Bounds on Condition Number of a Matrix

Abstract

For each vector norm ‖x‖, a matirx $A$ has its operator norm $‖A‖=\mathop{\rm min}\limits_{x≠0}\frac{‖Ax‖}{‖x‖}$ and a condition number $P(A)=‖A‖ ‖A^{-1}‖$. Let $U$ be the set of the whole of norms defined on $C^n$. It is shown that for a nonsingular matrix $A\in C^{n\times n}$, there is no finite upper bound of $P(A)$ whch ‖·‖ varies on $U$ if $A\neq \alpha I$; on the other hand, it is shown that $\mathop{\rm inf}\limits_{‖·‖\in U} ‖A‖ ‖A^{-1}‖ =ρ(A)ρ(A^{-1})$ and in which case this infimum can or cannot be attained, where $ρ(A)$ denotes the spectral radius of $A$.