On Approximation Problems With Zero-Trace Matrices

12 because the conditions formulated in Corollary 1 are satissed for the problem (??). Therefore we have for every z 2 C jjjI + zBjjj k jjjIjjj k : Hence for every unitarily invariant norm we have by the properties of the unitarily invariant norms jjI + zBjj jjIjj: This completes the proof. 2 The above considerations imply that the characterization of a zero-trace matrix by means of the problem (??) for the norm jj jj is possible if the subgradient of jjIjj is unique, because then F has to be equal to (1=jjIjj)I (see (??)). The condition tr(B) = 0 is suucient to have (B) = jjIjj for every unitarily invariant norm (see (??)). We now prove that it is also necessary if jj jj satisses the assumptions of Theorem 3.