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 $a \in G$.


Also see