13. (Bhatia-Davis) 设 $A,B,X\in M_n$, 则 $$\bex \sen{AXB^*}\leq \frac{1}{2}\sen{A^*AX+XB^*B} \eex$$ 对任何酉不变范数成立.

证明: 见 [R. Bhatia, C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl., 14 (1993), 132--136].

[詹兴致矩阵论习题参考解答]习题4.13