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Zero matrix

Summary

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 ${\displaystyle m\times n}$ matrices, and is denoted by the symbol ${\displaystyle O}$ or ${\displaystyle 0}$ followed by subscripts corresponding to the dimension of the matrix as the context sees fit.[1][2][3] Some examples of zero matrices are

${\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\ }$

Properties

The set of ${\displaystyle m\times n}$ matrices with entries in a ring K forms a ring ${\displaystyle K_{m,n}}$. The zero matrix ${\displaystyle 0_{K_{m,n}}\,}$ in ${\displaystyle K_{m,n}\,}$ is the matrix with all entries equal to ${\displaystyle 0_{K}\,}$, where ${\displaystyle 0_{K}}$ is the additive identity in K.

${\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &\ddots &\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}_{m\times n}}$

The zero matrix is the additive identity in ${\displaystyle K_{m,n}\,}$.[4] That is, for all ${\displaystyle A\in K_{m,n}\,}$ it satisfies the equation

${\displaystyle 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.}$

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 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 which sends all the vectors to the zero vector.[5] It is idempotent, meaning that when it is multiplied by itself, the result is itself.

The zero matrix is the only matrix whose rank is 0.

Occurrences

The mortal matrix problem is the problem of determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.[6]

In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.

References

1. ^ Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, We have a zero matrix in which aij = 0 for all ij. ... We shall write it O.
2. ^ "Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13.
3. ^ Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13.
4. ^ Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, The neutral element for addition is called the zero matrix, for all of its entries are zero.
5. ^ Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.
6. ^ Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].