How many positive eigenvalues for a symmetric doubly stochastic matrix

Question

one interesting question is here about spectrum of symmetric doubly stochastic matrix.

Given matrix $A\in R^{n \times n}$, which is a symmetric doubly stochastic matrix. and its spectrum is $\lambda_i$ for $1 \le i \le n$.

How many positive eigenvalue for $A$?

Is there some theory about that: such as at least half of eigenvalue are positive?

Some counter example exist?

Thanks.

Answer 1

There is at least one, because the stationary distribution has eigenvalue $1$. The example where all entries of $A$ are equal to $1/n$ has $0$ as an $(n-1)$-fold eigenvalue. So in general, you are only entitled to one positive eignenvalue.