On the ranks of skew centrosymmetric matrices over finite fields (Q1267008)

Let \(F\) be a field of characteristic different of 2. A skew centrosymmetric \(n\times n\) matrix \(A\) is a skew matrix which is also symmetric around the secondary diagonal, i.e., satisfying \(a_{ij}=a_{ji}\) and \(a_{ij}=a_{n+1-j,n+1-i}\). In this paper it is shown that any skew centrosymmetric matrix \(A\) is similar to a matrix of form \(\left(\begin{smallmatrix} 0&C\\ D&0\end{smallmatrix}\right)\). If \(n=2k\), then \(C\) is \(k\times k\) and \(D=-C^T\). If \(n=2k+1\), then \(C\) is \(k\times(k+1)\) and \(D=-C^T\cdot\text{diag}(1,\dots,1,1/2)\). Hence the rank of \(A\) is equal to twice the rank of \(C\). This result yields a formula for the number of skew centrosymmetric matrices of any particular rank over a finite field \(F\).