Diagonally scaled permutations and circulant matrices
The following results are obtained. (1) If an \(n \times n\) complex matrix \(R = D_ 1P\) is given, where \(D_ 1\) is a complex diagonal matrix, and \(P\) is a permutation matrix, then \(R\) is permutation similar to the direct sum of matrices of the form \(DC\), where \(D\) is a complex diagonal matrix and \(C\) is a circulant permutation matrix. (2) The authors consider two cases for \(R = DC\): when the scaling matrix \(D\) is nonsingular, and when \(D\) is singular. For the singular case, the authors prove that \(R\) is nilpotent, and obtain upper and lower bounds on the index of nilpotency of \(R\). In addition, the authors represent an arbitrary \(n\times n\) Toeplitz matrix as a sum of matrices of the form \(D(k, \alpha, \beta) C^ k\) for \(k = 1, \dots, n\), where \(D(k, \alpha, \beta)\) is a diagonal matrix.
scientific article; zbMATH DE number 720486
scientific article