# 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 $$a$$.