This paper presents an algebraic approach of fault diagnosis in linear dynamical system. Additive actuator and sensor faults are considered. Using tools and results of algebraic identification and pseudospectra analysis, a method is proposed for the generation of a residual sensitive to faults. The main advantage of this new approach is to realize fault diagnosis only from knowledge of input and output measurements and without identifying explicitly a model parameters. A numerical example is provided and discussed to illustrate the efficiency of the proposed fault diagnosis method.