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 −1 N) ofM −1 N 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.
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Literatur
Mangasarian, O. L.: A convergent splitting of matrices. Numer. Math.15, 351–353 (1970).
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York-London: Academic Press 1970.
Varga, R. S.: Matrix iterative analysis. Englewood Cliffs, New Jersey: Prentice Hall 1962.
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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet
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Alefeld, G. Über konvergente Zerlegungen von Matrizen. Numer. Math. 20, 312–316 (1972). https://doi.org/10.1007/BF01407373
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DOI: https://doi.org/10.1007/BF01407373