Regarding this matrix $\begin{matrix} 1 & 1 \\ 1 &-1 \\ \end{matrix}$. In the end I have to solve this equation system:
$(\sqrt2-1)x_1-x_2=0$
$-x_1+(\sqrt2+1)x_2=0$
While the eigenvector is $x_1=1$ and $x_2=\sqrt2-1$, I can't seem to get the result using Gaussian elemination. It is always $x_1=0$ and $x_2 =0$.
Why is that so?
Solving the system $(\sqrt 2 I - A)\vec x = \vec 0$ by row reducing the augmented matrix, we obtain: $$ \left[\begin{array}{cc|c} \sqrt 2 - 1 & -1 & 0 \\ -1 & \sqrt 2 + 1 & 0 \end{array}\right] \sim \left[\begin{array}{cc|c} 1 & -\sqrt 2 - 1 & 0 \\ \sqrt 2 - 1 & -1 & 0 \end{array}\right] \sim \left[\begin{array}{cc|c} 1 & -\sqrt 2 - 1 & 0 \\ 0 & 0 & 0 \end{array}\right] $$ Hence, since $x_1 = (\sqrt 2 + 1)x_2$, we obtain: $$ \vec x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} (\sqrt 2 + 1)x_2 \\ x_2 \end{bmatrix} = x_2\begin{bmatrix} \sqrt 2 + 1 \\ 1 \end{bmatrix} $$ where $x_2$ is any nonzero scalar. In particular, taking $x_2 = \sqrt 2 - 1$ yields the eigenvector that you want.