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Math Help - Nilpotent

  1. #1
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    Nilpotent

    An nxn matrix is said to be nilpotent if A^k=0 for some positive \mathbb{Z} k. Show that all eigenvalues of a nilpotent matrix are 0.

    I have proved by math induction that, for m\geq1, \lambda^m is an eigenvalue of A^m.

    I don't know if that should help.
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    An nxn matrix is said to be nilpotent if A^k=0 for some positive \mathbb{Z} k. Show that all eigenvalues of a nilpotent matrix are 0.

    I have proved by math induction that, for m\geq1, \lambda^m is an eigenvalue of A^m.

    I don't know if that should help.
    Yes, that certainly does help! I presume that in doing that you also showed that if v is an eigenvector of A corresponding to eigenvalue \lambda then v is also an eigenvector of A^m corresponding to eigenvalue \lambda^m.

    In particular, if \lambda an eigenvalue of A, then \lambda^k is an eigenvalue of A^k- that is, for some non-zero vector v, A^k v= \lambda^k v. But A^kv= 0 for any vector so we have \lambda^k v= 0, with v non-zero.
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    Quote Originally Posted by HallsofIvy View Post
    Yes, that certainly does help! I presume that in doing that you also showed that if v is an eigenvector of A corresponding to eigenvalue \lambda then v is also an eigenvector of A^m corresponding to eigenvalue \lambda^m.

    In particular, if \lambda an eigenvalue of A, then \lambda^k is an eigenvalue of A^k- that is, for some non-zero vector v, A^k v= \lambda^k v. But A^kv= 0 for any vector so we have \lambda^k v= 0, with v non-zero.
    So that is all it is then?
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  4. #4
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    Well, what do you conclude from \lambda^k v= 0?
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    I should probably conclude lambda is zero.
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