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Mathematics of Computation

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On the nonsingularity of real matrices

Author: A. J. Hoffman
Journal: Math. Comp. 19 (1965), 56-61
MSC: Primary 15.25; Secondary 15.20
MathSciNet review: 0174566
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Abstract: By exploiting the theory of linear inequalities, new bounds for the real eigenvalues of a real matrix are derived, along with sufficient conditions for matrix games to be completely mixed, for determinants to be positive, etc. The simple observation on which the derivation of new results and the unification of old results are based is that the typical conditions of diagonal dominance which insure the nonsingularity of matrices are essentially systems of linear inequalities on the rows of the matrices.

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  • H. F. Bohnenblust, S. Karlin, and L. S. Shapley, Solutions of discrete, two-person games, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N. J., 1950, pp. 51–72. MR 0039218
  • A. J. Goldman, Recognition of completely mixed games, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 23–29. MR 177818
  • L. Negrescu, A. Németh & T. Rus, "Sur les solutions positives d’un systeme d’équations linéaires," Mathematica (Cluj), v. 4 (27), 1962, p. 65–69.
  • Olga Taussky, A recurring theorem on determinants, Amer. Math. Monthly 56 (1949), 672–676. MR 32557, DOI

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Article copyright: © Copyright 1965 American Mathematical Society