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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.

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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.

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