Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

Xin Liu,Huajun Huang,Zhuo-Heng He

doi:10.1155/2019/3258349

Abstract

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

Highlights

Introduction e definitions of φHermitian and φ-skew-Hermitian quaternion matrices were first introduced by Rodman (Definition 3.6.1 in [1])
To well preserve the structure of φ-(skew)-Hermitian matrix, we present a new representation of quaternion matrix basing on the above one
It follows from eorem 1 and Lemma 2 that the real representation ση can map a φ-Hermitian matrix or φ-skewHermitian matrix over H into a skew-symmetric or symmetric matrix over R. is will be a critical technique in tackling the problem of the paper

Summary

In In

We can use Fn to define the new real representation. For X X0 + X1i + X2j + X3k ∈ Hm×n, X0, X1, X2, X3 ∈ Rm×n and η u1i + u2j + u3k ∈ H, which is unit and pure imaginary. It follows from eorem 1 and Lemma 2 that the real representation ση can map a φ-Hermitian matrix or φ-skewHermitian matrix over H into a skew-symmetric or symmetric matrix over R. is will be a critical technique in tackling the problem of the paper. We summarize the properties of real representations Xσ, Xση in the following proposition. Let A, B ∈ Hm×n, C ∈ Hn×s, b ∈ R. en (a) (A + B)σ Aσ + Bσ, (bA)σ bAσ, (A + B)ση Aση + Bση , (bA)ση bAση ;

In iIn jIn

Aσ η

By direct computation

Conclusion