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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: 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.

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Additional Information

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

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