Study Mathematics notes on Eigenvalues and Eigenvectors. Useful for revision, homework, and exam preparation in Zimbabwe.
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B6 (i) Determine the eigenvalues and corresponding eigenvectors of matrix $A$ :
Given the matrix:
\[
A=\left(\begin{array}{lll}
1 | 0 | 0
2 | 4 | 0
8 | 1 | 3
\end{array}\right)
\]
First, find the eigenvalues by solving the characteristic equation $\operatorname{det}(A-\lambda I)=0$ :
\[
\left|\begin{array}{ccc}
1-\lambda | 0 | 0
2 | 4-\lambda | 0
8 | 1 | 3-\lambda
\end{array}\right|=(1-\lambda)(4-\lambda)(3-\lambda)=0
\]
The eigenvalues are:
\[
\lambda_{1}=1, \quad \lambda_{2}=4, \quad \lambda_{3}=3
\]
Eigenvectors:
1. For $\lambda_{1}=1$ :
\[
(A-I) \mathbf{v}=\left(\begin{array}{ccc}
0 | 0 | 0
2 | 3 | 0
8 | 1 | 2
\end{array}\right)\left(\begin{array}{l}
v_{1}
v_{2}
v_{3}
\end{array}\right)=\mathbf{0}
\]
Solving gives the eigenvector $\mathbf{v}_{1}=\left(\begin{array}{c}1 -2 / 3 -11 / 3\end{array}\right)$ or any scalar multiple. 2. For $\lambda_{2}=4:$
\[
(A-4 I) \mathbf{v}=\left(\begin{array}{ccc}
-3 | 0 | 0
2 | 0 | 0
8 | 1 | -1
\end{array}\right)\left(\begin{array}{l}
v_{1}
v_{2}
v_{3}
\end{array}\right)=\mathbf{0}
\]
Solving gives the eigenvector $\mathbf{v}_{2}=\left(\begin{array}{l}0 1 1\end{array}\right)$. 3. For $\lambda_{3}=3$ :