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\textbf{First name:} Monika \\
\textbf{Last name:} Wunderlich\\
\textbf{Date:} 2.1.04\\
\textbf{Homeworknumber:} 4 \\
\textbf{Homework Title:} 4.2. \\
\newline
\newline
\textbf{Problem description:} 
\newline
What are the eigenvalues and corresponding eigenvectors of the following matrix?\\
$$
A = \left [
\begin{array}{c c c}
	$1$ & $2$ & $-4$ \\
	$0$ & $2$ & $1$ \\
	$0$ & $0$ & $3$ \\
\end{array}
\right ]
$$
\newline
\textbf{Problem solution:} 
\newline
The eigenvalues are the roots $\lambda_i$ of the characteristic polynomial\\ $det(A-\lambda I) = 0$
in $\lambda$ of degree $n$. \\
\par
Characteristic polynomial: $(1 - \lambda)(2-\lambda)(3- \lambda)$. \\
\newline
The eigenvector $x_i$ corresponding to $\lambda_i$ is the solution of $(A-\lambda_i I)x = 0. $\\
\newline
\textbf{Results:} 
\newline
The eigenvalues of A : $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3$.  \\
The eigenvectors of A:\\
\par $\lambda_1$:
	$$\left [
	 \begin{array}{c c c}
	$0$ & $2$ & $-4$ \\
	$0$ & $1$ & $1$ \\
	$0$ & $0$ & $2$ \\
	\end{array}
	\right ] \cdot
	\left [
	 \begin{array}{c}
	x \\
	y\\
	z\\
	\end{array}
	\right ] =
	\left [
	 \begin{array}{c}
	2y - 4z \\
	y + z \\
	2z \\
	\end{array}
	\right ] =^{!}
	\left [
	 \begin{array}{c}
	0 \\
	0 \\
	0 \\
	\end{array}
	\right ].$$
	$$x=a, \ y = 0,\ z =0. \ a  \in  \mathbf{R}. $$
\par $\lambda_2$:
	$$\left [
	 \begin{array}{c c c}
	$-1$ & $2$ & $-4$ \\
	$0$ & $0$ & $1$ \\
	$0$ & $0$ & $1$ \\
	\end{array}
	\right ] \cdot
	\left [
	 \begin{array}{c}
	x \\
	y\\
	z\\
	\end{array}
	\right ] =
	\left [
	 \begin{array}{c}
	-x +2y - 4z \\
	z \\
	z \\
	\end{array}
	\right ] =^{!}
	\left [
	 \begin{array}{c}
	0 \\
	0 \\
	0 \\
	\end{array}
	\right ].$$
	$$x=b,\ y = \frac{b}{2},\ z =0.\ b \in \mathbf{R}.$$
\par $\lambda_3$:
	$$\left [
	 \begin{array}{c c c}
	$-2$ & $2$ & $-4$ \\
	$0$ & $-1$ & $1$ \\
	$0$ & $0$ & $0$ \\
	\end{array}
	\right ] \cdot
	\left [
	 \begin{array}{c}
	x \\
	y\\
	z\\
	\end{array}
	\right ] =
	\left [
	 \begin{array}{c}
	-2x +2y - 4z \\
	-y + z \\
	0\\
	\end{array}
	\right ] =^{!}
	\left [
	 \begin{array}{c}
	0 \\
	0 \\
	0 \\
	\end{array}
	\right ].$$
	$$x=-c,\ y = c,\ z =c.\ c \in \mathbf{R}.$$
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