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

Abstract

A special Toeplitz matrix of the form View the MathML sourceTn(fn)=T0,n+fn(T0,n−1)⁎, where T0,nT0,n is the n×nn×n Toeplitz matrix with ones on the diagonal and negative ones on the first upper diagonal, and fnfn 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 View the MathML sourceTn−1(fn), ‖Tn(fn=1/n)‖‖Tn(fn=1/n)‖ and View the MathML source‖Tn−1(fn=1/n)‖ for any n