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We obtain a characterization of pair matrices A and B of order n such thatwheredenotes the j-th largest singular values ofIt can imply not only some well-known singular value inequalities for sums and direct sums of matrices but also Zhan’s result related to singular values of differences of positive semidefinite matrices. In addition, some related and new inequalities are also obtained.

A Characterization on Singular Value Inequalities of Matrices