Definition:Negative Matrix

Definition
Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\MM_R} {m, n}$ denote the $m \times n$ matrix space over $\struct {R, +, \circ}$.

Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition.

Then the negative (matrix) of $\sqbrk a_{m n}$ is denoted and defined as:
 * $-\mathbf A := \sqbrk {-a}_{m n}$

where $-a$ is the ring negative of $a$.

General Group
This can be defined in the more general context where the underlying structure of the $m \times n$ matrix space is an arbitrary group:

Also see

 * Negative Matrix is Inverse for Matrix Entrywise Addition