Further matrix algebra

Eigenvalues and diagonalisation

An eigenvector of $\mathbf M$ keeps its direction: $\mathbf M\mathbf v=\lambda\mathbf v$, where $\lambda$ is the eigenvalue. Eigenvalues solve $\det(\mathbf M-\lambda\mathbf I)=0$. Putting eigenvectors in $\mathbf P$ and eigenvalues in $\mathbf D$ gives $\mathbf M=\mathbf{PDP}^{-1}$, so $\mathbf M^n=\mathbf{PD}^n\mathbf P^{-1}$.