Cond(PA)=Cond(A)

Question

I want to proof the following equation. A is a nxn regular matrix and P a permutation Matrix

$$cond_2(PA)=cond_2(A)$$

So the spectral norm of $||P||=1$

so $cond(PA)=||PA|| \cdot ||(PA)^{-1}||$

I tried to show that $||A||\leq||PA||\leq||A||$ Can I do the same with $||(PA)^{-1}||$

Thanks

Answer 1

You are considering the changes of the euclidean norm and its operator norm under an orthogonal transformation. But orthogonal transformations are exactly those that leave this norm invariant...