# A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple eigenvalue of $A$, and every other eigenvalue of $A$ is smaller in modulus. If we let $\lambda_1(A),\ldots,\lambda_d(A)$ be a listing of the eigenvalues of $A$ in order of decreasing modulus (allowing repetition in the event of multiplicity) then the Perron-Frobenius theorem tells us in particular that if $A$ is positive, then $|\lambda_1(A)|>|\lambda_2(A)|$.

If we write $\sigma_1(A),\ldots,\sigma_d(A)$ for the singular values of $A$, then using the well-known formula $$|\lambda_k(A)|=\lim_{n \to \infty} \sigma_k(A^n)^{\frac{1}{n}}$$ we can restate the above result as follows: if $A$ is positive, then there exist $C>0$ and $\tau \in (0,1)$ such that $$\sigma_2(A^n) \leq C\tau^n\sigma_1(A^n)$$ for all $n \geq 1$. I am interested in generalisations of this property to sets of matrices (which in my situation happen to be invertible). In particular I need to use the following result: if $A_1,\ldots,A_r$ are positive $d \times d$ matrices, then there exist $C>0$ and $\tau \in (0,1)$ such that $$\sigma_2(A_{x_n}\cdots A_{x_1}) \leq C\tau^n\sigma_1(A_{x_n}\cdots A_{x_1})$$ for every $x_1,\ldots,x_n \in \{1,\ldots,r\}$ and every $n \geq 1$. I can see that this result follows from a much stronger theorem due to J. Bochi and N. Gourmelon (Some characterizations of domination, Mathematische Zeitschrift 263 (2009) 221--231). However, given that the Perron-Frobenius theorem is over 100 years old, I would be a little surprised if the oldest reference for the above result is as recent as 2009. Is anyone able to point out to me an older reference?