A real quaternion matrix equation with applications

摘要

Let i, j, k be the quaternion units and let A be a square real quaternion matrix. A is said to be η-Hermitian if -η A*η = A, where η ∈ {i, j, k} and A* is the conjugate transpose of A. Denote Aη* = - η A*η. Following Horn and Zhang's recent research on η-Hermitian matrices (A generalization of the complex AutonneTakagi factorization to quaternion matrices, Linear Multilinear Algebra, DOI:10.1080/03081087.2011.618838), we consider a real quaternion matrix equation involving η-Hermicity, i.e. A_1X + (A_1X)~η* + B_1YB_1 ~η* + C_1ZC_1 ~η* = D1, where Y and Z are required to be η-Hermitian. We provide some necessary and sufficient conditions for the existence of a solution (X, Y, Z) to the equation and present a general solution when the equation is solvable. We also study the minimal ranks of Y and Z satisfying the above equation.