Null Matrix
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly
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 matrix (mathemat ...
, a zero matrix or null matrix is a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
all of whose entries are
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
. It also serves as the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary ma ...
of the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structu ...
of m \times n matrices, and is denoted by the symbol O or 0 followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are : 0_ = \begin 0 \end ,\ 0_ = \begin 0 & 0 \\ 0 & 0 \end ,\ 0_ = \begin 0 & 0 & 0 \\ 0 & 0 & 0 \end .\


Properties

The set of m \times n matrices with entries in a
ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
K forms a ring K_. The zero matrix 0_ \, in K_ \, is the matrix with all entries equal to 0_K \, , where 0_K is the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary ma ...
in K. : 0_ = \begin 0_K & 0_K & \cdots & 0_K \\ 0_K & 0_K & \cdots & 0_K \\ \vdots & \vdots & \ddots & \vdots \\ 0_K & 0_K & \cdots & 0_K \end_ The zero matrix is the additive identity in K_ \, . That is, for all A \in K_ \, it satisfies the equation :0_+A = A + 0_ = A. There is exactly one zero matrix of any given dimension ''m''×''n'' (with entries from a given ring), so when the context is clear, one often refers to ''the'' zero matrix. In general, the
zero element In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An ''additive ide ...
of a ring is unique, and is typically denoted by 0 without any
subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, wh ...
indicating the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents the
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 pr ...
which sends all the
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s to the
zero vector In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An '' additive id ...
. It is
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
, meaning that when it is multiplied by itself, the result is itself. The zero matrix is the only matrix whose
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
is 0.


Occurrences

In
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...
regression, if there is a perfect fit to the data, the
annihilator matrix In statistics, the projection matrix (\mathbf), sometimes also called the influence matrix or hat matrix (\mathbf), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes ...
is the zero matrix.


See also

*
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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
, the multiplicative identity for matrices *
Matrix of ones In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one. For example: :J_2 = \begin 1 & 1 \\ 1 & 1 \end,\quad J_3 = \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end,\quad J_ = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & ...
, a matrix where all elements are one *
Nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear trans ...
* Single-entry matrix, a matrix where all but one element is zero


References

{{Matrix classes Matrices (mathematics) 0 (number) Sparse matrices