Characterization and properties of matrices with \(k\)-involutory symmetries
A matrix \(R\in \mathbb{C}^{n\times n}\) is \(k\)-involutory if its minimal polynomial is \(x^{k-1}\) for some \(k\geq 2\), so \(R^{k-1}=R^{-1}\) and the eigenvalues of \(R\) are \(1,\zeta ,\zeta ^{2},\dots ,\zeta ^{k-1}\), where \( \zeta =e^{2\pi i/k}\). Let \( \mu\in \{0,1,\dots ,k-1\}.\) 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 \((R,S, \mu)\)-symmetric if \(RAS^{-1}=\zeta ^{\mu}A\). If \(R,A\in \mathbb{C}^{n\times n}\), \(A\) is called \((R,\mu)\)-symmetric if \(RAR^{-1}=\zeta ^{\mu}A\). In this paper, the author shows that an \((R,S,\mu)\)-symmetric matrix \(A\) can be represented in terms of matrices \(F^{s}\in \mathbb{C}^{c_{s+\mu}\times d_{s}}\), \(0\leq s\leq k-1\), where \(c_{s}\) and \(d_{s}\) are, respectively, the dimensions of the \(\zeta ^{s}\)-eigenspaces of \(R\) and \(S \) and \(+\) denotes addition modulo \(k\). The system \(Az=w\) can be solved by solving \(k\) independent systems with the matrices \(F_{0},F_{1},\dots ,F_{k-1}\). If \(A\) is invertible then \(A^{-1}\) is can be expressed in terms of \(F_{0}^{-1},F_{1}^{-1},\dots ,F_{k-1}^{-1}.\) If \(R\) and \(S\) are unitary then the Moore-Penrose inverse \(A^{\dag }\) of \(A\) can be written in terms of \(F_{0}^{\dag},F_{1}^{\dag },\dots ,F_{k-1}^{\dag }\), and a singular value decomposition of \(A\) can be written simply in terms of singular value decompositions of \(F_{0},F_{1},\dots ,F_{k-1}\). If \(A\) is \((R,0)\)-symmetric then solving the eigenvalue problem for \(A\) reduces to solving the eigenvalue problems for \(F_{0},F_{1},\dots ,F_{k-1}\). The author also solves the eigenvalue problem for the more complicated case where \(A\) is \((R,\mu)\)-symmetric with \(\mu\in \{1,\dots ,k-1\}\).
scientific article; zbMATH DE number 5349189
scientific article