Kelly Nelson

2022-01-04

Determine the eigenvalues, eigenveltI and eigenspace of the follewing matix
$\left[\begin{array}{cc}1& 3\\ 5& 3\end{array}\right]$

William Appel

The characteristic equation can be written as:
$|A-\lambda I|=0$
Where $\lambda =$ eigen value
$\left[\begin{array}{cc}1-\lambda & 3\\ 5& 3-\lambda \end{array}\right]=0$
$\left(1-\lambda \right)\left(3-\lambda \right)-5×3=0$
$3-\lambda -3\lambda +{\lambda }^{2}-15=0$
${\lambda }^{2}-4\lambda -12=0$
$\lambda \left(\lambda -6\right)+2\left(\lambda -6\right)=0$
$\left(\lambda +2\right)\left(\lambda -6\right)=0$
$⇒\lambda +2=0,\lambda -6=0$
$⇒\lambda =-2,6$
${\lambda }_{1}=2,{\lambda }_{2}=6$
To find eigen vectors:
at $\lambda =-2$
$\left[\begin{array}{cc}1-\lambda & 3\\ 5& 3-\lambda \end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
$\left[\begin{array}{cc}1-\left(-2\right)& 3\\ 5& 3-\left(-2\right)\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
$\left[\begin{array}{cc}3& 3\\ 5& 5\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
$⇒3{x}_{1}+3{x}_{2}=0,5{x}_{1}+5{x}_{2}=0$
$⇒{x}_{1}+{x}_{2}=0,{x}_{1}+{x}_{2}=0$
Let ${x}_{2}={k}_{1}$
$⇒{x}_{1}=-{k}_{1}$
$\therefore \left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}-{k}_{1}\\ {k}_{1}\end{array}\right]={k}_{1}\left[\begin{array}{c}-1\\ 1\end{array}\right]$
$\therefore {\stackrel{\to }{v}}_{1}=\left[\begin{array}{c}-1\\ 1\end{array}\right]$
at $\lambda =6$
$\left[\begin{array}{cc}1-6& 3\\ 5& 3-6\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

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