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 of the
additive group of
matrices, and is denoted by the symbol
or
followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are
:
Properties
The set of
matrices with entries in a
ring K forms a ring
. The zero matrix
in
is the matrix with all entries equal to
, where
is the
additive identity in K.
:
The zero matrix is the additive identity in
. That is, for all
it satisfies the equation
:
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 of a ring is unique, and is typically denoted by 0 without any
subscript 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
vectors to the
zero vector.
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 is 0.
Occurrences
In
ordinary least squares regression, if there is a perfect fit to the data, the
annihilator matrix 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, 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