| Summary: | Given a complex idempotent matrix <i>A</i>, we derive simple, sufficient and necessary conditions for a matrix <i>X</i> being a nontrivial solution of the Yang-Baxter-like matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>A</mi><mo> </mo><mo>=</mo><mo> </mo><mi>X</mi><mi>A</mi><mi>X</mi></mrow></semantics></math></inline-formula>, discriminating commuting solutions from non-commuting ones. On this basis, we construct all the commuting solutions of the nonlinear matrix equation.
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