Bounds on condition number of singular matrix

Abstract

For each vector norm $\|x\|_\nu$, a matrix $A\in C^m\times n}$ has its operator norm $\|A\|_{\mu \nu}=\max_{\atop x\neq O}\frac{\|Ax\|_\mu}{\|x\|_\nu}$. If $A$ is  nonsingular, we can definite the condition number of $A\in C^{n\times n}$ as $P(A)=\|A\|_{\nu \nu}\|A^{-1}\|_{\nu \nu}$. If $A$ is singular, the condition number of matrix $A\in C^{m\times n}$ may be defined as $P_{\dag}(A)=\|A\|_{\mu \nu}\|A^{\dag}\|_{\nu \mu}$. Let $U$ be the set of the whole self-dual norms. It is shown that for a singular matrix $A\in C^{m\times n}$, there is no finite upper bound of $P_{\dag}(A)$, while $\|.\|$ varies on $U$. On the other hand, it is shown that $\inf_{\atop \|.\|\in U}\|A\|_{\mu \nu}\|A^{\dag}\|_{\nu \mu}=\frac{\sigma_1(A)}{\sigma_r(A)},$ where $\sigma_1(A)$ and $\sigma_r(A)$ are the largest and smallest nonzero singular values of $A$, respectively.