Given a $n \times n$ matrix $A$ with condition number $1$, prove or disprove $\mbox{cond} (A) = \mbox{cond} (PA)$, where $P$ is is any permutation matrix.
My attempt:
$cond(A) = ||A||*||A^{-1}||$
$cond(PA) = ||PA||*||(PA)^{-1}||$
(From here I am not sure I am doing it correctly:)
From the definition of the matrix norm:
${\displaystyle \left\|A\right\|_{2}\leq \left(\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{2}\right)^{1/2}=\left\|A\right\|_{F},} $
It clearly follows that ||PA|| = ||A||, since P has only the effect of permuting the values inside A without changing them.
Similarly, $||(PA)^{-1}|| = ||A^{-1}||.$
Thus, cond(A) = cond(PA)