Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-29T12:01:20.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities connecting the eigenvalues of a hermitian matrix with the eigenvalues of complementary principal submatrices

Published online by Cambridge University Press:  17 April 2009

Robert C. Thompson  and
S. Therianos
Robert C. Thompson
Affiliation:
University of California, Santa Barbara, California, USA.
S. Therianos
Affiliation:
University of California, Santa Barbara, California, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1972 , pp. 117 - 132
Copyright
Copyright © Australian Mathematical Society 1972

References

[7]Amir-Moéz, Ali R., “Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations”, Duke Math. J. 23 (1956), 463–476.CrossRefGoogle Scholar
[23]Amir-Moéz, A.R. and Perry, C.R., “Remarks on theorems of Thompson and Freede”, Bull. Austral. Math. Soc. 5 (1971), 221–225.CrossRefGoogle Scholar
[3]Aronszajn, N., “Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in a Hilbert space”, Proc. Nat. Acad. Sci. USA 34 (1948), 474–480.CrossRefGoogle Scholar
[4]Fan, Ky, “On a theorem of Weyl concerning eigenvalues of linear transformations I”, Proc. Nat. Acad. Sci. USA 35 (1949), 652–655.CrossRefGoogle ScholarPubMed
[5]Gohberg, I.C., Krei˘n, M.G., Introduction to the theory of linear nonselfadjoint operators (Translations of Mathematical Monographs, 18. Amer. Math. Soc., Providence, Rhode Island, 1969).Google Scholar
[6]Hamburger, H.L. and Grimshaw, M.E., Linear transformations in n-dimensional vector spaces (Cambridge University Press, Cambridge, 1951).Google Scholar
[7]Hersch, Joseph et Zwahlen, Bruno, “Évaluations par défaut pour une somme quelconque de valeurs propres γk, d'un operateur C = A + B à l'aide de valeurs propres α1 de A et βj de B”, C.R. Acad. Sci. Paris 254 (1962), 1559–1561.Google Scholar
[8]Lidskii˘, V.B., “О собственных значениях суммы произведения симметрических матриц”, Dokl. Akad. Nauk SSSR (N.S.) 75 (1950), 769–772 = “The proper values of the sum and product of symmetric matrices”, (translated by Benster, C.D.. National Bureau of Standards, Report 2248. U.S. Department of Commerce, Washington, D.C., 1953).Google Scholar
[9]Thompson, Robert C. and Freede, Linda J., “Eigenvalues of sums of hermitian matrices. II”, Aequationes Math. 5 (1970), 23–38.CrossRefGoogle Scholar
[10]Thompson, Robert C. and Freede, Linda J., “Eigenvalues of partitioned hermitian matrices”, Bull. Austral. Math. Soc. 3 (1970), 23–37.CrossRefGoogle Scholar
[11]Thompson, Robert C. and Freede, Linda J., “Eigenvalues of sums of hermitian matrices”, J. Linear Algebra Applic. (to appear).Google Scholar
[12]Thompson, Robert C. and Therianos, S., “The singular values of matrix products, I”, Scripta Math (to appear).Google Scholar
[13]Thompson, Robert C. and Therianos, S., “The eigenvalues of complementary principal submatrices of a positive definite matrix”, Canad. J. Math. (to appear).Google Scholar
[14]Thompson, Robert C. and Therianos, S., “On a construction of B.P. Zwahlen”, (in preparation).Google Scholar
[15]Weyl, Hermann, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)”, Math. Ann. 71 (1912), 441–479.CrossRefGoogle Scholar
[16]Wielandt, Helmut, “Einschliessung von Eigenwerten hermitischer Matrizen nach dem Abschnittsverfahren”, Arch. Math. 5 (1954), 108–114.CrossRefGoogle Scholar
[17]Wielandt, Helmut, “An extremum property of sums of eigenvalues”, Proc. Amer. Math. Soc. 6 (1955), 106–110.CrossRefGoogle Scholar
[18]Wielandt, Helmut, “Topics in the analysis of matrices”, (Notes prepared by R.S. Meyer. University of Wisconsin, Madison, 1967).Google Scholar
[19]Zwahlen, Bruno Peter, “Über die Eigenwerte der Summe zweier selbstadjungierter Operatoren”, Comment. Math. Helv. 40 (1965/1966), 81–116.CrossRefGoogle Scholar