Let $A \in \mathbb{R}^{n \times n}_{\geq 0}$ be a symmetric matrix with positive row sums $\mathbf{d} := A\mathbf{1} > 0$. I am interested in characterizing all those positive diagonal matrices $Z \in diag(\mathbb{R}^n_{> 0})$ for which the eigenvalue problem $\mathbf{x} = Z^{-1}A\mathbf{x}$ allows for an all-positive solution (some $\mathbf{x} \in \mathbb{R}^n_{>0}$).
By Perron-Frobenius-Theorem this holds if and only if the spectral radius $\rho(Z^{-1} A)$ equals $1$. Exactly in this case there is an all-positive eigenvector to the eigenvalue $1$.
For example, $Z := diag(\mathbf{d})$ is a valid choice, since then $Z^{-1} A$ is stochastic and thus has spectral radius 1.
However, in which other ways can $Z$ be chosen in order to let $\rho(Z^{-1} A)=1$?
How to nicely characterize the set of such matrices $Z$?
(Whenever helpful, it might be assumed that $A$ is irreducible.)