# How many positive eigenvalues for a symmetric doubly stochastic matrix?

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.

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.