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I just wanted to clarify my understanding of the sub-multiplicative property of the matrix operator norm.

On this Wikipedia page it says for any matrix:

$ \|AB\|_{\alpha, \gamma} \leq \|A\|_{\beta, \gamma} \|B\|_{\alpha,\beta} $

But then says for square matrices:

$\|AB\|_{\alpha,\alpha} \leq \|A\|_{\alpha,\alpha}\|B\|_{\alpha,\alpha}$

I do not understand the reason for this extra section for square matrices, is this not just a special case of what was written previously?

In addition, using this can we then say that for rectangular matrices $A$ and $B$. And for the spectral norm $\|\cdot\|_2$, i.e the induced norm from using the 2-norm for both vector spaces, that:

$\|AB\|_2 \leq \|A\|_2 \|B\|_2$

Dylan Dijk
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    Note that the definition of submultiplicativity only makes sense if you have a normed space with a product, so this only works for square matrices. – ronno Mar 15 '24 at 09:40
  • @ronno Does this then mean that this answer is wrong? That the spectral norm is not in general sub-multiplicative for non-square matrices. – Dylan Dijk Mar 15 '24 at 11:32
  • I wouldn't call that submultiplicative, but I guess there isn't a better name for this property. That answer seems correct since it specifies what it means by submultiplicative. – ronno Mar 15 '24 at 11:56
  • What? Could you please clarify your first comment then. Is what I have written wrong, if so which part exactly. Sorry im just a bit confused know. – Dylan Dijk Mar 15 '24 at 12:35
  • Maybe you should write down your definition of the sub-multiplicative property. But the square matrix version of the inequality is in fact a special case of the more general version with three (a priori distinct) norms. – ronno Mar 15 '24 at 12:44

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