The Real Solutions to a System of Quaternion Matrix Equations with Applications |
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| Authors: | Qing-Wen Wang Shao-Wen Yu Qin Zhang |
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| Institution: | 1. Department of Mathematics , Shanghai University , Shanghai , China wqw858@yahoo.com.cn;3. Department of Mathematics , East China University of Science and Technology , Shanghai , China;4. Department of Mathematics , Shanghai University , Shanghai , China |
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| Abstract: | In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2. Moreover, formulas of the maximal and minimal ranks of four real matrices X 1, X 2, X 3, and X 4 in solution X = X 1 + X 2 i + X 3 j + X 4 k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2, A 3 XB 3 = C 3 to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A 1 + B 1 i + C 1 j + D 1 k and A 2 + B 2 i + C 2 j + D 2 k, which can be expressed as E + Fi + Gj + Hk are also presented. |
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| Keywords: | Generalized inverse Maximal rank Minimal rank Quaternion matrix equation Real solution |
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