On the general solutions of a rank factorization problem
Roudy Dagher, Elisa Hubert, Alban Quadrat · HAL (Le Centre pour la Communication Scientifique Directe) · 2021
Vibration analysis aims at identifying potential failures of a rotating machinery from the monitoring of its vibration levels, i.e., by measuring the vibrations and comparing them to known failure vibration signals. For the diagnostic of gearboxes, new demodulation methods have recently been introduced in acoustic and signal processing. This new approach yields the problem of writing/factorizing a matrix M as D_1 u v_1 + ... + D_r u v_r = (D_1 u ... D_r u) (v_1^T ... v_r^T)^T,where the D_i's are fixed matrices, u (resp., v_i) is a row (resp., column) vector to be determined and i=1, ..., r. In this paper, using module theory and homological algebra, we study this rank factorization problem. More precisely, we characterize the general solutions of this family of polynomial systems. Finally, the results we develop are effective in the sense of computer algebra. Thus, they can be implemented in standard computer algebra systems handling polynomial systems and basic homological algebra methods (e.g., the Singular system, the GAP library CapAndHomalg, the Maple package OreModules).