Schur's Triangularization Theorem

Unfortunately not every matrix is diagonalizable by a similarity transformation. However, the next best thing possible: Any square matrix $$A$$ can be triangularized by a similarity transformation, as shown in Schur's Triangularization Theorem (see p508 ): $$A = U^* T U$$

where $$U$$ is a unitary matrix that is not unique. $$T$$ is a triangular matrix that is also not unique.