Zero matrix

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.

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.

• Identity matrix, the multiplicative identity for matrices
• Matrix of ones, a matrix where all elements are one
• Nilpotent matrix
• Single-entry matrix, a matrix where all but one element is zero