A unified extension of two results of Ky Fan

on the sum of matrices

Author:
Tin-Yau Tam

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2607-2614

MSC (1991):
Primary 15A60, 22E30

MathSciNet review:
1487343

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an Hermitian matrix with where are the ordered eigenvalues of . A result of Ky Fan (1949) asserts that if and are Hermitian matrices, then is majorized by . We extend the result in the framework of real semisimple Lie algebras in the following way. Let be a noncompact real semisimple Lie algebra with Cartan decomposition . We show that for any given , , where is the unique element corresponding to , in a fixed closed positive Weyl chamber of a maximal abelian subalgebra of in . Here the ordering is induced by the dual cone of . Fan's result corresponds to the Lie algebra . The compact case is also discussed. As applications, two unexpected singular values inequalities concerning the sum of two real matrices and the sum of two real skew symmetric matrices are obtained.

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Additional Information

**Tin-Yau Tam**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310

Email:
tamtiny@mail.auburn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04770-4

Keywords:
Eigenvalues,
singular values,
partial order

Received by editor(s):
February 13, 1997

Additional Notes:
Part of this work was done while the author was a visiting scholar in Mathematics Department of the University of Hong Kong, Dec. 1996-Jan. 1997. The travel was made possible by local subsistence provided by the department and travel grants from COSAM of Auburn University and NSF EPSCoR in Alabama.

Communicated by:
Lance W. Small

Article copyright:
© Copyright 1998
American Mathematical Society