Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

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...

We deal with two recent conjectures of R.-C. Li [Linear Algebra Appl. 278 (1998) 317– 326], involving unitarily invariant norms and Hadamard products. In the particular case of the Frobenius norm, the first conjecture is known to be true, whereas the second is still an open problem. In fact, in this paper we show that the Frobenius norm is essentially the only invariant norm which may comply wi...

A fundamental result of von Neumann’s identifies unitarily invariant matrix norms as symmetric gauge functions of the singular values. Identifying the subdifferential of such a norm is important in matrix approximation algorithms, and in studying the geometry of the corresponding unit ball. We show how to reduce many convex-analytic questions of this kind to questions about the underlying gauge...