Definition:Negative Matrix/General Group
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Theorem
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
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 $a \in G$.