On Matrix Unitarily Invariant Norm Condition Number

Abstract

In this paper, the unitarily invariant norm $\|\cdot\|$ on $\mathbb{C}^{m\times n}$ is used. We first discuss the problem under what case, a rectangular matrix $A$ has minimum condition number $K (A)=\| A \| \ \|A^+\|$, where $A^+$ designates the Moore-Penrose inverse of $A$; and under what condition, a square matrix $A$ has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of $K (A)=\|A\| \ \|A^{-1}\|_2$ in error estimation.