How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

Question

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$

First I got the eigenvalues which are $0$, $1$, $2$.

I tried to get the eigenvectors associated with the above eigenvalues but I cannot in case of the eigenvalue $1$ as I got the following matrix $$ \begin{bmatrix} 0& 0& 0\\ 0& 0& 1\\ 0& 1& 0 \end{bmatrix} $$ So, how can I get an eigenvector for this matrix?

Answer 1

The RREF of $[A-1I]v_1=0$ is:

$$\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}v_1 = 0$$

What if we chose:

$$v_1 = (1,0,0)$$

Update If we write:

$$ 0a + 1b + 0c = 0 \\ 0 a + 0b + 1 c = 0 \\ 0 a + 0 b + 0c = 0$$

What choices will actually make all three equations true and not be a zero eigenvector? What if we choose $(a , b , c) = (1, 0 , 0)$? Substitute those values back in and see if it satisfies the system.

Do you see how that satisfies the system? Recall that you cannot have a zero eigenvector. Would any other choice for $b = 1$, or $c=1$ or $b = c = 1$ work? No.

Also, see how this is the null space of the RREF?