Definition:Zero Matrix/General Monoid

Definition
Let $\struct {S, \circ}$ be a monoid whose identity is $e$.

Let $\map {\mathcal M_S} {m, n}$ be an $m \times n$ matrix space over $S$.

The zero matrix of $\map {\mathcal M_S} {m, n}$ is the $m \times n$ matrix whose elements are all $e$, and can be written $\sqbrk e_{m n}$.

If the monoid $S$ is a number field in which the additive identity is represented as $0$, the zero matrix is then given as $\mathbf 0 = \sqbrk 0_{m n}$.

Also known as
Some sources give this as null matrix.

Also see

 * Zero Matrix is Identity for Matrix Entrywise Addition
 * Definition:Zero Row or Column