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\author{Harald R�ck}

\begin{document}

\textbf{First name:} Harald \\
\textbf{Last name:} R"ock \\
\textbf{Date:} \today \\
\textbf{Homework number:} 4 \\
\textbf{Homework Title:} Exercise 4.3


\paragraph{Problem description:} ~\\
let 
\[ \mathbf{A} = \left[
\begin{array}{cc}
	1 & 4 \\
	1 & 1 
\end{array} \right] .
\]
(a) What is the characteristic polynomial of $\mathbf{A}$? \\
(b) What are the roots of the characteristic polynomial of $\mathbf{A}$? \\
(c) What are the eigenvalues of $\mathbf{A}$? \\
(d) What are the eigenvectors of $\mathbf{A}$?
\paragraph{Problem solution:} ~\\
(a)

\[ \mathrm{det}  \left[
\begin{array}{cc}
    1-r & 4 \\
    1 & 1-r 
\end{array} \right] = 0 \]
\[ \Longrightarrow (1-r)(1-r) - 4 = 0 \Rightarrow r^2 - 2r - 3 = 0 \]

(b)
\[ \Longrightarrow r_{1,2} = 1 \mp \sqrt{1 + 3} = 1 \mp 2 \]
\[ r_1 = 3 ;~ r_2 = -1 \]

(c)
\[ e_1 = 3 ;~ e_2 = -1 \]

(d)
\begin{multicols}{2}
$\mathbf{A} x = 3 x $
\begin{eqnarray*}
	x_1 + 4 x_2 & = & 3 x_1 \\
	x_1 + x_2 & = & 3 x_2 \\
	\rightarrow -x_1 + 2 x_2 & = & 0 
\end{eqnarray*}
$ \Rightarrow x = [ 2 ~~ 1 ]^T = v_1 $

$\mathbf{A} x = - x $
\begin{eqnarray*}
	x_1 + 4 x_2 & = & -x_1 \\
	x_1 + x_2 & = &  -x_2 \\
	\rightarrow x_1 + 2 x_2 & = & 0 
\end{eqnarray*}
$ \Rightarrow x = [ 2 ~ -1 ]^T = v_2 $
\end{multicols}
\end{document}
