Trace of a doubly-stochastic matrix 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 ?

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 $a$.