On the pure imaginary quaternionic solutions of the Hurwitz matrix equations (Q865424)

The number of \(n\times n\) pure imaginary quaternionic orthonormal matrices is analysed, i.e. the number \(p\) of the solutions of the Hurwitz matrix equations \[ T_iT_j^*+T_jT_i^*=2\delta_{ij}I,\quad i,j=1,\dots,p \] having the form \(sX\), where \(s\in\{1,i,j,k\}\) and \(X\) belonging to the real vector space of \(n\times n\) matrices with entries in the skew-field of real quaternions. The maximum number of these solutions is determined for \(n=2^m\) and \(m\neq0\)(mod 4). In the case \(m=0\) an evaluation of this number is given, and this is extended to a conjecture for the case \(m\equiv0\)(mod 4), \(m\neq0\). A construction of such numbers of pure imaginary matrices \(sX\) with \(X\) \(\{0,1,-1\}\)-matrices is provided.