Note on the Fundamental Theorem on Irreducible Non-Negative Matrices
Hans Schneider · Proceedings of the Edinburgh Mathematical Society · 1958
Let A = [ a ij ] be an n -th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n >l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).