| Summary: | In this paper, we consider the split quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>−</mo><mi>A</mi><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>X</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mi mathvariant="normal">H</mi></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">i</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">j</mi><mi mathvariant="normal">H</mi></mrow></msup><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">k</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.
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