The semigroup generated by a unitary orbit of a singular matrix

Abstract

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.

Keywords:

• Unitary orbit
• Matrix semigroup
• Generators
• Factorization
• 2001 Mathematics Subject Classifications:
• Primary: 15A30
• 15A23
• Secondary: 15A18
• 20M20

Acknowledgment

The authors are grateful to the referee for a thorough reading of the article and for discovering a few typographical errors. Research supported by NSERC grants and an internal research grant from the University of Victoria.