Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bases of Generalized Eigenvectors

  1. Apr 25, 2007 #1
    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? :eek:
  2. jcsd
  3. Apr 26, 2007 #2
    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.
  4. Apr 26, 2007 #3


    User Avatar
    Science Advisor

    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 [itex]\lamba[/itex] 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?
  5. May 5, 2008 #4
    I wonder if there is any online document available to show us how to compute all the generalized eigenvectors?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Bases Generalized Eigenvectors Date
I Bases, operators and eigenvectors Nov 15, 2017
I Free Modules, Bases and Direct Sums/Products May 21, 2016
Bases for Rn Nov 30, 2014
Bases of vector spaces and change of basis Aug 17, 2014
Need general algorithm for finding dual bases Jan 3, 2010