Proving Similarity of Transposed Matrices

[phyzmatix]

Homework Statement

Let A and B be similar matrices. Prove that A^T and B^T are also similar.

2. The attempt at a solution

We know that, since A and B are similar matrices, there exists an invertible matrix P such that

B=P^{-1}AP

so, I thought if we transposed both sides, this could lead to a proof, but this gives

B^T=\left(P^{-1}AP\right)^T

B^T=P^TA^T\left(P^{-1}\right)^T

B^T=P^TA^T\left(P^T\right)^{-1}

Which feels close, but how do I get it to the form

B^T=\left(P^T\right)^{-1}A^TP^T ?

Provided of course, that this is an actually viable route to take (I have virtually zero experience with mathematical proofs).

Thanks!
phyz

 

[winter85]

hi phyz,
you reached B^T=P^TA^T(P^T)^{-1}.
if you put Q = (P^T)^{-1} , what form would the above statement take?

 

[phyzmatix]

Ha! Easy as that is it?

Thanks winter85!

 

[winter85]

yep :)