Row and column rank partitions under small perturbations

Abstract

The concept of rank partition of a family of vectors v ₁, … , v_m is a generalization of that has been useful for studying problems in Multilinear Algebra, namely, establishing conditions for non-vanishing decomposable symmetrized tensors and conditions for the equality of decomposable symmetrized tensors. A previous paper has described the rank partitions that can be obtained with arbitrarily small perturbations of the vectors v ₁, … , v_m . The purpose of the present article is to describe the pairs of row rank partitions and column rank partitions that can be obtained with arbitrarily small perturbations of a matrix.

Keywords:

• rank partition
• perturbation of matrices

AMS Subject Classifications::

• 15A03
• 05B35

Acknowledgements

This research was done within the activities of the Centro de Estruturas Lineares e Combinatórias (CELC) and was supported by Fundação para a Ciência e a Tecnologia (FCT). The work of R. Simões was also supported by CELC grant BIC-CI/04/00, during 2000/2001 and FCT grant SFRH/BM/4422/2001, during 2001/2002.