Über konvergente Zerlegungen von Matrizen

Abstract

LetA, M, N ben × n real matrices, letA=M−N, letA andM be nonsingular. LetM′y≧0 implyN′y≧0 (where the prime denotes the transpose). ThenA′y≧0 impliesN′y≧0 if and only if the spectral radius ∂(M⁻¹N) ofM⁻¹N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA′, M′, andN′ are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.