On the spectra of a family of positive matrices

It is shown that every positive matrix A can be embedded in an analytic family of positive matrices {A(ν) : ν∈$\text{R}$} in such a way that A(1)=A, $\text{A(0)≡}\text{A}\text{?}$ is symmetric, and A(-1)=A^T. A necessary and sufficient condition that A and Å have the same maximal eigenvalue and that their ergodic limits have the same diagonal elements is stated and proved.