| Summary: | Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula> has been extensively studied. However, there is currently limited information on matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula> regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, and present the expression for the general solution when it is solvable. As an application, we derive the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-Hermitian solutions for dual quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><msup><mi>A</mi><mi>ϕ</mi></msup><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, where the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-Hermitian extends the concepts of Hermiticity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.
|
|---|