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

Abstract

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 X_i, Y_i, 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.