Let $A \in \mathbb{R}^{d \times d}$ be row stochastic (i.e., all entries of $A$ are $\ge 0$, and each row sums up to $1$.
Let $G(A)$ denote the directed graph that is associated to $A$, i.e., $G(A)$ has vertices $\{1, \dots, n\}$, and there is an edge from $j$ to $k$ iff $A_{jk} > 0$. Let us call a non-empty subset $S \subseteq \{1, \dots, n\}$ a sink of $G(A)$ if, from every vertex of $G(A)$, there is a path to at least one point in $S$.
Theorem. (a) One has $$ \dim \operatorname{Fix}(A) = \min \Big\{ \lvert S \rvert : \, S \subseteq \{1, \dots, n\} \text{ is a sink of } G(A) \Big\}, $$ where $\operatorname{Fix}(A) := \ker(\operatorname{id} - A)$ denotes the fixed space of $A$ (in other words: the eigenspace of $A$ for the eigenvalue $1$).
(b) If $S$ is a sink of $G(A)$ and $A_{jj} > 0$ for all $j \in S$, then $A$ is aperiodic, i.e., $A$ does not have any eigenvalues in the unit circle except for $1$.
The theorem can, for instance, be proved by combining the classical Perron-Frobenius theorem with a bit of vector lattice theory (but it seems likely that one can find several different proofs).
A few colleagues and myself would like to apply this result in an article. It seems very likely that this is known - so we would prefer giving a reference instead of including a proof -, but I couldn't find it in the classical textbooks on (nonnegative) matrices (more specifically, I checked in Gantmacher, Berman and Plemmons, and Minc).
Question: Do you know a reference that contains the theorem?
Remark. The theorem also holds if the assumption that $A$ be row stochastic is replaced with the weaker assumption that $A$ has nonnegative entries and a fixed vector whose entries are all strictly positive. A reference for this more general result would be even better - but it can also reduced to the theorem above by a similarity transformation.
