 Euclidea ...

$\backslash mathbb^$ ($\backslash mathbf\_\; =\; \backslash mathbf^,\; \backslash mathbf\_\; \backslash cdot\; \backslash mathbf^\; =\; \backslash delta\_i^j$), then using 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 hyperco ...

(or Geometric Algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...

) we compute the reciprocal (sometimes called dual) column vectors:
:$\backslash mathbf^\; =\; x\_\; \backslash mathbf^\; =\; (-1)^\; (\backslash mathbf\_\; \backslash wedge\backslash cdots\backslash wedge\; ()\_\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\_)\; \backslash cdot\; (\backslash mathbf\_\; \backslash wedge\backslash \; \backslash mathbf\_\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\_)^$
as the columns of the inverse matrix $\backslash mathbf^\; =;\; href="/html/ALL/l/\_.html"\; ;"title="\_">\_$ Note that, the place "$()\_$" indicates that "$\backslash mathbf\_$" is removed from that place in the above expression for $\backslash mathbf^$. We then have $\backslash mathbf\backslash mathbf^\; =\; \backslash left;\; href="/html/ALL/l/\backslash mathbf\_\_\backslash cdot\_\backslash mathbf^\_\backslash right.html"\; ;"title="\backslash mathbf\_\; \backslash cdot\; \backslash mathbf^\; \backslash right">\backslash mathbf\_\; \backslash cdot\; \backslash mathbf^\; \backslash right$, where $\backslash delta\_^$ is the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...

. We also have $\backslash mathbf^\backslash mathbf\; =\; \backslash left;\; href="/html/ALL/l/left(\backslash mathbf\_\backslash cdot\backslash mathbf^\backslash right)\backslash left(\backslash mathbf^\backslash cdot\backslash mathbf\_\backslash right)\backslash right.html"\; ;"title="left(\backslash mathbf\_\backslash cdot\backslash mathbf^\backslash right)\backslash left(\backslash mathbf^\backslash cdot\backslash mathbf\_\backslash right)\backslash right">left(\backslash mathbf\_\backslash cdot\backslash mathbf^\backslash right)\backslash left(\backslash mathbf^\backslash cdot\backslash mathbf\_\backslash right)\backslash right$, as required. If the vectors $\backslash mathbf\_$ are not linearly independent, then $(\backslash mathbf\_\; \backslash wedge\; \backslash mathbf\_\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\_)\; =\; 0$ and the matrix $\backslash mathbf$ is not invertible (has no inverse).
Derivative of the matrix inverse

Suppose that the invertible matrix A depends on a parameter ''t''. Then the derivative of the inverse of A with respect to ''t'' is given by : $\backslash frac\; =\; -\; \backslash mathbf^\; \backslash frac\; \backslash mathbf^.$ To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse $\backslash mathbf^\backslash mathbf=\backslash mathbf$ and then solve for the inverse of A: : $\backslash frac\; =\; \backslash frac\backslash mathbf\; +\; \backslash mathbf^\backslash frac\; =\; \backslash frac\; =\; \backslash mathbf.$ Subtracting $\backslash mathbf^\backslash frac$ from both sides of the above and multiplying on the right by $\backslash mathbf^$ gives the correct expression for the derivative of the inverse: : $\backslash frac\; =\; -\; \backslash mathbf^\; \backslash frac\; \backslash mathbf^.$ Similarly, if $\backslash varepsilon$ is a small number then : $\backslash left(\backslash mathbf\; +\; \backslash varepsilon\backslash mathbf\backslash right)^\; =\; \backslash mathbf^\; -\; \backslash varepsilon\; \backslash mathbf^\; \backslash mathbf\; \backslash mathbf^\; +\; \backslash mathcal(\backslash varepsilon^2)\backslash ,.$ More generally, if : $\backslash frac\; =\; \backslash sum\_i\; g\_i\; (\backslash mathbf)\; \backslash frach\_i\; (\backslash mathbf),$ then, : $f\; (\backslash mathbf\; +\; \backslash varepsilon\backslash mathbf)\; =\; f\; (\backslash mathbf)\; +\; \backslash varepsilon\backslash sum\_i\; g\_i\; (\backslash mathbf)\; \backslash mathbf\; h\_i\; (\backslash mathbf)\; +\; \backslash mathcal\backslash left(\backslash varepsilon^2\backslash right).$ Given a positive integer $n$, : $\backslash begin\; \backslash frac\; \&=\; \backslash sum\_^n\; \backslash mathbf^\backslash frac\backslash mathbf^,\backslash \backslash \; \backslash frac\; \&=\; -\backslash sum\_^n\; \backslash mathbf^\backslash frac\backslash mathbf^.\; \backslash end$ Therefore, : $\backslash begin\; (\backslash mathbf\; +\; \backslash varepsilon\; \backslash mathbf)^\; \&=\; \backslash mathbf^\; +\; \backslash varepsilon\; \backslash sum\_^n\; \backslash mathbf^\backslash mathbf\backslash mathbf^\; +\; \backslash mathcal\backslash left(\backslash varepsilon^2\backslash right),\backslash \backslash \; (\backslash mathbf\; +\; \backslash varepsilon\; \backslash mathbf)^\; \&=\; \backslash mathbf^\; -\; \backslash varepsilon\; \backslash sum\_^n\; \backslash mathbf^\backslash mathbf\backslash mathbf^\; +\; \backslash mathcal\backslash left(\backslash varepsilon^2\backslash right).\; \backslash end$Generalized inverse

Some of the properties of inverse matrices are shared bygeneralized 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 in ...

s (for example, the Moore–Penrose inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Ro ...

), which can be defined for any ''m''-by-''n'' matrix.
Applications

For most practical applications, it is ''not'' necessary to invert a matrix to solve asystem of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in t ...

; however, for a unique solution, it ''is'' necessary that the matrix involved be invertible.
Decomposition techniques like 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 p ...

are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.
Regression/least squares

Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.Matrix inverses in real-time simulations

Matrix inversion plays a significant role incomputer 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 de ...

, particularly in 3D graphics rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.
Matrix inverses in MIMO wireless communication

Matrix inversion also plays a significant role in theMIMO
In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wi ...

(Multiple-Input, Multiple-Output) technology in wireless communications. The MIMO system consists of ''N'' transmit and ''M'' receive antennas. Unique signals, occupying the same frequency band, are sent via ''N'' transmit antennas and are received via ''M'' receive antennas. The signal arriving at each receive antenna will be a linear combination of the ''N'' transmitted signals forming an ''N'' × ''M'' transmission matrix H. It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information.
See also

References

Further reading

* * * *External links

* *Moore-Penrose Inverse Matrix

{{DEFAULTSORT:Invertible Matrix Linear algebra Matrices Determinants Matrix theory