Some Properties of the Solution to a System of Quaternion Matrix Equations

Abstract

This paper investigates the properties of the ϕ-skew-Hermitian solution to the system of quaternion matrix equations involving ϕ-skew-Hermicity with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4. We present the general ϕ-skew-Hermitian solution to this system. Moreover, we derive the β(ϕ)-signature bounds of the ϕ-skew-Hermitian solution X1 in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have β(ϕ)-positive semidefinite, β(ϕ)-positive definite, β(ϕ)-negative semidefinite and β(ϕ)-negative definite solutions.