Bases of spaces of matrices satisfying rank conditions

Abstract

Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one.

Keywords:

• spaces of matrices
• rank condition
• non-singular bilinear maps

AMS subject classifications::

• 15A03
• 15A69

Acknowledgements

This work was partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020. The author would like to thank the anonymous referee for his/her valuable comments which have not only improved the presentation but have also helped in extending the results of this article.