Definition:Negative Matrix

Definition
Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $\Bbb F$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.

Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\map \MM {m, n}$.

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

where:
 * $-1 \mathbf A$ denotes the matrix scalar product of $-1$ with $\mathbf A$
 * $-a$ 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