Rank Reducing Matrix Norms

Authors: Okubo K.¹; Woerdeman H.J.²

Source: Linear and Multilinear Algebra, Volume 50, Number 2, 1 March 2002 , pp. 185-197(13)

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Abstract:

We consider approximation numbers for some norms on matrices, and look at the question when a closest rank

p approximant can be chosen to reduce the rank of a matrix by p. If the latter is always possible, we call the norm rank p reducing. It is easily seen that any unitarily invariant norm is rank p reducing. We show that any absolute norm on $shadCˆ{n times m}$ is rank n - 1 reducing and that the numerical radius norm on $ shadCˆ{ntimes n}$ is rank n - 1 reducing as well. Non-examples and computations of approximation numbers are also presented.

Keywords: Approximation number; Closest rank p approximant; Rank p reducing; Unitarily invariant norm; Absolute matrix norm; Numerical radius norm

Document Type: Research article

Affiliations: 1: Mathematics Laboratory, Hokkaido University of Education Sapporo, Sapporo, 002-8502 Japan 2: Department of Mathematics, The College of William and Mary, P.O. Box 8795 Williamsburg, VA 23187-8795

Publication date: 2002-03-01