On solutions to the quaternion matrix equation AXB+CYD=E

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Qing-Wen Wang
Hua-Sheng Zhang
Shao-Wen Yu


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 AX B+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, ··· , 4, in a solution pair X = X1 +X2i+X3j +X4k and Y = Y1 +Y2i+Y3j +Y4k 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.

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