Abstract

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.

keywords: Kronecker products; Matrix least-squares problems; Generalized inverses; Matrix norms; Iterative methods

Mathematical subject codes: 15A24; 15A69; 15A09

Article Outline

1. Introduction and preliminary results

2. Restricted singular matrix equations

3. Coupled restricted singular matrix equations

4. Weighted least-squares iterative solutions

5. Concluding remarks

Acknowledgements

References