Geodesic
On solutions to the quaternion matrix equation $AXB+CYD=E$
The electronic journal of linear algebra, Tome 17 (2008), pp. 343-358.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices X i , Y i , i = 1, $\cdot \cdot \cdot , 4,$ in a solution pair X = X 1 + X 2 i + X 3 j + X 4 k and Y = Y 1 + Y 2 i + Y 3 j + Y 4 k to this equation. Moreover, necessary and sufficient conditions are derived for all solution pairs X and Y of this equation to be real or pure imaginary, respectively. Some known results can be regarded as special cases of the results in this paper.
Classification : 15A03, 15A09, 15A24, 15A33
Keywords: quaternion matrix equation, extreme rank, generalized inverse 
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     title = {On solutions to the quaternion matrix equation $AXB+CYD=E$},
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Wang, Qing-Wen; Zhang, Hua-Sheng; Yu, Shao-Wen. On solutions to the quaternion matrix equation $AXB+CYD=E$. The electronic journal of linear algebra, Tome 17 (2008), pp. 343-358. http://geodesic.mathdoc.fr/item/ELA_2008__17__a20/