Contractive matrices of Hua type

Abstract

This is continuation of the recent work by Xu, Xu and Zhang [Revisiting Hua–Marcus–Bellman–Ando inequalities on contractive matrices, Linear Algebra Appl. 430 (2009), pp. 1499–1508] on contractive matrices. We study the relations of block matrices of Hua type, present some properties that the eigenvalues of Hua matrices possess, especially for the 2 × 2 block case, discuss the analogues for higher dimensions and estimate the closeness of two Hua matrices. At the end, we propose a conjecture on the eigenvalues of Hua matrices and an open problem on the symmetric functions of the eigenvalues of contractive matrices.

Keywords:

• contractions
• contractive matrices
• determinant inequalities
• eigenvalues
• elementary symmetric functions
• Hua's determinant inequality
• Hua's matrix inequality
• matrix inequalities
• positive semidefinite matrix

AMS Subject Classifications:

• 15A15
• 15A24
• 15A45

Acknowledgements

This project was supported in part by the Fund for the International Cooperation from the International Department of Zhejiang Province, by NNSF of China (No. 10671074 and No. 10871230), NSF of Zhejiang (Y607480, Y7080364, and Y807332), an NSU-FCAS Minigrant and NSU President's Faculty Research and Development Grant.