A real quaternion matrix equation with applications

作者：He Zhuo Heng; Wang Qing Wen^*

来源：Linear and Multilinear Algebra, 2013, 61(6): 725-740.

DOI：10.1080/03081087.2012.703192

• linear matrix equation
• eta-Hermitian solution
• quaternion matrix
• Moore-Penrose inverse
• ranks

Let i, j, k be the quaternion units and let A be a square real quaternion matrix. A is said to be ?-Hermitian if -? A*??=?A, where ????{i,?j,?k} and A* is the conjugate transpose of A. Denote A ?*?=?-?? A*?. Following Horn and Zhang's recent research on ?-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 ?-Hermicity, i.e.
where Y and Z are required to be ?-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.

• 出版日期2013-6-1
• 单位上海大学