In the second paragraph on Page 71 of the book [Matrix Analysis by
Bhatia, 1997](http://books.google.com/books?id=F4hRy1F1M6QC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=71&f=false), it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
(III.12) for $\Phi\left(x\_{1},\cdots,x\_{n}\right)=\left|x\_{1}\right|+\cdots+\left|x\_{n}\right|$? 



A question on gauge functions  
Inequality III.12 is the famous Lidskii majorization for Hermitian matrices $A$ and $B$, which says that


$$\lambda^\downarrow(A) - \lambda^\downarrow(B) \prec \lambda(A-B) \prec \lambda^\downarrow(A) - \lambda^\uparrow(B).$$


Now, recall the following simple but crucial fact:


**Fact.** $x \prec y \implies$ $|x|\quad \prec\_w\quad |y|$, where $\prec\_w$ denotes weak majorization, and $|x|$ denotes the vector obtained from $x$ by taking elementwise absolute values.


Now Problem II.5.11-(vi) asks you to prove that $x\ \prec\_w\ y\ $ iff $\Phi(x) \le \Phi(y)$ for every symmetric gauge function $\Phi$. Once you have proved / believed this, the inequality that you allude to follows after invoking the abovementioned "fact".


