Séminaire Calcul Formel

# On the general solutions of a rank factorization problem arising in vibration analysis

## by Alban Quadrat (Ouragan, Inria Paris & IMJ - PRG)

Europe/Paris
https://bbb.unilim.fr/b/vac-m6r-7dv

#### https://bbb.unilim.fr/b/vac-m6r-7dv

Description

Given a field K, r matrices D_1, …, D_r \in K^{n \times n} and a matrix M \in K^{n \times m}, in this talk, we shall study the problem of factoring M as follows

M=\sum_{i=1}^r D_i  u  v_i,

where u \in K^{n \times 1} and v_i \in K^{1 \times m} for i=1, …, r.

This rank factorization problem arises in modulation-based mechanical models studied in gearbox vibration analysis. It amounts to solving a family of bilinear polynomial systems.

In this talk, using module theory and homological algebra methods, we shall characterize the complete set of solutions of this rank factorization problem.

The results will be illustrated with the package CapAndHomalg developed in GAP by Mohamed Barakat (Univ. Siegen) and his collaborators.

This work was done in collaboration with Yacine Bouzidi, Axel Barrau (Safran Tech), Roudy Dagher (Inria Lille) and Elisa Hubert (LASPI, Univ. Lyon).