Theorem

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

Theorem

If $\vec{v}_1, \dots, \vec{v}_r$ are eigenvectors that correspond to distinct eigenvalues $\lambda_1, \dots, \lambda_r$ of an $n \times n$ matrix $A$, then the set $\{ \vec{v}_1, \dots, \vec{v}_r \}$ is linearly independent.