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The Eigenvalues of Complementary Principal Submatrices of a Positive Definite Matrix

Published online by Cambridge University Press:  20 November 2018

R. C. Thompson  and
S. Therianos
R. C. Thompson
Affiliation:
The University of California, Santa Barbara, California
S. Therianos
Affiliation:
The University of California, Santa Barbara, California
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Let C be an n-square Hermitian matrix, presented in partitioned form as

where A is a-square and B is b-square. Let denote the eigenvalues of C, A, B, respectively. In a recent paper [10] the following inequality was established:

1.1

if

1.2

This inequality is a simplification and a sharpening of an inequality established earlier in [6], and is a wide generalization of an inequality of Aronszajn [4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 658 - 667
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Amir-Moez, A. 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.Google Scholar
2. Amir-Moez, A. R. and Perry, C., Remarks on Thompson, Freede theorems, Bull. Australian Math. Soc. 5 (1971), 221–226.Google Scholar
3. Amir-Moez, A. R. and Perry, C., Positive transformations restricted to subspaces and inequalities among their proper values, Proc. Amer. Math. Soc. 32 (1972), 363–367.Google Scholar
4. Aronszajn, N., Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in a Hilbert space. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 474–480.Google Scholar
5. Hersch, J. and Zwahlen, B. P., Evaluations par defaut pour une somme quelconque de valeurs propres yk d'un opérateur C = A + Bà l'aide de valeurs αi de A et βi de B, C. R. Acad. Sci. Paris 254 (1962), 1559–1561.Google Scholar
6. Thompson, R. C. and Freede, L. J., Eigenvalues of partitioned Hermitian matrices, Bull. Australian Math. Soc. 3 (1970), 23–37.Google Scholar
7. Thompson, R. C. and Freede, L. J., On the eigenvalues of sums of Hermitian matrices, Linear Algebra and Appl. 4 (1971), 369–376.Google Scholar
8. Thompson, R. C. and Freede, L. J., On the eigenvalues of sums of Hermitian matrices II, Aequationes Math. 5 (1970), 103–115.Google Scholar
9. Thompson, R. C. and Therianos, S., The singular values of a matrix product. I (to appear in Scripta Math.).Google Scholar
10. Thompson, R. C. and Therianos, S., Inequalities connecting the eigenvalues of a Hermitian matrix with the eigenvalues of complementary principal submatrices, Bull. Australian Math. Soc. 6 (1972), 117–132.Google Scholar
11. Zwahlen, B. P., Über die Eigenwerte der Summe zweier selbstadjungierter Operatoren, Comment. Math. Helv. 40 (1966), 81–116.Google Scholar