A peculiar permutation phenomenon arising from the singular vector entries of a special class of Toeplitz matrices

A special Toeplitz matrix of the form Tn(fn)=T0,n+fn(T0,n−1)⁎, where T0,n is the n×n Toeplitz matrix with ones on the diagonal and negative ones on the first upper diagonal, and fn some sequence of positive numbers is considered. We explain the presence of a peculiar permutation phenomenon occurring in its singular vector entries. This is achieved by finding explicit formulas for its singular values and the entries of its singular vectors. In addition, these explicit formulas lead to determining Tn−1(fn), ‖Tn(fn=1/n)‖ and ‖Tn−1(fn=1/n)‖ for any n.