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Eigenvalue of 0

  1. Aug 10, 2007 #1
    This is just a general question:

    If, when you are calculating the eigenvalues for a matrix, you get a root of 0 (eg. x^3 - x) --> x(x-1)(x+1), what does that mean for the eigenvectors?

    thanks,
    w.
     
  2. jcsd
  3. Aug 10, 2007 #2

    olgranpappy

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    Nothing. It just means that one of the eigenvalues is zero, it doesn't mean anything special about eigenvectors... When diagonalized the matrix of the operator looks like
    [tex]
    \left(
    \begin{array}{ccc}
    -1 & 0 & 0 \\
    0 & 0 & 0 \\
    0& 0 & 1
    \end{array}
    \right)\;.
    [/tex]
    In this basis, the eigenvector with eigenvalue -1 is (1,0,0) and the eigenvector with eigenvalue 0 is (0,1,0) and the eigenvector with eigenvalue 1 is (0,0,1).
     
  4. Aug 11, 2007 #3

    CompuChip

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    There is nothing wrong with an eigenvalue being zero, and it is not more special than an eigenvalue being -1, i or [itex]\pi[/itex].

    Only an eigenvector cannot be zero. Which makes sense, because the zero vector trivially satisfies A 0 = [itex]\lambda[/itex] 0 for any number [itex]\lambda[/itex].
     
  5. Aug 11, 2007 #4

    D H

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    A zero eigenvalue means the matrix in question is singular. The eigenvectors corresponding to the zero eigenvalues form the basis for the null space of the matrix.
     
  6. Aug 11, 2007 #5
    According to definition: Ax=cx,x is nonzero vector,then
    we have Ax=0,
    which means it has nonzero solutions,
    also means A is signular,
    also means the corresponding eigenvector is not uniquely determined.
     
  7. Aug 11, 2007 #6

    Hurkyl

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    Eigenvectors are never1 uniquely determined; at the very least, any nonzero scalar multiple of an eigenvector is an eigenvector.

    1: Unless your scalar field is GF(2).
     
  8. Aug 11, 2007 #7

    quasar987

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    And who, might I ask, is GF(2) ? :eek:
     
  9. Aug 11, 2007 #8
    Thank Hurkyl for pointing out my misunderstanding (I was thinking about sth else.)
     
  10. Aug 11, 2007 #9

    Hurkyl

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    GF(2) is the finite field with two elements. It's isomorphic to the integers modulo 2.
     
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