EigenVector I don't understand I know how to calculate the eigenvalues and get eigenvectors but there are some Matrix I don't understand?


When Eigenvalue is $1$ we have two eigenvectors i understand and have don gauss-eleminationsand only get $(1,0,1)$ when I have done Gauss-Eleminations  but where does $(0,1,0)$ comes from?

And here is the Matrix above picture:
You can divide the middle row with $1/3$ so you get $(1,0,1)$
and remove the upper and bottom row but where does the eigenvector $(0,1,0)$ comes from
and how two vectors span this eigenspace I don't understand??
 A: I think the best way to understand eigenvectors is to look at the definition of eigenvectors:
$M\underline{v}=\lambda\underline{v}$ where $M$ is the matrix and $\underline{v}$ the eigenvector and $\lambda$ the associated eigenvalue.
So what I am guessing you are doing is that you rearranged the equation a bit and got the following $(M-\lambda I)\underline{v}=\underline{0}$. So looking at the matrix you got at the end you need to find vectors that gives you the zero vector and looking at it if you have the matrix operating on $(0,1,0)$ you get $(0,0,0)$ so everything is in order. And thus $(0,1,0)$ is an eigenvector.
A: The row reduced form is
$$\begin{pmatrix} 
1 & 0 & -1 \\ 
0 & 0 & 0 \\ 
0 & 0 & 0 \\ 
\end{pmatrix}.$$
The null space of this matrix is
$$\{(x_3, x_2, x_3) : x_2, x_3 \in \mathbb{R}\} = \{x_2(0, 1, 0) + x_3(1, 0, 1) : x_2, x_3 \in \mathbb{R}\} = \text{span}\{(0, 1, 0), (1, 0, 1)\}.$$
Since row operations don't change the null space of a matrix, this is the null space of the original matrix.
