Spectral properties of some matrices close to the Toeplitz triangular form (Q1323624)

A class of matrices denoted as nearly Toeplitz is considered (a lower triangular matrix is almost Toeplitz if the \(i\), \(j\) entries outside the main diagonal depend on the difference \(i-j\) only, and if the elements along the main diagonal form an arithmetic progression with positive ratio \(d)\). It is shown that nearly Toeplitz matrix \(A\) can be diagonalized by means of a lower triangular Toeplitz similarity transformation, i.e. there exists a nonsingular matrix \(T\) which is lower triangular Toeplitz and such that \(T^{-1} AT\) is diagonal. If \(A\) is a nearly Toeplitz matrix with ratio 1, then the elements of \(T\), viewed as polynomials in the elements of \(A\), give detailed information on the structure of the symmetric group, i.e. the group of permutations of a finite set. The \((k + 1,1)^{th}\) element of \(T^{- 1}\) is the cycle index of the symmetric group \(S_ k\). The entries of both \(T\) and \(T^{-1}\) can be used to compute the character table of \(S_ k\). Such information is important in combinatorial mathematics, namely in Polya's theory of counting [cf. \textit{G. Polya}, \textit{R. E. Tarjan}, and \textit{D. R. Woods}, Notes on introductory combinatorics (1983; Zbl 0632.05001)] and in the character theory of the symmetric group [cf. \textit{W. Ledermann}, Introduction to group characters (1977; Zbl 0384.20001)].