Let $K$ denote a proper cone in $\Bbb R^n, n>1$, $\pi(K)$ denote the set of all matrices which leaves the cone invariant. Let $\rho(A)$ denote the spectral radius of the matrix $A$.
Def: Let $\lambda $ be an eigenvalue of $A$. The degree of $\lambda $, deg$\lambda $, is the size of the largest diagonal block in the Jordan canonical form of A, which contains $\lambda $.
Then I want to show that if $\lambda $ is an eigenvalue of A such that $|\lambda| = \rho(A)$, then deg$\lambda \leq $ deg $\rho(A)$.
In other words degree of $\lambda $ can also be thought as the multiplicity of $\lambda $ in the minimal polynomial of $A$.