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How are the the singular values & eigenvalues of A and PAP^T related? P is a permutation matrix.
Singular values of $A$ and $PAP^T$: Suppose $A=U\Sigma V^T$. Then $PA=(PU)\Sigma V^T$ so $PA$ and $A$ have the same singular values. Similarly, $A$ and $AP$ have the same singular values. The same reasoning can be applied to $A$ and $PAP^T = (PU)\Sigma (PV)^T$. Is this reasoning correct?
Eigenvalues of $A$ and $PAP^T$: Suppose $Ax=\lambda x$. Then, $PAx=\lambda Px$. What about $AP$ and $PAP^T$?