On the Minimum Rank of a Generalized Matrix Approximation Problem in the Maximum Singular Value Norm
Kin Cheong Sou, Anders Rantzer · 2010
In this paper theoretical results regarding a gen- eralized minimum rank matrix approximation problem in the maximum singular value norm are presented. Using the idea of projection, the considered problem can be shown to be equiva- lent to a classical minimum rank matrix approximation which can be solved efficiently using singular value decomposition. In addition, as long as the generalized problem is feasible, it is shown to have exactly the same optimal objective value as that of the classical problem. Certain comments and extensions of the presented theorem are included in the end of the paper. I. I NTRODUCTION Let m, n, mX , nX be positive integers. Define the following three matrices and assume that A ∈ R m×n , B ∈ R m×mX , C ∈ R nX ×n m > mX and B has full column rank n > nX and C has full row rank