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Inequalities Involving the Inverses of Positive Definite Matrices1

Published online by Cambridge University Press:  20 January 2009

Russell Merris
Russell Merris
Affiliation:
California State University, Hayward CA 94542
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Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined by

For example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 1 , February 1979 , pp. 11 - 15
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

(1) Richard, Bellman, Introduction to Matrix Analysis, 2nd ed. (McGraw-Hill, New York, 1970).Google Scholar
(2) Richard, Bellman, Notes on matrix theory II, Amer. Math Monthly 60 (1953), 173–175.Google Scholar
(3De Bruijn, N. G., Inequalities concerning minors and eigenvalues, Nieuw Arch. Wiskunde. 4 (1956), 18–35.Google Scholar
(4) Marvin, Marcus, Finite Dimensional Multilinear Algebra, Part 1 (Marcel Dekker, New York, 1973).Google Scholar
(5) Marvin, Marcus and Henryk, Minc, An inequality for Schur functions, Linear Algebra Appl. 5 (1971), 19–28.Google Scholar
(6) Marvin, Marcus and Henryk, Minc, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316–329.Google Scholar
(7) Marvin, Marcus and Herbert, Robinson, On exterior powers of endomorphisms, Linear Algebra Appl. 14 (1976), 219–225.Google Scholar
(8) Russell, Merris and Stephen, Pierce, Monotonicity of positive semidefinite Hermitian matrices, Proc. Amer. Math. Soc. 31 (1972), 437–440.Google Scholar
(9) Leon, Mirsky, An inequality for positive definite matrices, Amer. Math. Monthly 62 (1955), 428–130.Google Scholar
(10) Muir, W. W., Inequalities concerning the inverses of positive definite matrices, Proc. Edinburgh Math. Soc. (2) 19 (1974), 109–113.CrossRefGoogle Scholar
(11) Morris, Newman, Matrix Representations of Groups (U.S. National Bureau of Standards Applied Math. Series 60, Washington D.C., 1968).Google Scholar
(12) William, Watkins, Convex matrix functions, Proc. Amer. Math. Soc. 44 (1974), 31–34.Google Scholar