Solving a system of linear equations to find two eigenvectors.

Question

So I have \begin{align*} x - 2y + z & = 0 \\ -2x + 4y - 2z & = 0 \\ x - 2y + z & = 0 \end{align*}

I know I need to find two eigenvectors for the eigenspace with eigenvalue 2 as I know the matrix is diagonalizable and I've already found the eigenvector for the eigenspace with eigenvalue 8...

I can't seem to solve this. Please help!

Answer 1

$$ \begin{align*} x - 2y + z & = 0 \\ -2x + 4y - 2z & = 0 \\ x - 2y + z & = 0 \end{align*} $$

$$ \implies \begin{align*} x - 2y + z & = 0 \\ 0 & = 0 \\ 0 & = 0. \end{align*}$$

So you have two free variables. Assuming $y=t, z=s,\, t,s\in \mathbb{R} $, then the solution is given by

$$ \begin{bmatrix} x \\ y\\ z\end{bmatrix} = \begin{bmatrix} 2t-s \\ t\\ s\end{bmatrix} = \begin{bmatrix} 2t \\ t\\ 0\end{bmatrix} + \begin{bmatrix} -s \\ 0\\ s\end{bmatrix} = t\begin{bmatrix} 2 \\ 1\\ 0\end{bmatrix} + s\begin{bmatrix} -1 \\ 0\\ 1\end{bmatrix}.$$

Can you see the two eigenvectors now?