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.


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 $$ \left( P^{-1} \right)^T = \left( P^T \right)^{-1}. $$


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