Is there anything special about the trace of a doubly-stochastic matrix ?

Formally, let $\mathbf{A}$ be doubly-stochastic of size $n$, and write $\mathrm{Tr}(\mathbf{A}) = \sum_{i = 1}^{n} \mathbf{A}_{ii}$. Are there additional properties to $\mathrm{Tr}(\mathbf{A})$, than if $\mathbf{A}$ was stochastic but not doubly-stochastic ?

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In both cases (stochastic/ doubly stochastic) the trace is a number in $[0,n]$ and nothing more can be said about the trace. In other words, for any $a \in [0,n]$ there exists a doubly stochastic matrix with trace $a$.

  • So that means that there is nothing special about the sum of the eigenvalues. Which leads me to wonder if there are additional spectral gap properties (except all the ones that arise from being a stochastic matrix, having all values in (-1, 1) and for the leading value to be equal to 1). – ippiki-ookami 2 days ago

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