Let $k$ be a field and let $n\in\mathbb{N}$. Let $M_n(k)$ denote the $k$-[vector space](/page/Vector%20Space) of $n\times n$ matrices with entries in $k$. For $A\in M_n(k)$, write $A^\top\in M_n(k)$ for the transpose matrix, defined by $(A^\top)_{ij}=A_{ji}$ for all $1\le i,j\le n$. Define