| Summary: | The solution space is explicitly constructed for all 2x2 complex matrices using Basis Groebner techniques. When A is a 2 x 2 matrix, the equation X²AX=AXA is equivalent to a system of four polynomial equations. The solution space is then the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In the procedure for solving these equations, Grobner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classify all solution for 2 x 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Toeplitz matrices are used to construct the solution of the matrix equation.
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