(<i>R</i>, <i>S</i>)-(Skew) Symmetric Solutions to Matrix Equation <i>AXB</i> = <i>C</i> over Quaternions

(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></semantics></math></inline-formula>)-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></semantics></math></inline-formula>)-(skew) symmetric solutions to the 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>. We use a real representation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mi>τ</mi></msup></semantics></math></inline-formula> to obtain the necessary and sufficient conditions for <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> to have (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></semantics></math></inline-formula>)-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></semantics></math></inline-formula>)-(skew) symmetric solution to the above matrix equation.