An example for a case where $\left \| AB \right \|_{HS}\ > \left \| A \right \|_{HS}\left \| B \right \|_{HS}$

Question

I need to disprove that $\left \| AB \right \|_{HS}\ \leq \left \| A \right \|_{HS}\left \| B \right \|_{HS}$. (HS= Hilbert-Schmidt norm)

I've tried several random $2\times 2$ matrices without much success.

Is there any smart way of coming up with such matrices, apart from guessing?

Can someone give an example? Thank you...

Answer 1

The Hilbert-Schmidt norm is submultiplicative so no such example exists. This follows by applying the Cauchy-Schwarz inequality to all pairs of row vectors of $A$ and column vectors of $B$; concretely

$$\lVert AB \rVert_{\mathrm{HS}}^2 = \sum_{i, j} \left(\sum_k a_{ik} b_{kj}\right)^2 \leq \sum_{i, j} \sum_k a_{ik}^2 \sum_m b_{mj}^2 = \sum_{i, k} a_{ik}^2 \sum_{m, j} b_{mj}^2 = \lVert A \rVert_{\mathrm{HS}}^2 \lVert B \rVert_{\mathrm{HS}}^2.$$