# Definition:Negative Matrix

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## Definition

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $\GF$ denote one of the standard number systems.

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

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:

$-\mathbf A := \sqbrk {-a}_{m n}$
 $\ds -\mathbf A$ $=$ $\ds -1 \mathbf A$ $\ds$ $=$ $\ds \sqbrk {-a}_{m n}$

where:

$-1 \mathbf A$ denotes the matrix scalar product of $-1$ with $\mathbf A$
$-a$ is the negative of $a$.

### Ring

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 $\map {\MM_R} {m, n}$.

Then the negative (matrix) of $\mathbf A$ 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:

Let $\struct {G, \cdot}$ be a group.

Let $\map {\MM_G} {m, n}$ denote the $m \times n$ matrix space over $\struct {G, \cdot}$.

Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\struct {\map {\MM_G} {m, n}, \circ}$, where $\circ$ is the Hadamard product.

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

$-\mathbf A := \sqbrk {a^{-1} }_{m n}$

where $a^{-1}$ is the inverse element of $a \in G$.