Inequalities for Contraction Matrices

Abstract

Let $A,B,C,X,$ and Y be n× n matrices such that A and B are positive definite contractions. It is shown that if $r\ge s_n\left(A\right)$ and $t\ge s_n\left(B\right),$ then$${\Vert A^{-r}X+X{B}^{-t}\Vert }_2^2+{\Vert \mathit{AX}+\mathit{XB}\Vert }_2^2\le 4{\Vert \mathit{AX}{B}^{-1}+A^{-1}XB\Vert }_2^2.$$Moreover, if $0<Y\le X\le C+Y\le 2C,$ then $$s_j\left({(C+X)}^{-1/2}A{(C+Y)}^{-1/2}\right)\le \frac{\kappa \left(C\right)}{\Vert C\Vert +\sqrt{s_{n-j+i}\left(X\right)s_{n-j+i}\left(Y\right)}}$$for $i,j=1,\dots ,n$ with $i\le j\le 2i-1,$ where ${\Vert T\Vert }_2,\Vert T\Vert ,s_j\left(T\right),$ and $\kappa \left(T\right)$ denote the Hilbert-Schmidt norm, the spectral matrix norm, the j th singular value, and the condition number of the n×n matrix T, respectively.

Keywords:

• A contraction matrix
• condition number of a matrix
• Hilbert–Schmidt norm
• norm inequality
• positive definite matrix
• singular value
• spectral norm

2000 Mathematics Subject Classification:

• 15A18
• 15A42
• 15A45
• 15A60

Acknowledgment

The authors would like to thank the editor and the referees for their comments and suggestions.