Linear mapping/Trigonalizable/Characterizations/Method/Remark

The method to find an invariant flag, which is described in the proof of the implication (4) ${}\Rightarrow$ (2) of fact and which rests on fact, is constructive. If the restriction ${}\varphi _{|}U_{i}$ to some invariant linear subspace ${}U_{i}$ already constructed does not have an eigenvalue, then we know that the linear mapping is not trigonalizable.