A family of doubly stochastic matrices involving Chebyshev polynomials

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).