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

Abstract

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.

Scaled total least squares fundamentals

“…We see that we could have proved our Theorem 3.1 directly using the ideas of Wilkinson, but we prefer our way above, because it is logically simpler, and it also provides some algebraic relations that we use later in the paper. Some related questions were also studied in [6], but a statement similar to our Theorem 3.1 was not considered there.…”