Characterization and properties of matrices with \(k\)-involutory symmetries. II
A matrix is \(k\)-involutory if its minimal polynomial is \(x^{k-1}\) for some \(k^{2}\), so \(R^{k-1}=R-1\) and the eigenvalues of \(R\) are 1, \(\xi,\xi ^{2},\dots,\xi^{k-1}\), where \(\xi=e^{2\pi i/k}\). Let \(\alpha,\mu \in\{0,1,\dots,k-1\}\). If \(R\in\mathbb{C}^{m\times m},A\in\mathbb{C}^{m\times n},S\in\mathbb{C}^{n\times n},\)and \(R\) and \(S\) are \(k\)-involutory, A is called \((R,S,\alpha,\mu)\)-symmetric if \(RAS^{-\alpha}=\xi^{\mu}A\). The author shows that an \((R,S,\alpha,\mu)\)-symmetric matrix \(A\) can be usefully represented in terms of matrices \(F_{\ell}\in\mathbb{C}^{c_{\alpha^{\ell}+u}\times d_{\ell}}\), \(0\leq\ell\leq k-1,\) where \(c^{\ell}\) and \(d_{\ell}\) are respectively the dimensions of the \(\xi^{\ell}\)-eigenspaces of \(R\) and \(S\). This continues a theme initiated part I [Linear Algebra Appl. 429, No. 8-9, 2278--2290 (2008; Zbl 1167.15020)], in which the author assumed that \(\alpha=1\). A \(k\)-involution is equidimensional with width \(d\) if all of its eigenspaces have dimension \(d\). The author shows that if \(R\) and \(S\) are equidimensional \(k\)-involutions with widths \(d_{1}\) and \(d_{2}\) respectively, then \((R,S,\alpha,\mu)-\)symmetric matrices are closely related to generalized \(\alpha\)-circulants \([\xi^{\mu}A_{s-\alpha r}]_{r,s=0}^{k-1}\), where \(A_{0},\dots,A_{k-1}\in \mathbb{C}^{d_{1}\times d_{2}}\). For this case the results are new even if \(\alpha=1\). The author also gives an explicit formula for the Moore-Penrose inverse of a unilevel block circulant \([A_{s-\alpha r}]_{r,s=0}^{k-1}\) for any \(\alpha\in\{0,1,\dots,k-1\}\), generalizing a result previously obtained for the case where gcd \((\alpha,k)=1\).
scientific article; zbMATH DE number 5702821
scientific article