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Zero as an element of an eigenvector

  1. Apr 18, 2013 #1
    Quick question on eigenvectors;

    Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector?

    Thank you!
     
  2. jcsd
  3. Apr 18, 2013 #2

    AlephZero

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    Suppose the zero is the last element of the eigenvector. Then you can partition the matrix and write
    ##\begin{bmatrix} A & u \\ v^T & s \end{bmatrix}\begin{bmatrix} x \\ 0 \end{bmatrix} = \lambda
    \begin{bmatrix} x \\ 0 \end{bmatrix}## where ##u## and ##v## are vectors and ##s## is a single matrix element.

    Multiplying out you get the two equations ##Ax = \lambda x## and ##v^t x = 0##. Interestingly, the last column of the original matrix doesn't appear in those equations.
     
  4. Apr 19, 2013 #3
    Thank you for your reply!

    Is there a way to determine from the matrix whether a zero will appear without calculating the eigenvector explicitly?
     
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