Singular Values of Differences of Positive Semidefinite Matrices
Xingzhi Zhan · SIAM Journal on Matrix Analysis and Applications · 2001
Let M n be the space of n × n complex matrices. For $A\in M_n,$ let $s(A)\equiv (s_1(A),\dotsc,\break s_n(A)),$ where $s_1(A)\ge\cdots\ge s_n(A)$ are the singular values of A. We prove that if $A,B\in M_n$ are positive semidefinite, then (i) $s_j(A-B)\le s_j(A\oplus B), j=1,2, . . . ,n, and (ii) the weak log-majorization relations $s(A-|z|B)\prec_{wlog} s(A+zB)\prec_{wlog} s(A+|z|B)$ hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.