Multilevel matrices with involutory symmetries and skew symmetries (Q2484478)

The involutory symmetries and skew symmetries of the matrices \(A\in\mathbb{C}^{n\times n}\), the so-called centrosymmetric and skew-centrosymmetric matrices are studied in detail. As a generalization of the known investigations \textit{A. R. Collar} [Q. R. Mech. Appl. Math. 15, 265--281 (1962; Zbl 0106.01205)], \textit{I. J. Good} [Technometrics 12, 925--928 (1970; Zbl 0194.05903)], \textit{A. L. Andrew} [ibid. 15, 405-407 (1973; Zbl 0261.65027)], and so on, the matrices with \(k\)-level block structure [cf. \textit{E. E. Tyrtyshnikov}, Linear Algebra Appl. 232, 1--43 (1996; Zbl 0841.15006)] are considered. As a result, the class of all \((\mathbb{R},\mu)\)-symmetric matrices, in addition to the known \(\mathbb{R}\)-symmetric matrices, is introduced and characterized. Two different cases, with \(\mu= 0\) and \(\mu\neq 0\), are distinguished. It is shown that the problem involving an \((\mathbb{R}, 0)\)- symmetric matrix and the problems involving \((\mathbb{R},\mu)\)-symmetric matrix with \(\mu\neq 0\) split into corresponding problems for \(2^k- 1\) matrices and for \(2^{k-1}- 1\) matrices respectively, with orders summing to \(n\) in both cases. The latter is also true for sum of \((\mathbb{R}, 0)\)- and \((\mathbb{R},\mu)\)- symmetric matrices.