I need help finding an eigenspace corresponding to each eigenvalue of A = $\begin{bmatrix}
1 & -1 & 0 \\
2 & 4 & 0 \\
9 & 5 & 4
\end{bmatrix}$ ?
I followed standard eigen-value finding procedures, and I was able to find that $\lambda = 4, 2, 3$. I was even able to find the basis corresponding to $\lambda = 4$:
$\begin{bmatrix}
0\\
0\\
1
\end{bmatrix}$
However, I am unable to find the basis corresponding to $\lambda = 2, 3$. I would really appreciate it if someone could please help me with this.
$\lambda = 2$:
First we compute A – I$\lambda$ to get:
$\begin{bmatrix}
-1 & -1 & 0 \\
2 & 2 & 0 \\
9 & 5 & 2
\end{bmatrix}$
Best Answer
$\begin{bmatrix} -1 & -1 & 0 \\ 2 & 2 & 0 \\ 9 & 5 & 2 \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \\ x_3 \end{bmatrix}=\mathbf 0$
$x_1 + x_2 = 0\\ x_2 = -x_1\\ 9x_1 + 5(-x_1) + 2x_3 = 0\\ x_3 = -2x_1 $
$\mathbf x = \begin{bmatrix} 1\\ -1\\ -2 \end{bmatrix}$