四元数方程AXA^H=B厄米特解中的复矩阵极秩

The Extremal Ranks of Complex Matrices in Hermitian Solutions of Quaternion Equation AXA^H=B

通过四元数矩阵的复表示X=X0+X1j和矩阵秩的许多性质,确定出四元数矩阵方程AXAH=B厄米特解集{X}的复表示矩阵集{X0}和{X1}的最大秩、最小秩公式.作为这些极秩公式的应用,探讨了四元数厄米特矩阵广义逆的一些性质,得出任意一个四元数厄米特矩阵M的广义逆中存在纯复矩阵、广义逆全部为纯复矩阵、广义逆中存在纯非复矩阵、广义逆全部为纯非复矩阵这4种情形的充要条件.

By using the complex representation of quaternion matrix X=X0 ＋X1j and several properties regarding ranks of matrices,we establish formulas of the maximal and minimal ranks of complex matrices {X0 } and {X1 },where X0 and X1 are the complex representation matrices of the Hermitian solutions {X}of the quaternion matrix equation AXA^H=B. As an application,we consider properties of the generalized inverse of a Hermitian quaternion matrix M,and give necessary and sufficient conditions for these special cases： （i） There is at least a complex matrix N0 ∈ {M^-}. （ii） All the matrices of {M^-} are complex. （iii） There is at least a pure non complex matrix with the form N1j∈{M^-}. （iv） All the matrices of {M^-} are pure non complex with the form N1j.