Problem:
Find the eigenvectors of the following matrix:
$\left(
\begin{array}{cc}
 \frac{19}{2} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{39}{4} \\
\end{array}
\right)$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find all the eigenvalues and eigenvectors of the matrix }M: \\
 M=\left(
\begin{array}{cc}
 \frac{19}{2} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{39}{4} \\
\end{array}
\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Find }\lambda \in \mathbb{C} \text{such }\text{that }M v=\lambda  v \text{for }\text{some }\text{nonzero }\text{vector }v: \\
 M v=\lambda  v \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Rewrite }\text{the }\text{equation }M v=\lambda  v \text{as }(M-\mathbb{I} \lambda ) v=0: \\
 (M-\mathbb{I} \lambda ) v=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Find }\text{all }\lambda  \text{such }\text{that }| M-\mathbb{I} \lambda | =0: \\
 | M-\mathbb{I} \lambda | =0 \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{lll}
 M-\mathbb{I} \lambda  & = & \fbox{$\left(
\begin{array}{cc}
 \frac{19}{2} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{39}{4} \\
\end{array}
\right)$}-\fbox{$\left(
\begin{array}{cc}
 1 & 0 \\
 0 & 1 \\
\end{array}
\right)$} \lambda  \\
 \text{} & = & \left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right) \\
\end{array}
 \\
 \left| 
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right| =0 \\
\end{array}
 \\
 
\begin{array}{l}
 \left| 
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right| =\lambda ^2+\frac{\lambda }{4}-97: \\
 \lambda ^2+\frac{\lambda }{4}-97=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\text{97 }\text{to }\text{both }\text{sides}: \\
 \lambda ^2+\frac{\lambda }{4}=97 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{1}{64} \text{to }\text{both }\text{sides}: \\
 \lambda ^2+\frac{\lambda }{4}+\frac{1}{64}=\frac{6209}{64} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{left }\text{hand }\text{side }\text{as }\text{a }\text{square}: \\
 \left(\lambda +\frac{1}{8}\right)^2=\frac{6209}{64} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Take }\text{the }\text{square }\text{root }\text{of }\text{both }\text{sides}: \\
 \lambda +\frac{1}{8}=\frac{\sqrt{6209}}{8}\text{ or }\lambda +\frac{1}{8}=-\frac{\sqrt{6209}}{8} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{1}{8} \text{from }\text{both }\text{sides}: \\
 \fbox{$\lambda =\frac{\sqrt{6209}}{8}-\frac{1}{8}$}\text{ or }\lambda +\frac{1}{8}=-\frac{\sqrt{6209}}{8} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{1}{8} \text{from }\text{both }\text{sides}: \\
 \lambda =\frac{\sqrt{6209}}{8}-\frac{1}{8}\text{ or }\fbox{$\lambda =-\frac{1}{8}-\frac{\sqrt{6209}}{8}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Find }\text{all }v \text{such }\text{that }(M-\mathbb{I} \lambda ) v=0 \text{for }\text{some }\text{eigenvalue }\lambda : \\
 (M-\mathbb{I} \lambda ) v=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Substitute }\left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right) \text{for }(M-\mathbb{I} \lambda ): \\
 \fbox{$\left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right)$} v=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }v \text{as }\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right) \text{and }0 \text{as }\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right): \\
 \left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right).\fbox{$\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)$}=\fbox{$\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right)$} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{First, }\text{substitute }\frac{\sqrt{6209}}{8}-\frac{1}{8} \text{for }\lambda  \text{in }\text{the }\text{matrix }\left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right) \text{and }\text{solve }\text{the }\text{system}: \\
 \fbox{$\left(
\begin{array}{cc}
 \frac{77}{8}-\frac{\sqrt{6209}}{8} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{77}{8}-\frac{\sqrt{6209}}{8} \\
\end{array}
\right)$}.\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{In }\text{augmented }\text{matrix }\text{form, }\text{the }\text{system }\text{is }\text{written }\text{as}: \\
 \left(
\begin{array}{ccc}
 \frac{77}{8}-\frac{\sqrt{6209}}{8} & \frac{5}{4} & 0 \\
 \frac{7}{2} & -\frac{77}{8}-\frac{\sqrt{6209}}{8} & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Swap }\text{row }1 \text{with }\text{row }2: \\
 \left(
\begin{array}{ccc}
 \frac{7}{2} & -\frac{77}{8}-\frac{\sqrt{6209}}{8} & 0 \\
 \frac{77}{8}-\frac{\sqrt{6209}}{8} & \frac{5}{4} & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }-\frac{2}{7} \left(\frac{77}{8}-\frac{\sqrt{6209}}{8}\right)\, \times \, \text{(row }1) \text{to }\text{row }2: \\
 \left(
\begin{array}{ccc}
 \frac{7}{2} & -\frac{77}{8}-\frac{\sqrt{6209}}{8} & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply }\text{row }1 \text{by }\frac{2}{7}: \\
 \left(
\begin{array}{ccc}
 1 & \frac{1}{28} \left(-77-\sqrt{6209}\right) & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Translated }\text{back }\text{to }\text{a }\text{matrix }\text{equation, }\text{the }\text{reduced }\text{system }\left(
\begin{array}{ccc}
 1 & \frac{1}{28} \left(-77-\sqrt{6209}\right) & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \text{is}: \\
