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Using the Kronecker product of matrices, the Moore-Penrose generalized inverse, and the complex representation of quaternion matrices, we derive the expressions of least squares solution with the least norm, least squares pure imaginary solution with the least norm, and least squares real solution with the least norm of the quaternion matrix equation AXB+CXD=E, respectively

journal of applied mathematics

Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>A</mml:mi><mml:mi>X</mml:mi><mml:mi>B</mml:mi><mml:mo>+</mml:mo><mml:mi>C</mml:mi><mml:mi>X</mml:mi><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math>with the Least Norm

Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix EquationAXB+CXD=Ewith the Least Norm