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

Elsner L, He C, Mehrmann V. Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion. Numerical Linear Algebra with Applications. 1995;2(2):155-171.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