On least squares solutions subject to a rank restriction

Abstract

In this paper,we discuss rank-constrained least squares solutions to the matrix equation under the rank restriction in the Frobenius norm. We derive the rank range and expression of these least squares solutions by applying generalized inverses, singular value decomposition and the Eckart–Young–Mirsky theorem.

Keywords:

• singular value decomposition
• least squares
• rank-constrained

AMS Subject Classifications:

• 15A24
• 15A29

Acknowledgements

The author is grateful to Professor Ren-Cang Li and the referees for their valuable comments and suggestions which helped to greatly improve this paper.

Notes

The work was supported in part by the National Natural Science Foundation of China [grant number 11171226], [grant number 11201193]; the Foundation of Anhui Educational Committee [grant number KJ2012B175] and the Scientific Research Foundation of Huainan Normal University.