This may sound like a strange question, but I've actually been having some difficulty finding a general algorithm to create eigenvectors from a given eigenvalue + the associated matrix A.
Trying to solve $(A-λI)X =\begin{bmatrix}0\\ 0\\0\end{bmatrix}$ with my Gaussian Elimination or Least Squares algorithms doesn't work because $A-λI$ is singular by definition.
I find the eigenvalues in the first place using the iterative QR algorithm, and I've heard that the columns in the final product are the eigenvectors, but that doesn't seem to generalize.
Obviously there's $n$ unknowns in $n$ equations so an Algebra II student could do it using elimination or substitution, but those methods are basically Gaussian Elimination in the first place.
Feel like I'm missing something basic. Any thoughts?