Matrix Quasinorms Induced by Maximal and Minimal Vector Norms

Jong-Do Park

doi:10.1155/2016/5240218

Abstract

In the set of all vector norms inCn, there exist maximal and minimal complex norms which coincide with the real Euclidean norm inRn. The purpose of this paper is to introduce new quasinorms defined on complex matrices. These two matrix quasinorms are induced by maximal and minimal complex vector norms. We also prove the dual relation between these two quasinorms.

Highlights

The standard Euclidean norm in Rn is ‖x‖ fl √ n ∑ 󵄨󵄨󵄨󵄨󵄨xj󵄨󵄨󵄨󵄨󵄨2, (1)j=1 where x = (x1, . . . , xn) ∈ Rn
It is known that L(z) and N∗(z) have the following properties [1,2,3]: (i) L(⋅) and N∗(⋅) are norms. (ii) N∗(z) = L(z) = ‖z‖ for all z ∈ Rn. (iii) N∗(z) ≤ ‖z‖ ≤ L(z) for all z ∈ Cn. (iv) They are dual in the sense that |z ∙ w| ≤ L(z)N∗(w) for all z, w ∈ Cn
L(⋅) and N∗(⋅) to complex matrices satisfying (ii), (iii), and (iv). We show that these two extensions are quasinorms

Summary

Introduction

We could extend this vector norm to matrices, just by taking a matrix Z = (zij)1≤i≤m,1≤j≤n as a vector in Rmn (or Cmn). This natural extension is called the Frobenius norm ‖Z‖F ( called Hilbert-Schmidt norm or Schur norm) defined by. (iii) N∗(z) ≤ ‖z‖ ≤ L(z) for all z ∈ Cn. (iv) They are dual in the sense that |z ∙ w| ≤ L(z)N∗(w) for all z, w ∈ Cn. L(z) and N∗(z) are maximal and minimal norms satisfying {z ∈ Cn (ii) : and (iii). L(⋅) and N∗(⋅) to complex matrices satisfying (ii), (iii), and (iv).

Statements of Main Results

Proofs