Spectral decomposition of selfadjoint matrices in positive semidefinite inner product spaces and its applications

Abstract

Given a positive semidefinite matrix A, we obtain spectral decomposition of A-selfadjoint matrices which act on a positive semidefinite inner product space in which the inner product is induced by A. As applications, we improve the singular value decomposition of square matrices given by Zheng and Li (2014) and give polar decomposition of square matrices in positive semidefinite inner product spaces, and these two decompositions are extended to rectangular matrices.

Keywords:

• Positive semidefinite inner
• spectral decomposition
• singular value decomposition
• polar decomposition

AMS Subject Classifications:

• 15A23
• 15A18
• 65F35
• 15A09

Acknowledgements

The authors thank the referee and the handling editor for their valuable comments and suggestions which improve the readability of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11601054], [grant number 11671060]; the Basic and Advanced Research Project of CQ CSTC [grant number cstc2016jcyjA0466]; the Science and Technology Research Project of Chongqing Municipal Education Commission [grant number KJ1600403]; the China Scholarship Council [grant number CSC201607845012].