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 ∑i aij =∑j aij =1. The Chebyshev polynomial of the first kind is defined by the recurrence relation T₀ (x) = 1, T₁ (x) = x, and Tn+1(x) = 2xTn(x) − Tn−1(x). In this paper, we show a 2ᵏ ×2ᵏ (for each integer k > 1) doubly stochastic matrix whose characteristic polynomial is x² − 1 times a product of irreducible Chebyshev polynomials of the first kind (upto rescaling by rational numbers).