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Suppose we have two matrices $A\in \mathbb{R}^{m \times n}$ and $B\in \mathbb{R}^{n \times p}$, then what's the relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$? The notation $\|\cdot\|_*$ means the nuclear norm (aka trace norm).

If I need to bound $\|AB\|_*$, what restrictions should be added on $A$ and $B$?

Thanks for your answer!

Rodrigo de Azevedo
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Ryan
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1 Answers1

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The nuclear norm is submultiplicative, which is to say that it satisfies $\|AB\|_* \leq \|A\|_* \|B\|_*$.

For a direct proof of this fact, it suffices to note that $$ \|AB\|_* \leq \|A\| \cdot \|B\|_* \leq \|A\|_* \cdot \|B\|_*, $$ where $\|A\|$ denotes the spectral norm.

Ben Grossmann
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