# Finding eigenvectors with irrational eigenvalues.

$$\begin{bmatrix} 4 & \frac {69}{28} \\ \frac {69}{28} & \frac {15}{7} \\ \end{bmatrix}$$ This is my initial matrix. So, I calculated my eigenvalues with no problem but got stuck with eigenvectors. My eigenvalues are 5.7049 and 0.4380 (they are right - I checked with Matlab).

Then I performed row manipulation and got

$$\begin{bmatrix} -1.70631 & 2.464 \\ 0 & 0 \\ \end{bmatrix}$$

And now I don't know what to do to get the correct eigenvectors. There's one free variable so I set $x_1$ as 't' and tried solving for $x_2$ but that gives me the correct answer. According to Matlab, the correct eigenvector should be $$\begin{bmatrix} -0.8224 \\ 0.5689\\ \end{bmatrix}$$

What do I have to do to get that??

• Since the eigenvector is only unique upto multiplication with a scalar, you can choose one coordinate free, but not $0$ (for example $x_2=1$). This gives an eigenvector colinear with the one given by Matlab. – Peter Oct 14 '17 at 17:15