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Remarks on theorems of Thompson and Freede

Published online by Cambridge University Press:  17 April 2009

A.R. Amir-Moéz  and
C.R. Perry
A.R. Amir-Moéz
Affiliation:
Texas Tech University, Lubbock, Texas, USA.
C.R. Perry
Affiliation:
Texas Tech University, Lubbock, Texas, USA.
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Abstract

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Let A be a hermitian transformation on an n-dimensional unitary space En, with proper values a1 ≥ … ≥ an. Let M be a proper subspace of En. Suppose b1 ≥ … ≥ bh are the proper values of A/M and c1 ≥ … ≥ ck are the proper values of . Let i1. < … < ir and j1 < … < jr be sequences of positive integers, with ir ≤ k and jr ≤ h. then

.

This is a special case of one of the Thompson-Freede theorems which is proved by use of certain invariants.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 221 - 225
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]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
[2]Fan, Ky, “On a theorem of Weyl concerning eigenvalues of linear transformations. I”, Proa. Nat. Acad. Sci. U.S.A. 35 (1949), 652–655.CrossRefGoogle ScholarPubMed
[3]Hamburger, H.L. and Grimshaw, M.E., Linear transformations in n-dimensional vector spaces (Cambridge University Press, Cambridge, 1951).Google Scholar
[4]Thompson, Robert C. and Freede, Linda J., “Eigenvalues of partitioned hermitian matrices”, Bull. Austral. Math. Soc. 3 (1970), 23–37.CrossRefGoogle Scholar
[5]Weyl, Hermann, “Inequalities between the two kinds of eigenvalues of a linear transformation”, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 408–411.CrossRefGoogle ScholarPubMed