Choose $\rho$ such that $\rho$-norm minimizes the matrix condition number I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes.
Question
Let $A$ a symmetric matrix. Which $\rho$-norm minimizes $A$'s condition number: $\kappa(A,\rho)$?
Edit
Well, my solution was clearly wrong.
I would love any suggestion or direction for solution you might have.
Thanks! :)

$k(A,2)=\|A\|_2\|A^{-1}\|_2= \max | \lambda_A|.\max|\lambda_{A^{-1}}|\leq\|A\|\|A^{-1}\|=k(A,\|.\|)$ for each operator norm $\|.\|$.
That $\max|\lambda_A|\leq \|A\|$ follows from:
Let $x$ be an eigenvector of $A$ for the eigenvalue $\lambda$, where $\max\limits_{i}|\lambda_i|=|\lambda|$. Then for arbitrary matrix norm $\|.\|$, subordinate to the vector norm $\|.\|$, we have $\|A\|=\max\limits_{y\neq 0} \frac{\|Ay\|}{\|y\|}\ge \frac{\|Ax\|}{\|x\|}=\frac{\|\lambda x\|}{\|x\|}=|\lambda|$
