A family of doubly stochastic matrices involving Chebyshev polynomials

Abstract

A doubly stochastic matrix is a square matrix \(A=(a_{ij})\) of non-negative real numbers such that \(\sum_{i}a_{ij}=\sum_{j}a_{ij}=1\). The Chebyshev polynomial of the first kind is defined by the recurrence relation \(T_0(x)=1, T_1(x)=x\), and \[T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).\]In this paper, we show a \(2^k\times 2^k\) (for each integer \(k\geq 1\)) doubly stochastic matrix whose characteristic polynomial is \(x^2-1\) times a product of irreducible Chebyshev polynomials of the first kind (up to rescaling by rational numbers).