Bases of Generalized Eigenvectors

[quasar_4]

Hi all.

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?

 

[lalbatros]

Could you show us first a small example of an operator that is not diagonalizable.
Take then the opportunity to illustrate the steps 1 and 2.
Then we can discuss maybe step 3.

 

[HallsofIvy]

I recommend you do what lalbatross suggests- looking at a specific simple example might help you see what is happening.

You first find the lowest integer p such that (T-λI)^p has the entire eigenspace of $\lamba$ as kernel. But I don't understand what you mean by "(T-λI)^(p-1) is an eigenvector of T". (T-λI)^(p-1) isn't a vector, it is a linear transformation. Did you mean to apply it to something?

 

[bchui]

I wonder if there is any online document available to show us how to compute all the generalized eigenvectors?