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Characterization of the subdifferential of some matrix norms. (English) Zbl 0751.15011
The subdifferential $\partial\Vert A\Vert$ is the set of matrices $G$ satisfying $\Vert A\Vert=\text{trace}(G\sp TA)$, and $\max\sb{\Vert B\Vert\le 1}$ $\text{trace}(B\sp TG)\le 1$, where $\Vert\centerdot\Vert$ is a given norm on $M\sb n(\bbfR)$. This paper is concerned with a characterization of the subdifferential of some important matrix norms. For an operator norm, a set $\Phi(A)$ of vector pairs is defined so that $\lim\sb{\gamma\to 0\sb +}[(\Vert A+\gamma R\Vert-\Vert A\Vert)/\gamma]=\max\sb{(v,w)\in\Phi(A)}w\sp TR v$, and then $\partial\Vert A\Vert=\text{conv}\{wv\sp T:(v,w)\in\Phi(A)\}$.

15A60Applications of functional analysis to matrix theory
Full Text: DOI
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