3
$\begingroup$

Taken from a previously answered question on matrix similarities:

Similar matrices share many properties:

  • Rank
  • Determinant
  • Trace
  • Eigenvalues (though the eigenvectors will in general be different)
  • Characteristic polynomial
  • Minimal polynomial (among the other similarity invariants in the Smith normal form)
  • Elementary divisors

What I'm wondering is if this implies that the transposes will be similar.

$\endgroup$
5
$\begingroup$

If $$ P^{-1} A P = B, $$ $$ P^T A^T \left( P^{-1} \right)^T = B^T $$

Just to be sure, $$ I = I^T = \left( P P^{-1} \right)^T = \left( P^{-1} \right)^T P^T. $$ This means that, once $P$ is invertible, so is $P^T,$ and $$ \left( P^{-1} \right)^T = \left( P^T \right)^{-1}. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.