Pre-order between diagonal elements and singular values of real matrices

Abstract

Thompson obtained necessary and sufficient conditions on the existence of a real square matrix with prescribed diagonal elements, prescribed singular values and non-negative determinant. This result gave a Thompson’s type pre-order on ${\mathbb{R}}^n$. In this paper, a new pre-order on ${\mathbb{R}}^n$ motivated by two special classes of real matrices called pinching matrices and mirror sliding matrices will be introduced. We show that this pre-order is equivalent to Thompson’s type pre-order. We further apply it to study convexity and inclusion relations on linear images of the special orthogonal orbits of real matrices. Moreover, inclusion relations among matrix sets motivated by Thompson’s type pre-order, pinching matrices, mirror sliding matrices and orthostochastic matrices will be studied.

Keywords:

• Diagonal elements
• singular values
• pinching matrices
• special orthogonal matrices

AMS Subject Classifications:

• 15A23
• 15A18
• 15B51
• 15B10

Acknowledgements

The author would like to thank Dr N. K. Tsing for the helpful discussion. The author would also like to thank the referee for his/her helpful suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the author was supported by a HK RGC [grant number PolyU 502512] and a PolyU central research [grant number G-YBNS].