Section 35.1 Diagonal form
What.
\(\inv{P}AP = D\text{,}\) a diagonal matrix.
When.
Each eigenvalue of \(A\) has its algebraic multiplicity equal to its geometric multiplicity. Additionally, if \(A\) is a real matrix and we would like both \(P\) and \(D\) to be real matrices, then we also need \(A\) to not have any complex eigenvalues.
How.
Compute a basis for each eigenspace. Put these bases together into the columns of \(P\text{.}\)
Result.
If the \(\nth[j]\) column of \(P\) is an eigenvector for eigenvalue \(\lambda\text{,}\) then the \(\nth[j]\) diagonal entry of \(D\) will be \(\lambda\text{.}\)