For each vector norm $\|x\|_v$, a matrix $A\in C^{m\times n}$ has its operator norm $\|A\|_{\mu v}=\max_{x\neq O}\frac{\|A\|_\mu}{\|x\|_v}$. If $A$ is nonsingular, we can define the condition number of $A\in C^{n\times n}$ as $P(A)=\|A\|_{vv}\|A^{-1}\|_{vv}$. If $A$ is singular, the condition number of matrix $A\in C^{m\times n}$ may be defined as $P_\dag(A)=\|A\|_{\mu v}\|A^\dagger\|_{v\mu}$. 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 $\operatorname{inf}_{\|.\|\in U}\|A\|_{\mu v}\|A^\dagger\|_{v\mu}=\frac{\sigma_1(A)}{\sigma_r(A)}$, where $\sigma_1(A)$ and $\sigma_r(A)$ are the largest and smallest nonzero singular values of $A$, respectively.