# Math Help - Eigen Space

1. ## [Solved] Eigen Space

$\begin{bmatrix}
3 & 2\\
4 & 1
\end{bmatrix}$

$det(A-\lambda I)=\begin{vmatrix}
3-\lambda & 2\\
4 & 1-\lambda
\end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5$

$\lambda_{1}=5$ and $\lambda_{2}=-1$
When $\lambda=5$, $\begin{bmatrix}
-2 & 2\\
4 & -4
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & -1\\
0 & 0
\end{bmatrix}$

The eigenspace for $\lambda_{1}$ is $\begin{bmatrix}
1\\
1
\end{bmatrix}$

When $\lambda=-1$, $\begin{bmatrix}
4 & 2\\
4 & 2
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & \frac{1}{2}\\
0 & 0
\end{bmatrix}$

The eigenspace for $\lambda_{2}$ is $\begin{bmatrix}
1\\
-2
\end{bmatrix}$

Solved nevermind