Eigenvalues

Eigenvalues, eigenvectors & eigenspaces

Eigenvector $x$ corresponding to eigenvalue $\lambda$ of an $n \times n$ matrix $A$:

$$A\vec{x} = \lambda\vec{x}$$

The eigenvalues of a triangular matrix are the entries on its main diagonal

$\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if and only if the equation $$(A - \lambda I)\vec{x}=\vec{0}$$ has a nontrivial solution / $\det A - \lambda I = 0$ / the matrix is linearly dependent.

The set of all solutions of $A - \lambda I$ is called the eigenspace

Note: eigenvalues & eigenvectors can be complex.

Diagonalization #

Rewrite $A$ as $PDP^{-1}$

$PDP^{-1}$ where D is the diagonal matrix

1. Compute eigenvalues
2. Compute eigenvectors
3. Create $D$ by filling in the eigenvalues over the diagonal
4. $P$ has the eigenvectors of the corresponding eigenvalues in the same order
5. Compute $P^{-1}$

Diagonal matrices can be used to compute matrix powers faster since:

$$A^K = PD^{k}P^{-1}$$

Since $D$ is a diagonal matrix, computing its power is easy.