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So I'm a bit confused about finding a basis of generalized eigenvectors for an operator that is not diagonalizable. I have some "steps" in mind, but maybe someone can help me out here:

1) Find the eigenvalues of the matrix/operator

2) Find the eigenspaces corresponding to each eigenvector; for those which do not have a "big enough" basis, we must compute the generalized eigenspace for that eigenvalue

3) Here's where I'm confused! I know that the generalized eigenspace is given by kernel((T-λI)^p) for some positive integer p, and that (T-λI)^(p-1) is an eigenvector of T. I guess I'm lost with calculating this nullspace.

Are we literally taking the matrix (T-λI) and raising it to the power p? Because it doesn't seem to work on any problems I've attempted so far... anyone know?