Matrix Estimation using Matrix Forgetting Factor and Instrumental Variable for Nonstationary Sequences with Time Variant Matrix Gain José de Jesús Medel Juárez 1 , Pedro Guevara López 2 , Alberto Flores Rueda 3 1, 3 Centro de Investigación en Computación Instituto Politécnico Nacional Av. Juan de Dios Bátiz S/N esq. Miguel Othon de Mendizábal C. P. 07738, México D. F. 1 jjmedel@pollux.cic.ipn.mx , 2 aflores@pollux.cic.ipn.mx 2 Centro de Investigación en Tecnologías de Información y Sistemas Universidad Autónoma del Estado de Hidalgo Carretera Pachuca-Tulancingo km. 4.5 Ciudad Universitaria, Hidalgo 2 pguevara@df1.telmex.net.mx , Abstract. Consider us the problem of time-varying parameter estimation. The most immediate and simple idea is to include a discounting procedure in an estimation algorithm i.e., a procedure for discarding (forgetting) old information. The most common way to do is to introduce an exponential forgetting factor (FF) into the corresponding estimation procedure (to see: Ljung and Gunnarson (1990)). In this paper, the authors going to describe a good enough estimator considering a system with nonstationary time variant properties with respect to input and output qualities. The techniques used are Instrumental Variable (IV) and Matrix Forgetting Factor (MFF). The results previously obtained by (Poznyak and Medel 1999 a , 1999 b ) were the basis of this paper. The theoretical description illustrates the advantages with respect to others filters below cited. Keywords: Filtering, simulation, estimation, signal processing. 1 INTRODUCTION In many papers used a constant scalar Forgetting Factor (FF) for to filter a non-stationary system in SISO case, for example: Marco Campiy (1994) exposed in his paper, that the systems with unknown time-varying parameters and subject to stochastic disturbances have a problem for tracking parameters because in each parameter evolution, resorting to a class of adaptive recursive least squares algorithms, equipped with variable FF. The basic assumption in the analysis is that the observation vector, the noise and the parameter drift are stochastic processes satisfying a mixing condition. Furthermore, the observation vector satisfies an excitation condition imposed on its minimum power. In this paper, the author shown that the algorithm estimates with bounded error whenever the so-called covariance matrix of the algorithm keeps bounded. Finally, the size of such a matrix by a suitable choice of the feasible range for FF is possible to control. George V. Moustakides (1997) investigated the convergence properties of the FF into RLS algorithm by stationary data environment. He used the settling time as a performance measure and