Let $k\geq 1$, and let $M$ be an irreducible non-negative $k\times k$ matrix, where irreducible means that for every $i,j \in \{1,\dots,k\}$ there exists $n \in \mathbb{N}$ such that $(M^n)_{ij}>0$. Then there is a real number $\lambda>0$ and vectors $r,l\in\mathbb R^k$ with all coordinates strictly positive such that

\begin{align*} Mr=\lambda r, \qquad l^\top M=\lambda l^\top. \end{align*}

The number $\lambda$ is the spectral radius of $M$, and the corresponding positive right and left eigenvectors are unique up to positive scalar multiples. If $M$ is aperiodic, meaning that $\gcd\{n \in \mathbb{N}: (M^n)_{ii}>0\}=1$ for one, equivalently every, index $i$, then after normalising $l^\top r=1$ one has $\lambda^{-n}M^n\to rl^\top$, a positive rank-one matrix.