Thedifference between diagonalazation and basis transformation

[transgalactic]

why when we want to trasform a matrix to a diagonolized form
D=p^-1*A*P

but when we want to change a matrix to a new basis
$$[A]_v=P*[A]_E*p^-1$$
??

why the transformation matrices are flipped??

 

[HallsofIvy]

They are not. It's just a different choice where which matrix is P and which is P^-1.

 

[transgalactic]

but there is no choise
p is always a transformation matrix and P^-1 is its inverse

why the other formula flips them
??