- #1
Dustinsfl
- 2,281
- 5
[tex]\begin{bmatrix}
3 & 2\\
4 & 1
\end{bmatrix}[/tex]
[tex]det(A-\lambda I)=\begin{vmatrix}
3-\lambda & 2\\
4 & 1-\lambda
\end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5[/tex]
[tex]\lambda_{1}=5[/tex] and [tex]\lambda_{2}=-1[/tex]
When [tex]\lambda=5[/tex], [tex]\begin{bmatrix}
-2 & 2\\
4 & -4
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & -1\\
0 & 0
\end{bmatrix}[/tex]
The eigenspace for [tex]\lambda_{1}[/tex] is [tex]\begin{bmatrix}
1\\
1
\end{bmatrix}[/tex]
When [tex]\lambda=-1[/tex], [tex]\begin{bmatrix}
4 & 2\\
4 & 2
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}[/tex]
The eigenspace for[tex]\lambda_{2}[/tex] is [tex]\begin{bmatrix}
0\\
0
\end{bmatrix}[/tex]
I don't know what is going wrong but my second Eigenspace is wrong compared to the books answer which is [tex]\begin{bmatrix}
1\\
-2
\end{bmatrix}[/tex]
3 & 2\\
4 & 1
\end{bmatrix}[/tex]
[tex]det(A-\lambda I)=\begin{vmatrix}
3-\lambda & 2\\
4 & 1-\lambda
\end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5[/tex]
[tex]\lambda_{1}=5[/tex] and [tex]\lambda_{2}=-1[/tex]
When [tex]\lambda=5[/tex], [tex]\begin{bmatrix}
-2 & 2\\
4 & -4
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & -1\\
0 & 0
\end{bmatrix}[/tex]
The eigenspace for [tex]\lambda_{1}[/tex] is [tex]\begin{bmatrix}
1\\
1
\end{bmatrix}[/tex]
When [tex]\lambda=-1[/tex], [tex]\begin{bmatrix}
4 & 2\\
4 & 2
\end{bmatrix}\Rightarrow \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}[/tex]
The eigenspace for[tex]\lambda_{2}[/tex] is [tex]\begin{bmatrix}
0\\
0
\end{bmatrix}[/tex]
I don't know what is going wrong but my second Eigenspace is wrong compared to the books answer which is [tex]\begin{bmatrix}
1\\
-2
\end{bmatrix}[/tex]