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Show that r is repeated root for characteristic equation iff

  1. Feb 6, 2014 #1
    1. The problem statement, all variables and given/known data
    A:B→B a linear operator

    Show r is multiple root for minimal polynomial u(x) iff

    >$$\{0\}\subset \ker(A - rI) \subset \ker(A - rI)^2$$

    note: it is proper subset


    2. Relevant equations

    3. The attempt at a solution
    1. The problem statement, all variables and given/known data

    My thought:

    I know ker(A−rI) is basically {{0} and {eigenvectors associated with r}}.

    what is ker((A−rI)^2) with respect to above and/or r? How is eigenvector of (A−rI)^2 related to that of (L−rI)?
     
    Last edited: Feb 6, 2014
  2. jcsd
  3. Feb 7, 2014 #2
    Have you ever heard of a generalized eigenvector?

    http://en.wikipedia.org/wiki/Generalized_eigenvector

    Suppose v is an eigenvector corresponding corresponding to r. If you could find a vector v2 such that (A - rI) v2 = v. Then:

    (A - rI)2 v2 = (A - rI) v = 0​

    So v2 [itex]\in[/itex] ker( (A - rI)2). But v2 [itex]\notin[/itex] ker(A - rI).

    See if you can use the fact that (x-r)2 divides the minimal polynomial to show that such a v2 exists.
     
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