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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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.
Author information
Authors and Affiliations
Additional information
Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet
Rights and permissions
About this article
Cite this article
Alefeld, G. Über konvergente Zerlegungen von Matrizen. Numer. Math. 20, 312–316 (1972). https://doi.org/10.1007/BF01407373
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01407373