I am trying to prove the following:
For any diagonal matrix $D = \mbox{diag}(d_i)$, we have $$\mbox{cond}(D)=\frac{\max|d_i|}{\min|d_i|}$$ The matrix norm is a norm induced by a vector norm, i.e., $$\|A\| := \max_{\|x\|=1} \|Ax\|$$ for any vector norm $\|\cdot\|$.
If the vector norm in the problem is induced by an inner product, I have managed to prove the above. Is it also true for any vector norm?