# Finding Eigenvectors and Eigenvalues

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?

## 1 Answer

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.

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