Which one of the diagonalization is correct?

Question

I have sometimes seen diagonalization written as:

$A=P^{-1}DP$,

and I have also seen it as

$A=PDP^{-1}$.

Can anyone tell me which one of these is correct, and why?

Answer 1

The two are equivalent; if $A$ can be written in one form, it can also be written in the other (just replace $P$ by $P^{-1}$).

(Also, note that the question in the body doesn't match the title . . .)

Answer 2

Either one is correct, one can simply redefine $P$ to be what was called $P^{-1}$ to switch between the forms. I think people usually prefer the form $A=PDP^{-1}$ because then $P$ is the matrix of (right) eigenvectors. One exception might be if I want $P$ to be the matrix of left eigenvectors, which sometimes happens when talking about Markov chains.