. . A PROPOSAL FOR RESOLUTION OF LYAPUNOV EQUATION. . . Cosme Rafael Marcano Gamero. Systems Engineer, (Universidad de los Andes, Mérida – 1986). Magíster Scientiarum in Electronic Engineering, (Universidad Nacional Experimental Politécnica “Antonio José de Sucre” – 2004). Professor at UNEXPO “Antonio José de Sucre”. Estado Bolívar – Venezuela firstname.lastname@example.org tlf. 58-0286-9619965 . . . Abstract. The Lyapunov Second Stability Method consists of select ing a so-called Lyapunov Candidate Function, which satisfies certain conditions that permit us to utilize it in the stability analysis of a mathematical model synthesized from a process which we want to put under action of a given Control Law. In linear cases, it is always possible to find a candidate function of quadratic kind, , that satisfies the desired cond5itions. By applying the Lyapunov Second Stability Method to this function, it appears an algebraic, ordinary matrix equations system of the kind , where P and Q are positive-definite matrices. In this work, the solution of this algebraic system by solving a lineal system of unknowns and same number of equations is proposed. After some elementary manipulation of original equations, solution can be achieved with some traditional method, like Gauss Inverse Deletion or any of its variants. This paper presents some easy algorithms that allow us to re-accommodate the original matrix system into a conventional algebraic, ordinary, linear equations system. Ax=b. . Key words: Lyapunov Second Stability Method, Matrix System, Gaussian Back Deletion Algorithm. . . . . . . . . . . . . . . . ….