Is this inequality on Schatten p-norm $\|AB\|_{S_p} \le \|A\|_{S_p}\|B\|_{S_p}$ true?

Question

The Schatten-$p$ norm ($p >0$) of a matrix $X$ is defined as

$\|X\|_{S_p} = (\sum_{i=1}^{n} \sigma_i^p(X))^{1/p}$

Where $\sigma_i(X)$ denotes the $i$-th singular value of $X$

Is the follow inequalities true for any two matrices $A$ and $B$

$\|AB\|_{S_p} \le \|A\|_{S_p}\|B\|_{S_p}$

And how to proof? Thanks

Answer 1

Note that the Schatten $p$-norm is only a norm when $p \geq 1$.

The inequality indeed holds for any matrices $A,B$ when $p \geq 1$. To see that this is the case, it suffices to use the fact that $$ \sigma_i(AB) \leq \sigma_1(A) \sigma_i(B) $$