Definition:Negative Matrix
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 $a \in G$.
Also see
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices