Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion

✍ Scribed by Ludwig Elsner; Chunyang He; Volker Mehrmann

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 640 KB
• Volume: 2
• Category: Article
• ISSN: 1070-5325
• DOI: 10.1002/nla.1680020207

✦ Synopsis

We study the problem of minimizing the norm, the norm of the inverse and the condition number with respect to the spectral norm, when a submatrix of a matrix can be chosen arbitrarily. For the norm minimization problem we give a different proof than that given by Davis/Kahan/Weinberger. This new approach can then also be used to characterize the completions that minimize the norm of the inverse. For the problem of optimizing the condition number we give a partial result.

KEY

WORDS condition number; norm of a matrix; matrix completion; dilation theory; robust regularization of descriptor systems 1. Introduction We study the following optimization problem: Given integers n, m, N > n, m and matrices A E Prn, B E CniN-"', C E CN-n,m, find X E CN-n,N-rn such that the matrix satisfies CCC 1070-5325/95/020155-17 81995 by John Wiley & Sons, Ltd.