Zero as an element of an eigenvector

  • Thread starter ekkilop
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  • #1
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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!
 

Answers and Replies

  • #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.
 
  • #3
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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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