Features of permutation matrix

Question

Given a permutation matrix $P$ and a symmetric matrix $A$ (of compatible order), I am looking for the clarification why the diagonal elements of $PAP^T$ are same as the diagonal elements of$A$.

Answer 1

yes though order of diagonal elements would change

Let P = $[e_{\pi(i)}]$ as in http://en.wikipedia.org/wiki/Permutation_matrix

Then $(PA)_{ij}=\Sigma e_{\pi(i),k}a_{kj}=a_{\pi(i)j}$

As $PAP^T=(P(PA)^T)^T = \Sigma e_{\pi(i),k}a_{k\pi(j)}=a_{\pi(i)\pi(j)}$

Therefore the new diagonal elements are $a_{\pi(i)\pi(i)} $ which is a permutation of $a_{ii}$

If A is symmetric , $a_{ij}=a_{ji} \Rightarrow a_{\pi(i)\pi(j)}=a_{\pi(j)\pi(i)} $ , the new matrix is also symmetric.