Eigenvectors and Eigenvalues

To find the eigenvalues of a matrix, $A$, we need to find $det(A-\lambda I_n)=0$ To find the eigenvectors of a matrix, $A$, we need to find $A{q}_i={\lambda }_i{q}_i$ → $ker({\lambda }_i{I}_n-A)$

Examples

$A=\left[\begin{array}{cc}0& 1\\ 0& -3\end{array}\right]$

Eigenvalues $det(A-\lambda I_2)=0$ → ${\lambda }^2+3\lambda =0$ → ${\lambda }_1=0.{\lambda }_2=-3$

Eigenvectors $A{q}_1={\lambda }_1{q}_1$ → $ker(\left[\begin{array}{cc}3& 1\\ 0& 0\end{array}\right])$ ~ $ker(\left[\begin{array}{cc}0& 1\\ 0& -3\end{array}\right])$ → $q_1$ = $ker(\left[\begin{array}{c}1\\ 0\end{array}\right])$ Same for $q_2$