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Depth of an eigen value (generalized eigenspaces)

  1. Oct 13, 2009 #1
    Let [tex]A[/tex] be a matrix and [tex]\mu[/tex] be an eigenvalue of that matrix. Suppose that for some [tex]k[/tex], [tex]\tex{ker}\left(A-\mu I\right)^k=\tex{ker}\left(A-\mu I\right)^{k+1}[/tex]. Then show that [tex]\tex{ker}\left(A-\mu I\right)^{k+r}=\tex{ker}\left(A-\mu I\right)^{k+r+1}[/tex] for all [tex]r\geq0[/tex].
     
  2. jcsd
  3. Oct 13, 2009 #2

    Mark44

    Staff: Mentor

    What have you tried? Do you know what the kernel of a matrix is?
     
  4. Oct 13, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Important point: for any matrix A, A0= 0!
     
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