primitive matrix

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

A sufficient condition for a matrix to be a primitive matrix is for the matrix tobe a nonnegative, irreducible matrix with a positive element on the main diagonal.