Singular value inequalities for convex functions of positive semidefinite matrices

Abstract

In this paper, we give new singular value inequalities for matrices. It is shown that if A, B, X are $n\times n$ matrices such that X is positive semidefinite, and if $f:[0,\infty )\rightarrow {\mathbb {R}}$ is an increasing nonnegative convex function, then

$$\begin{aligned} s_{j}\left( f\left( \frac{\left| AXB^{*}\right| }{\left\| X\right\| }\right) \right) \le \frac{\left\| f\left( \frac{A^{*}A+B^{*}B}{2}\right) \right\| }{\left\| X\right\| }s_{j}\left( X\right) \end{aligned}$$

and

$$\begin{aligned} s_{j}\left( AXB^{*}\right) \le \frac{1}{2}\left\| \frac{A^{*}A}{ \left\| A\right\| ^{2}}+\frac{B^{*}B}{\left\| B\right\| ^{2}} \right\| \left\| A\right\| \left\| B\right\| s_{j}\left( X\right) \end{aligned}$$

for $j=1,2,...,n$. Some of our inequalities present refinements of some known singular value inequalities.