A family of doubly stochastic matrices involving chebyshev polynomials

Abstract

A doubly stochastic matrix is a square matrix A = (aij ) of non-negative real numbers such that P i aij = P j aij = 1.

The Chebyshev polynomial of the first kind is defined by the recur- rence relation T0(x) = 1, T1(x) = x, and

Tn+1(x) = 2xTn(x) − Tn−1(x).

In this paper, we show a 2 k × 2 k (for each integer k > 1) doubly stochastic matrix whose characteristic polynomial is x 2 − 1 times a product of irreducible Chebyshev polynomials of the first kind (upto rescaling by rational numbers).