I'm trying to find the value of this matrix, $A = \begin{pmatrix} 1 & 4 \\ 3 & 2\end{pmatrix}$ to the power of $10$.
I've determined (and confirmed on Wolfram) that the eigenvalues are $5$, and $-2$.
I started looking for P by using the eigenvalue $5$, and found the eigenvector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
I did the same for the eigenvalue $-2$, and found eigenvector $\begin{bmatrix} -4 \\ 3 \end{bmatrix}$
I then put these together to get $P: \begin{bmatrix} 1 & -4\\ 1 & 3 \end{bmatrix}$. I followed that by finding $P^{-1}: \dfrac{1}{7} \begin{bmatrix} 3 & 4\\ -1 & 1 \end{bmatrix}$.
The problem is, that $PAP^{-1}$ is not diagonalizing. I checked Wolfram, and I found that the reason is because my $P$ should have been $\begin{bmatrix} -4 & 1\\ 3 & 1 \end{bmatrix}$.
My question is, what is the general rule about what eigenvalue to plug in first and how to arrange the columns of $P$?
Best Answer
The problem does not lie in the order of the eigenvalues or the order of the columns of $P$. What should be diagonal is $P^{-1}AP$, not $PAP^{-1}$. The order of the eigenvalues or eigenvectors do not matter, as long as the ordering of eigenvalues is consistent with the ordering of eigenvectors.