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