Let $N\in\mathbb{N}$. Define the matrix $Q\in\mathbb{R}^{(N+1)\times(N+1)}$ by $$Q_{ij}=\binom{N}{j}\Big(\frac{i}{N}\Big)^j\Big(1-\frac{i}{N}\Big)^{N-j}$$ for all $0\le i,j\le N$.
The sum of each row is $1$, so $(1,\dots,1)$ certainly is an eigenvector with eigenvalue $1$.
Is there a way to directly read the other eigenvectors / eigenvalues of this matrix $Q$?