Eigenvalues

An eigenvector of a square matrix $A$ is a nonzero vector $v$ such that $Av=\lambda v$ for some scalar $\lambda$, called an eigenvalue. To find eigenvalues, solve $\det (A-\lambda I)=0$, where $I$ is the identity matrix. For a $2\times 2$ matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, this gives $(a-\lambda )(d-\lambda )-bc=0$. Once an eigenvalue $\lambda$ is known, find an eigenvector by solving $(A-\lambda I)v=0$. If $A$ is diagonalizable as $A=PD{P}^{-1}$ with $D$ diagonal, then $A^n=P{D}^n{P}^{-1}$, so powers of $A$ are easy to compute by raising the diagonal entries of $D$ to the power $n$.