linear algebra - Understanding the formula for a diagonal matrix $A = P^{-1}DP$.

I'm trying to understand the procedure for finding the powers of a matrix using the diagonal relation $A^n = P^{-1}D^nP$. Here's what I understand so far.

• We find eigenvalues of A. The matrix D is formed with eigenvalues in
  the diagonal line and zeros everywhere else. The order of entering
  diagonal values doesn't matter.


• The matrix $P$ is a matrix that contains eigenvectors of $A$. Again, the order does not matter.

Is this right? I'm assuming matrix is nice (invertible etc). Am I right in thinking that the diagonal matrix itself isn't useful (i.e. doesn't give you $A^2$ unless you find $P$ and $P^{-1}$ too).

Answer

Indicating the eigenvalues along the diagonal $D$ with $\lambda_i=D_{ii}$ we need that the corresponding eigenvectors $\vec v_i$ are placed as the i-th column of the matrix $P$.

Therefore the order of the eigenvalues in $D$ doesn't matter but the corresponding eigenvectors must be placed in $P$ accordingly and viceversa.