Let there be a matrix $$\begin{pmatrix} 1 & 3 & 5 \\ 1 & 3 & 3 \\ 1 & 5 & 1 \end{pmatrix}$$ Give an example for a vector how is not an Eigenvector (and not zero)

  1. The straightforward way is to look for the Eigenvectors and Eigenvalues, is there a shorter way?
  2. Can I row reduce the matrix to find the characteristic polynomial?

Most non-zero vectors are not eigenvectors of this matrix. For example, $(1,0,0)^T$ is not an eigenvector.

Row-reducing the matrix will change the characteristic polynomial. So no, you can't find the eigenvalues of this matrix by row-reducing it.

  • $\begingroup$ As for the eigenvector, should I look for linear combination of the matrix columns, how did you find $(1,0,0)^T$? $\endgroup$ – gbox May 5 '15 at 17:44
  • $\begingroup$ I guessed. As I indicated, most guesses will be correct here. $\endgroup$ – Omnomnomnom May 5 '15 at 17:45
  • $\begingroup$ and on any other matrix are they things that I should look for? $\endgroup$ – gbox May 5 '15 at 17:46
  • 1
    $\begingroup$ For similar problems: note that if $(1,0,0),(0,1,0),(0,0,1)$ are all eigenvectors, then the matrix must be diagonal. So, for any non-diagonal matrix, one of these will work. $\endgroup$ – Omnomnomnom May 5 '15 at 17:46
  • $\begingroup$ If you have a diagonal matrix, then the eigenvalues/eigenvectors are obvious anyway. $\endgroup$ – Omnomnomnom May 5 '15 at 17:47

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