hi there!

I´m having some trouble with calculating the eigenvectors for the following matrix:

$\displaystyle \left(\begin{array}{ccc}3&2&1\\0&1&2\\0&1&-1\end{array}\right)$

the char. polynomial can easily be calculated to:

$\displaystyle (3-x)((1-x)(-1-x)-2)=(3-x)(x^2-3)$

hence, 3 is an eigenvalue! But trying to determine the associated eigenvector I got stuck:

$\displaystyle \left(\begin{array}{ccc}0&2&1\\0&-2&2\\0&1&-4\end{array}\right)\left(\begin{array}{c}x_1\\x_2\ \x_3\end{array}\right)=\left(\begin{array}{c}0\\0\ \0\end{array}\right)$

There are 3 linearly independent lines, but 3 as an eigenvalue has the algebraic multiplicity 1, which requires an geometrical multiplicity (number of basis vectors for the eigenspace) less or equal to 1.

Can someone help please help me finding this eigenvector?

thanks a lot in advance!