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

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Abstract: Let $A$ be an $n\times n$ Hermitian matrix with $\lambda (A) = (\lambda _1(A), \dots , \lambda _n(A))$ where $\lambda _1(A) \ge \cdots \ge \lambda _n(A)$ are the ordered eigenvalues of $A$. A result of Ky Fan (1949) asserts that if $A$ and $B$ are $n\times n$ Hermitian matrices, then $\lambda (A+B)$ is majorized by $\lambda (A) + \lambda (B)$. We extend the result in the framework of real semisimple Lie algebras in the following way. Let $\frak g$ be a noncompact real semisimple Lie algebra with Cartan decomposition ${\frak g} = {\frak t} + {\frak p}$. We show that for any given $p, q\in \frak p$, $a_+(p+q)\le a_+(p) + a_+(q)$, where $a_+(x)$ is the unique element corresponding to $x\in \frak p$, in a fixed closed positive Weyl chamber ${\frak a}_+$ of a maximal abelian subalgebra ${\frak a}$ of ${\frak g}$ in ${\frak p}$. Here the ordering $\le$ is induced by the dual cone ${\frak a}_+^*$ of ${\frak a}_+$. Fan’s result corresponds to the Lie algebra ${\frak {sl}}(n, {\Bbb C})$. 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.

- M. F. Atiyah and R. Bott,
*The Yang-Mills equations over Riemann surfaces*, Philos. Trans. Roy. Soc. London Ser. A**308**(1983), no. 1505, 523–615. MR**702806**, DOI https://doi.org/10.1098/rsta.1983.0017 - N. Irimiciuc, I. Rusu, and C. Huiu,
*Exposition of a plane-axial geometric representation*, Bul. Inst. Politehn. Iaşi (N.S.)**8(12)**(1962), no. fasc. 3-4, 65–74 (Romanian, with Italian and Russian summaries). MR**184116** - Fan K. (1949), On a theorem of Weyl concerning eigenvalues of linear transformatioins I., Proc. Nat. Acad. Sci. U.S.A.
**35:**652-655. - Fan K. (1951), Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A.
**37:**760-766. - Fan K. and Hoffman A. (1955), Some metric inequalities in the space of matrices, Proc Amer. Math. Soc.
**6:**111-116. - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie D. Lawson,
*Lie groups, convex cones, and semigroups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR**1032761** - Anthony W. Knapp,
*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239** - Bertram Kostant,
*On convexity, the Weyl group and the Iwasawa decomposition*, Ann. Sci. École Norm. Sup. (4)**6**(1973), 413–455 (1974). MR**364552** - Albert W. Marshall and Ingram Olkin,
*Inequalities: theory of majorization and its applications*, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**552278** - A. L. Onishchik and È. B. Vinberg,
*Lie groups and algebraic groups*, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR**1064110** - Tam T. Y. (1997), Kostant’s convexity theorem and the compact classical groups, Linear and Multilinear Algebra 43 (1997), 87–113.
- Robert C. Thompson and Linda J. Freede,
*On the eigenvalues of sums of Hermitian matrices*, Linear Algebra Appl.**4**(1971), 369–376. MR**288132**, DOI https://doi.org/10.1007/bf01817787

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Additional Information

**Tin-Yau Tam**

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

Email:
tamtiny@mail.auburn.edu

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