# Least squares solutions to the rank-constrained matrix approximation problem in the Frobenius norm

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## Abstract

In this paper, we discuss the following rank-constrained matrix approximation problem in the Frobenius norm: \(\Vert C-AX\Vert =\min \) subject to \( \text{ rk }\left( {C_1 - A_1 X} \right) = b \), where *b* is an appropriate chosen nonnegative integer. We solve the problem by applying the classical rank-constrained matrix approximation, the singular value decomposition, the quotient singular value decomposition and generalized inverses, and get two general forms of the least squares solutions.

## Keywords

Matrix approximation problem Rank-constrained matrix SVD Q-SVD## Mathematics Subject Classification

15A09 15A24## Notes

### Acknowledgements

The authors wish to extend their sincere gratitude to Professor Eugene E. Tyrtyshnikov and the referees for their precious comments and suggestions.

### Funding

This work is supported by Guangxi Natural Science Foundation (Grant Number 2018GXNSFAA138181), the Special Fund for Scientific and Technological Bases and Talents of Guangxi (Grant Number 2016AD05050) and the Special Fund for Bagui Scholars of Guangxi. Funding was provided by National Natural Science Foundation of China (Grant No. 11401243).

### Compliance with ethical standards

### Conflict of interest

No potential conflict of interest was reported by the author.

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