Monotonicity of positive semidefinite Hermitian matrices
Authors:
Russell Merris and Stephen Pierce
Journal:
Proc. Amer. Math. Soc. 31 (1972), 437-440
DOI:
https://doi.org/10.1090/S0002-9939-1972-0285556-7
MathSciNet review:
0285556
Full-text PDF
Abstract | References | Additional Information
Abstract: Inequalities which compare elements of the convex cone of positive semidefinite hermitian matrices with products of roots of elements are proved. They yield inequalities for Schur functions (generalized matrix functions) which, when specialized to the determinant, give a result of R. Bellman and L. Mirsky.
- [1] E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 30, Springer-Verlag, Berlin, 1961. MR 28 #1266. MR 0158038 (28:1266)
- [2] R. Bellman, Notes on matrix theory. II, Amer. Math. Monthly 60 (1953), 173-175. MR 14, 731. MR 0053170 (14:731e)
- [3] Ralph Freese, Inequalities for generalized matrix functions based on arbitrary characters, Linear Algebra and Appl. (to appear). MR 0340273 (49:5028)
- [4] Marvin Marcus and Henryk Mine, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, Mass., 1964. MR 29 #112. MR 0162808 (29:112)
- [5] -, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316-329. MR 33 #2655. MR 0194445 (33:2655)
- [6] Marvin Marcus and Paul J. Nikolai, Inequalities for some monotone matrix functions, Canad. J. Math. 21 (1969), 485-494. MR 38 #5815. MR 0237534 (38:5815)
- [7] L Mirsky, An inequality for positive definite matrices, Amer. Math. Monthly 62 (1955), 428-430. MR 17, 338. MR 0072833 (17:338y)
- [8] I. Schur, Über endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), 184-207. MR 1544291
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1972-0285556-7
Keywords:
Symmetric group,
character,
tensor power,
Kronecker power
Article copyright:
© Copyright 1972
American Mathematical Society