# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A unified extension of two results of Ky Fan on the sum of matricesHTML articles powered by AMS MathViewer

by Tin-Yau Tam
Proc. Amer. Math. Soc. 126 (1998), 2607-2614 Request permission

## 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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