Is it possible to show that the Schatten 1-norm is convex by the definition of convexity? I can't seem to find any way to derive an expression of the sum or matrix of singular values of a convex combination.
Asked
Active
Viewed 271 times
1
-
1I would not expect there to be any formula for the norm of a convex combination. One way to proceed is through the dual characterization of the trace norm: $|A|_1 = \sup \operatorname{tr}(AB)$ where the supremum is taken over all matrices $B$ with operator norm at most one. This identifies $|A|_1 $ as the supremum of linear functions, hence convex. – user127096 Mar 18 '14 at 00:46