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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 307036, 19 pages
Research Article

The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 9 October 2011; Accepted 17 November 2011

Academic Editor: Qing-Wen Wang

Copyright Β© 2012 Shao-Wen Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equations 𝐴1𝑋=𝐢1,𝑋𝐡1=𝐢2,and𝐴3π‘‹π΄βˆ—3=𝐢3. Moreover, formulas of the maximal and minimal ranks of four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations 𝐴1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3, and𝐴4π‘‹π΄βˆ—4=𝐢4 to have real and complex Hermitian solutions.