I see different ways to diagonalise matrix A, what is correct and when?

Question

I see different ways to diagonalize a matrix.

sometimes they use the inverse like: $A = P^T D P$ and sometimes they use transverse like: $A = P^{-1} D P$

I've read that when a matrix is orthogonal, then $ Q^T = Q^{-1}$

$P$ should be the eigenvectors of matrix $A$, but also sometimes they make $P$ orthonormal to each other. I also don't know when and why would you do that.

Thanks in advance,

Answer 1

It is always the inverse.

Diagonalisation is about changing basis so that the transformation represented by $A$ becomes diagonal (and represented by $D$ instead). And when changing basis, the matrix representation of linear transformations is always sandwiched between the change-of-basis matrix $P$ and its inverse $P^{-1}$ (in the appropriate order).

However, if the transformation / matrix in question is symmetric, then one can take the change-of-basis matrix to be orthogonal. And in those cases, one might instead write its inverse as $P^T$, using transpose notation, to make it clear that this is a special case (and also make it clear how easy the inverse is to calculate in this case).

Answer 2

This is a matter of approach: some problems will be easier to solve using the first method, and other problems will be a better fit for the second method. Notice that this depends whether or not the matrix $A$ is invertable, which for many cases would not be the issue.

As both methods produce the same result for invertable matrices, it shouldn't matter which way you choose, as long as you will preform it correctly (try check the result with one method using the other method!)