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

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

MathSciNet review:
1487343

Full-text PDF

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.

**[A]**Atiyah M. F. and Bott R. (1983), The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A**308:**523-615. MR**85k:14006****[B]**Bourbaki, N. (1968),*Elements de mathematique, groupes et algébres de Lie, ch. 4-6.*Paris, Hermann. MR**32:1590****[F1]**Fan K. (1949), On a theorem of Weyl concerning eigenvalues of linear transformatioins I., Proc. Nat. Acad. Sci. U.S.A.**35:**652-655. MR**11:600e****[F2]**Fan K. (1951), Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A.**37:**760-766. MR**13:661e****[F3]**Fan K. and Hoffman A. (1955), Some metric inequalities in the space of matrices, Proc Amer. Math. Soc.**6:**111-116. MR**16:784j****[He]**Helgason S. (1978),*Differential Geometry, Lie Groups and Symmetric Space*, New York: Academic. MR**80k:53081****[Hi]**Hilgert J., Hofmann K.H. and Lawson J.D. (1989),*Lie Groups, Convex Cones and Semi-groups*, Oxford Science Publications. MR**91k:22020****[Ka]**Knapp A. W. (1986),*Representation Theory of Semisimple Groups*, Princeton University Press, Princeton, New Jersey. MR**87j:22022****[Ko]**Kostant B. (1973), On convexity, the Weyl group and Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup. (4)**6:**413-460. MR**51:806****[M]**Marshall A. W. and Olkin I. (1979)*Inequalities: Theory of Majorization and Its Applications*, Academic Press, New York. MR**81b:00002****[O]**Onishchik A.L. and Vinberg E. B. (1990),*Lie Groups and Algebraic Groups,*Springer-Verlag, Berlin. MR**91g:22001****[T]**Tam T. Y. (1997), Kostant's convexity theorem and the compact classical groups, Linear and Multilinear Algebra**43**(1997), 87-113.**[Th]**Thompson R. C. and Freede L. (1971), On the eigenvalues of sums of Hermitian matrices, Linear Algebra and Its Applications**4:**369-376. MR**44:5330**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
15A60,
22E30

Retrieve articles in all journals with MSC (1991): 15A60, 22E30

Additional Information

**Tin-Yau Tam**

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

Email:
tamtiny@mail.auburn.edu

DOI:
https://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