The monotonicity of the eigenvalues of doubly symmetric matrices (Q1323954)

Recently \textit{W. F. Trench} [SIAM J. Matrix Appl. 11, 601-611 (1990; Zbl 0717.15008)] proved six theorems concerning the dependence on \(t_{n - 1}\) of the eigenvalues and eigenvectors of the real symmetric Toeplitz matrix \((t_{| i - j |)})_{i,j = 1}^ n\). In this paper a different, and somewhat simpler, proof is used to extend the first two of these theorems to the wider class of real symmetric centrosymmetric matrices.