Eigenvalues and Eigenvectors

An eigenvector of $A$ is a nonzero vector $\mathbf{x}$ such that:

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

$\lambda$ is the eigenvalue — the scaling factor.

Finding Eigenvalues

Solve the characteristic equation:

$$\det(A - \lambda I) = 0$$

This gives a polynomial in $\lambda$ of degree $n$.

Finding Eigenvectors

For each eigenvalue $\lambda_i$, solve:

$$(A - \lambda_i I)\mathbf{x} = \mathbf{0}$$

Key Facts

• $\text{trace}(A) = \sum \lambda_i$
• $\det(A) = \prod \lambda_i$
• Eigenvalues of $A^T$ are the same as $A$
• If $A$ is invertible, eigenvalues of $A^{-1}$ are $1/\lambda_i$