Zero matrix

In linear algebra, a zero matrix is a real or complex matrix whose entries are all equal to the number zero . More generally, a matrix over a body or ring is called a zero matrix if all matrix elements correspond to the neutral element of addition in the body or ring. The zero matrix represents the zero mapping between finite-dimensional vector spaces and is itself the neutral element in the vector space or ring of matrices. The most important parameters of a zero matrix, such as determinant , track and rank , are each zero. The transposed , adjoint or complementary matrix of a zero matrix is ​​again a zero matrix.

definition

If a ring has zero elements , then the zero matrix is defined as ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle 0_{mn}\in R^{m\times n}}$

${\displaystyle 0_{mn}={\begin{pmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{pmatrix}}}$.

The entries of a zero matrix are therefore all equal to the zero element of the ring. The zero matrix, provided its dimension is irrelevant and there is no possibility of confusion, is simply marked with or . A matrix with no content, in which the number of rows or columns is zero, is called an "empty matrix". Such a matrix is always a zero matrix and, if square, at the same time identity matrix . ${\displaystyle 0}$ ${\displaystyle \mathbf {0} }$

Examples

If the field of real numbers and denotes the number zero , then examples of zero matrices are: ${\displaystyle R}$ ${\displaystyle 0}$

${\displaystyle 0_{0,0}={\begin{pmatrix}\end{pmatrix}},0_{1,1}={\begin{pmatrix}0\end{pmatrix}},0_{2,2}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},0_{3,4}={\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}}}$

properties

neutrality

Between two finite-dimensional vector spaces over the same body, the zero matrix represents the zero mapping , i.e. the linear mapping that maps all vectors onto the zero vector . If the zero vector is the target space, then applies to all vectors ${\displaystyle 0_{m}\in K^{m}}$ ${\displaystyle x\in K^{n}}$

${\displaystyle 0_{mn}\cdot x=0_{m}}$.

In the vector space of the matrices , the zero matrix itself represents the zero vector with regard to the matrix addition, that is, it applies to all matrices ${\displaystyle A\in K^{m\times n}}$

${\displaystyle 0_{mn}+A=A+0_{mn}=A}$.

Absorbent element

In the die ring is the zero matrix the zero element and corresponding identity matrix identity element. With regard to the matrix multiplication affects the zero matrix as absorbent element , because for all matrices applies ${\displaystyle A\in K^{n\times n}}$

${\displaystyle 0_{nn}\cdot A=A\cdot 0_{nn}=0_{nn}}$.

A -Null matrix therefore has no (multiplicative) inverse , because the product of the zero matrix with any matrix cannot result in the identity matrix. The ring of square matrices is also not free from zero divisors , because it does not necessarily follow or . ${\displaystyle n\times n}$ ${\displaystyle n>0}$ ${\displaystyle A\cdot B=0_{nn}}$ ${\displaystyle A=0_{nn}}$ ${\displaystyle B=0_{nn}}$

Parameters

The following applies to the determinant of a square zero matrix

${\displaystyle \operatorname {det} (0_{nn})={\begin{cases}1&n=0\\0&n>0.\end{cases}}}$

The following applies to the trace of a square zero matrix

${\displaystyle \operatorname {spur} (0_{nn})=0}$.

The same applies to the rank of a zero matrix over a body

${\displaystyle \operatorname {rang} (0_{mn})=0}$,

where zero matrices are the only matrices with rank zero. The characteristic polynomial of a square zero matrix over a body has the form

${\displaystyle \chi (\lambda )=\lambda ^{n}}$.

Thus the only eigenvalue and the associated eigenspace is the whole space. A square zero matrix over the real or complex numbers is both positive semidefinite and negative semidefinite. ${\displaystyle 0}$

Illustrations

Every zero matrix can be represented as the dyadic product of two zero vectors of corresponding length, i.e.

${\displaystyle 0_{mn}=0_{m}\otimes 0_{n}}$.

The transposed matrix , adjoint matrix or complementary matrix of a zero matrix is ​​again a zero matrix in which only the dimensions are exchanged:

${\displaystyle (0_{mn})^{T}=(0_{mn})^{H}=0_{nm}}$   and   . ${\displaystyle \operatorname {adj} (0_{nn})=0_{nn}}$

The matrix exponential of a real or complex square zero matrix is ​​the identity matrix of equal size, in short ${\displaystyle I}$

${\displaystyle e^{0}=I}$.

See also

• One matrix
• Zero ring
• Zero vector space

literature

• Siegfried Bosch : Linear Algebra . 3. Edition. Springer, Berlin et al. 2006, ISBN 3-540-29884-3 .
• Gilbert Strang : Linear Algebra . Springer, Berlin et al. 2003, ISBN 3-540-43949-8 .

Individual evidence

1. ^ Bosch: Linear Algebra . 2006, p. 91 .

Web links

• Eric W. Weisstein : Zero Matrix . In: MathWorld (English).
• Will Jennings, matte: Zero matrix . In: PlanetMath . (English)