I have a matrix
$$\begin{bmatrix}1 &-1 &2\\2 &-2 &4\\0 &1 &1\end{bmatrix}$$
Its eigenvalues are $0$, $\sqrt{5}$ and $-\sqrt{5}$
(These are checked in MATLAB to be correct). I have found its eigenvectors for $0$, and also don't seem to have a problem in similar situations when the eigenvalues are not "root". But when I have root, and are not allowed to turn into descimal-numbers, I have no clue how to proceed.
Does anyone know how to find eigenvectors when the values are $\sqrt{x}$? I'd be happy to watch a youtube video, or an example of how to proceed.
Best Answer
The method to find an eigenvector is the same, regardless of what the eigenvalue is. The only difference between different eigenvalues is the appearance of the computation involved. For example, you will need to add, multiply, divide by real numbers like $(1-\sqrt{5})$.
More details:
To compute an eigenvector, you need to solve the system $Av=\lambda v$. For $\lambda=\sqrt{5}$, you need to solve the system $\left[\begin{smallmatrix}1 & -1 & 2\\2 & -2 & 4\\ 0 & 1 & 1\end{smallmatrix}\right]\left[\begin{smallmatrix}x \\ y \\ z\end{smallmatrix}\right]=\left[\begin{smallmatrix}\sqrt{5} x\\ \sqrt{5} y \\ \sqrt{5} z\end{smallmatrix}\right]$, which becomes the system:
$(1-\sqrt{5})x -y +2z=0\\ 2+(-2-\sqrt{5})y + 4z=0\\ 0x+y+(1-\sqrt{5})z=0$