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Zbl 1188.15016
Wang, Qingwen; Yu, Shaowen; Xie, Wei
Extreme ranks of real matrices in solution of the quaternion matrix equation $AXB = C$ with applications. (English)
[J] Algebra Colloq. 17, No. 2, 345-360 (2010). ISSN 1005-3867

Summary: For a consistent quaternion matrix equation $AXB = C$, the formulas are established for maximal and minimal ranks of real matrices $X_{1}, X_{2}, X_{3}, X_{4}$ in solution $X = X_{1} + X_{2}i + X_{3}j + X_{4}k$. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices $E, F, G, H$ in a generalized inverse $(A +Bi + Cj + Dk)^{-} = E + Fi + Gj + Hk$ of a quaternion matrix $A + Bi + Cj + Dk$ are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations $A_{1}XB_{1} = C_{1}$ and $A_{2}XB_{2} = C_{2}$ to have a common real solution.

MSC 2000:: *15A24 Matrix equations
15A33 Matrices over special rings
15A03 Vector spaces
15A09 Matrix inversion

Keywords: quaternion matrix equation; minimal rank; maximal rank; generalized inverse; real solution; complex solution; pure imaginary solution

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