Row/column swaps on $AA^T$ (Effect on Eigenvalues?) Let $A$ be a matrix.
I am trying to study how row/column swaps on $A$ affect $AA^T$, in particular its eigenvalues?
I was thinking that row/column swaps amount to pre- and post- multiplying  permutation matrices (https://en.wikipedia.org/wiki/Permutation_matrix) on $A$.
Hence, $A$ becomes the matrix $B=P_1AP_2$ after row/column swaps.
Then $BB^T=P_1AP_2P_2^TA^TP_1^T=P_1AA^TP_1^T$ since permutation matrices are orthogonal.
Is there any way I can proceed further?
Thanks.

Actually, given this form you can easily see that the eigenvectors of $AA^T$ have simply seen their components permuted too:
Let $u$ an eigenvector of $AA^T$ for eigenvalue $\lambda$, then if $v = P_1 u$:
$$BB^T v = P_1 AA^T P_1^T v = P_1 AA^T P_1^T P_1 u = P_1 AA^T u = P_1 \lambda u = \lambda v$$
Thus, $v$ is an eigenvector of $BB^T$ for the same eigenvalue $\lambda$.
This means that the permutation operation does not change the eigenvalues or their multiplicities, and only changes the order of the eigenvector components.
