A Bauer-Hausdorff Matrix Inequality

Zizong Yan,Shanhe Wu

doi:10.1155/2013/894824

Abstract

We present a biorthogonal process for two subspaces ofℂn. Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality.

Highlights

The Cauchy-Bunyakovsky-Schwarz, or for short the C.B.S.inequality, plays an important role in different branches of Modern Mathematics including Hilbert Space Theory, Probability & Statistics, Classical Real and Complex Analysis, Numerical Analysis, and Qualitative Theory of Differential Equations and their applications.Given an n-dimensional complex space Cn and two linear subspaces U and V such that U ∩ V = {0}, (1)there exists γ = γ (U, V) ∈ [0, 1) (2)such that for all x ∈ U and y ∈ V the following strengthened C.B.S.-inequality holds:󵄨󵄨󵄨󵄨x∗y󵄨󵄨󵄨󵄨 ≤ γ ‖x‖ ⋅ 󵄩󵄩󵄩󵄩y󵄩󵄩󵄩󵄩, (3)where ‖ ⋅ ‖ denotes the standard Euclidian norm
The main result of this paper is stated in the following theorem
Let γ be the C.B.S.-ratio of two subspaces U and V of Cn satisfying (1)

Summary

Introduction

The Cauchy-Bunyakovsky-Schwarz, or for short the C.B.S.inequality, plays an important role in different branches of Modern Mathematics including Hilbert Space Theory, Probability & Statistics, Classical Real and Complex Analysis, Numerical Analysis, and Qualitative Theory of Differential Equations and their applications. Given an n-dimensional complex space Cn and two linear subspaces U and V such that. Such that for all x ∈ U and y ∈ V the following strengthened C.B.S.-inequality holds (see [1]):. Among them two important extensions of the Wielandt inequality were given by Bauer and Householder [3], and Wang and Ip [4]. The aim of this paper is to present a matrix version of the Bauer-Hausdorff inequality like one of the Wielandt inequality given by Wang and Ip in [4]. The survey by Eijkhout and Vassilevski [1] attributes the basic theory of this inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations. For two n × n positive semidefinite Hermitian matrices A, and B, we say that A is below B with respect to Abstract and Applied Analysis the Lowner partial ordering, and we write A ≤L B, if B − A is positive semidefinite

A Matrix Inequality

Findings

Proof of the Main Result