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I read in a paper called "Characterization of the subdifferential of some matrix norms" that it defines the subdifferential of the matrix norm like this:
$$\partial ||A||=\{G \in R^{m\times n} : \forall B \in R^{m\times n}, ||B|| \geq ||A|| + \operatorname{Tr}[(B-A)^TG] \}$$ Can someone tell me why they define it like this? I really don't know the intuition behind it! Thanks!

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This is the standard definition of a subdifferential, the only possible confusion is that they are using the inner product $\langle A, B \rangle = \text{tr }A^TB$ (which is consistent with the Frobenius norm). The intuition is that it describes supporting hyperplanes to the graph of $\|\cdot\|$ at $A$. –  copper.hat Jan 2 '13 at 5:38

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