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.

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