 \left(
\begin{array}{cc}
 1 & \frac{1}{28} \left(-77-\sqrt{6209}\right) \\
 0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{As }\text{a }\text{scalar }\text{equation, }\text{the }\text{system }\left(
\begin{array}{cc}
 1 & \frac{1}{28} \left(-77-\sqrt{6209}\right) \\
 0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \text{translates }\text{to}: \\
 v_1+\frac{1}{28} \left(-\sqrt{6209}-77\right) v_2=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Rewrite }\text{the }\text{equation }\text{as}: \\
 v_1=-\frac{1}{28} \left(-77-\sqrt{6209}\right) v_2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{According }\text{to }\text{the }\text{above }\text{equation}: \\
 v\text{ = }\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)\text{ = }\left(
\begin{array}{c}
 -\frac{1}{28} \left(-77-\sqrt{6209}\right) v_2 \\
 v_2 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Letting }v_2=1 \text{in }\left(
\begin{array}{c}
 -\frac{1}{28} \left(-77-\sqrt{6209}\right) v_2 \\
 v_2 \\
\end{array}
\right), \text{we }\text{find }\text{that }\left(
\begin{array}{c}
 \frac{1}{28} \left(77+\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \text{is }\text{an }\text{eigenvector }\text{of }\text{the }\text{matrix }\left(
\begin{array}{cc}
 \frac{19}{2} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{39}{4} \\
\end{array}
\right) \text{associated }\text{with }\text{the }\text{eigenvalue }\frac{\sqrt{6209}}{8}-\frac{1}{8}: \\
 v=\left(
\begin{array}{c}
 \frac{1}{28} \left(77+\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Substitute }-\frac{1}{8}-\frac{\sqrt{6209}}{8} \text{for }\lambda  \text{in }\text{the }\text{matrix }\left(
\begin{array}{cc}
 \frac{19}{2}-\lambda  & \frac{5}{4} \\
 \frac{7}{2} & -\lambda -\frac{39}{4} \\
\end{array}
\right) \text{and }\text{solve }\text{the }\text{system}: \\
 \fbox{$\left(
\begin{array}{cc}
 \frac{77}{8}+\frac{\sqrt{6209}}{8} & \frac{5}{4} \\
 \frac{7}{2} & \frac{\sqrt{6209}}{8}-\frac{77}{8} \\
\end{array}
\right)$}.\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{In }\text{augmented }\text{matrix }\text{form, }\text{the }\text{system }\text{is }\text{written }\text{as}: \\
 \left(
\begin{array}{ccc}
 \frac{77}{8}+\frac{\sqrt{6209}}{8} & \frac{5}{4} & 0 \\
 \frac{7}{2} & \frac{\sqrt{6209}}{8}-\frac{77}{8} & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{7}{2 \left(\frac{77}{8}+\frac{\sqrt{6209}}{8}\right)}\, \times \, \text{(row }1) \text{from }\text{row }2: \\
 \left(
\begin{array}{ccc}
 \frac{77}{8}+\frac{\sqrt{6209}}{8} & \frac{5}{4} & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide }\text{row }1 \text{by }\frac{77}{8}+\frac{\sqrt{6209}}{8}: \\
 \left(
\begin{array}{ccc}
 1 & \frac{1}{28} \left(\sqrt{6209}-77\right) & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Translated }\text{back }\text{to }\text{a }\text{matrix }\text{equation, }\text{the }\text{reduced }\text{system }\left(
\begin{array}{ccc}
 1 & \frac{1}{28} \left(\sqrt{6209}-77\right) & 0 \\
 0 & 0 & 0 \\
\end{array}
\right) \text{is}: \\
 \left(
\begin{array}{cc}
 1 & \frac{1}{28} \left(\sqrt{6209}-77\right) \\
 0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{As }\text{a }\text{scalar }\text{equation, }\text{the }\text{system }\left(
\begin{array}{cc}
 1 & \frac{1}{28} \left(\sqrt{6209}-77\right) \\
 0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
 0 \\
 0 \\
\end{array}
\right) \text{translates }\text{to}: \\
 v_1+\frac{1}{28} \left(\sqrt{6209}-77\right) v_2=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Rewrite }\text{the }\text{equation }\text{as}: \\
 v_1=-\frac{1}{28} \left(\sqrt{6209}-77\right) v_2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{According }\text{to }\text{the }\text{above }\text{equation}: \\
 v\text{ = }\left(
\begin{array}{c}
 v_1 \\
 v_2 \\
\end{array}
\right)\text{ = }\left(
\begin{array}{c}
 -\frac{1}{28} \left(\sqrt{6209}-77\right) v_2 \\
 v_2 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Letting }v_2=1 \text{in }\left(
\begin{array}{c}
 -\frac{1}{28} \left(\sqrt{6209}-77\right) v_2 \\
 v_2 \\
\end{array}
\right), \text{we }\text{find }\text{that }\left(
\begin{array}{c}
 \frac{1}{28} \left(77-\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \text{is }\text{an }\text{eigenvector }\text{of }\text{the }\text{matrix }\left(
\begin{array}{cc}
 \frac{19}{2} & \frac{5}{4} \\
 \frac{7}{2} & -\frac{39}{4} \\
\end{array}
\right) \text{associated }\text{with }\text{the }\text{eigenvalue }-\frac{1}{8}-\frac{\sqrt{6209}}{8}: \\
 v=\left(
\begin{array}{c}
 \frac{1}{28} \left(77-\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{We }\text{found }\text{the }\text{following }\text{eigenvalue/eigenvector }\text{pair}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 
\begin{array}{c|c}
 \text{Eigenvalue} & \text{Eigenvector} \\
\hline
 \frac{\sqrt{6209}}{8}-\frac{1}{8} & \left(
\begin{array}{c}
 \frac{1}{28} \left(77+\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \\
 -\frac{1}{8}-\frac{\sqrt{6209}}{8} & \left(
\begin{array}{c}
 \frac{1}{28} \left(77-\sqrt{6209}\right) \\
 1 \\
\end{array}
\right) \\
\end{array}
 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/linear_algebra/eigenvectors_w_steps/1648.txt'}