| Summary: | This paper investigates the Sylvester-transpose matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>+</mo><msup><mi>X</mi><mi>T</mi></msup><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, and the classical Sylvester-transpose matrix equation.
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