It is shown among other inequalities that if A, B and X are \(n\times n\) complex matrices such that A and B are positive semidefinite, then \(s_{j}(AX-XB)\le \) \(s_{j}\left( \left( \frac{1}{2}A+\frac{1}{2}A^{1/2}\left| X^{*}\right| ^{2}A^{1/2}\right) \oplus \left( \frac{1}{2}B+\frac{1}{2} B^{1/2}\left| X\right| ^{2}B^{1/2}\right) \right) \) for \(j=1,2,\ldots ,2n\). Several related singular value inequalities and norm inequalities are also given.

中文翻译：

---

奇异值不等式及其应用

除其他不等式外，还表明如果A，  B和X是\（n乘以n \）个复矩阵，使得A和B为正半定数，则\（s_ {j}（AX-XB）\ le \） \ （s_ {j} \ left（\ left（\ frac {1} {2} A + \ frac {1} {2} A ^ {1/2} \ left | X ^ {*} \ right | ^ {2} A ^ {1/2} \ right）\ oplus \ left（\ frac {1} {2} B + \ frac {1} {2} B ^ {1/2} \ left | X \ right | ^ {2} B ^ {1/2} \ right）\ right）\）为\（j = 1,2，\ ldots，2n \）。还给出了几个相关的奇异值不等式和范数不等式。