I stumbled on the following inequality for singular values (stated without proof), and would like to understand it better:
Let $A,B$ be two $n\times n$ real matrices. Denoting by $\mu_i(C)$ the $i$-th singular value of a matrix $C$ and $by$ $\|\cdot\|$ the operator norm we have for $i=1,\dots,n$
$\mu_i(AB) \ \leq \|A\| \ \mu_i(B)$
$\mu_i(AB) \ \leq \|B\| \ \mu_i(A)$
Why is this true? Standard references? Is this inequality a specific property of singular values or does it work also for eigenvalues?