On matrix approximation problems with Ky Fank norms

Abstract

LetA be a realm xn matrix whose elements depend onl free parameters forming the vectorx. Then a class of approximation problems can be defined by the requirement thatx be chosen to minimize ∥A(x)∥, for a given matrix normon m ×n matrices. For example, it may be required to approximate a given matrix by a particular type of matrix, or by a linear combination of matrices. In the derivation of effective algorithms for such problems, a prerequisite is the provision of appropriate conditions satisfied by a solution, and the subdifferential of the matrix norm plays a crucial role in this. Therefore a characterization of the subdifferential is important, and this is considered for a class of orthogonally invariant norms known as Ky Fank norms, which include as special cases the spectral norm and the trace norm. The results lead to a consideration of efficient algorithms.