On the singular values of a product of matrices

W. Watkins
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引用次数: 2

Abstract

The purpose of this note is to give necessary and sufficient conditions for the singular values of a product of matrices to be equal to certain products of their singular values. We then analyze the case of equa]jty in a matrix inequality of Os trows ki . Th e s ingular values of an n·square co mplex matrix X are the positive square roots of the eigen· values of X*X, where X* is the conjugate tran spose of X. Denote the singular values of X by a t (X) , ... , an (X), arranged so that al (X) ~ ... ~ an(X) > 0 (all matrices are assumed to be nonsi ngular). Let A and B be n,·square complex matri ces and let A = UH, B = VK be the polar factorizations of A a nd B. In the factorization s U and V are unitary matrices and Hand K are positive·definit e hermitian matri ces. THEOREM 1: Let k be a positive integer less than n. Then
关于矩阵乘积的奇异值
本文的目的是给出矩阵之乘积的奇异值等于其奇异值之乘积的充分必要条件。然后,我们分析了一个矩阵不等式中0个矩阵的不等式的情况。n·平方复矩阵X的奇异值是X*X的特征值的正平方根,其中X*是X的共轭变换。用t (X),…,一个(X),安排使所有(X) ~…~ an(X) > 0(所有矩阵都假定是非奇异矩阵)。设A和B为n,·平方复矩阵,设A = UH, B = VK为A, A, B的极分解。在分解过程中,U和V是酉矩阵,K是正的·定义e厄米矩阵。定理1:设k为小于n的正整数,则
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