Does anyone have a good resource for the proof behind primitive matrices. PlanetMath gives the definition:

A nonnegative square matrix A=(a_{ij} is said to be a primitive matrix if there exists k such that A^k > 0, i.e., if there exists k such that for all i,j, the $ (i,j) entry of A^k is positive.

I believe there's a theorem that states what the largest k is that you'll have to check to determine if a matrix is primitive...I wonder if there's a theorem that states what the smallest k is that you have to check..