Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   A=PQ T      A  =  P    Q    T         , where   P     P        and   Q     Q        are two   n×2     n  ×  2        complex matrices of full column rank such that   detQ T P≠0     det    Q    T    P  ≠  0        and so   0     0        is a semisimple eigenvalue of   A     A        with multiplicity   n−2     n  −  2       .  solve the quadratic matrix equation   AXA=XAX     A  X  A  =  X  A  X        completely.
['Duanmei Zhou', 'Guoliang Chen', 'Jiu Ding']
I am writing a research paper on a idea Solving the Yang-Baxter-like matrix equation for rank-two matrices
[{"from": "human", "value": "I am writing a research paper on a idea Solving the Yang-Baxter-like matrix equation for rank-two matrices"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   A=PQ T      A  =  P    Q    T         , where   P     P        and   Q     Q        are two   n\u00d72     n  \u00d7  2        complex matrices of full column rank such that   detQ T P\u22600     det    Q    T    P  \u2260  0        and so   0     0        is a semisimple eigenvalue of   A     A        with multiplicity   n\u22122     n  \u2212  2       .  solve the quadratic matrix equation   AXA=XAX     A  X  A  =  X  A  X        completely."}]