Note on the Fundamental Theorem on Irreducible Non - Negative Matrices
نویسنده
چکیده
1. Let A = [aii] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root p (provided that n> I), which is simple and maximal in the sense that every characteristic root A satisfies I A I ~ p, and the characteristic vector x belonging to p 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). We shall sketch part of the proof. For each non-negative column vector Y we set p*( y)=sup r : ry~Ay, .............................. (1)
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