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A unified extension of two results of Ky Fan on the sum of matrices

Authors:	Tin-Yau Tam

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

Abstract:	Let 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 result of Ky Fan (1949) asserts that if and 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 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.

Keywords:	Eigenvalues singular values partial order

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