Optimal Conditioning of Matrices

Abstract

The optimal condition number c of a matrix S is the minimum, over all diagonal matrices D, of $\| {DS} \|\| {(DS)^{ - 1} } \|$. It arises in many applications of scaling in linear systems, and has been computed in the maximum norm by Bauer. In the $\ell _2 $-norm, we establish the inequalities $b \leqq c \leqq 2b$ and disprove—by an example of order seven !—the minimax conjecture that always $b = c$. Here b is the supremum over diagonal unitary matrices of $\| {S^{ - 1} US} \|$.