How to enumerate the number of solutions in sylvester equations and how to solve

Question

how to solve a, b, c and d for below equations:

Both known Matrix and solution Matrix Properties

Matrix([[a,b],[c,d]]);
a*b = 1
c*d = 1

conjugate(a*c)-b*d = 0
Determinant(Matrix([[a,b],[c,d]])) = sqrt(3)*I
a*d-b*c = sqrt(3)*I

How to solve system AA when known Jesus7 to find Jesus2 which is Matrix([[a,b],[c,d]]) ?

Sylvestersolve(Jesus7,Jesus7,matrix([[0,0],[0,0]]))

It return 0 matrix but actually has solution How many solutions can it have? Is sylvester equation isomorphism? How to test?

Jesus7 := Matrix(2, 2, {(1, 1) = (1296309/5)*(22243/5+((7358/5)*I)*sqrt(3))/((-384813/10+((33701837/10)*I)*sqrt(3))*(-2277/10+((13/10)*I)*sqrt(3))), (1, 2) = (1296309/5)*(22243/5-((7358/5)*I)*sqrt(3))/((-384813/10-((33701837/10)*I)*sqrt(3))*(-2277/10-((13/10)*I)*sqrt(3))), (2, 1) = -(1/5)*(-25372581+(8329256*I)*sqrt(3))/(-384813/10+((33701837/10)*I)*sqrt(3)), (2, 2) = -(1/5)*(-25372581-(8329256*I)*sqrt(3))/(-384813/10-((33701837/10)*I)*sqrt(3))});

AA := simplify(MatrixMatrixMultiply(Matrix([[a,b],[c,d]]),Jesus7)- MatrixMatrixMultiply(Jesus7,Matrix([[a,b],[c,d]]))):

I am using different properties and even full properties , still can not find the solution equal to exact solution, but number of equations had already been more than number of variables? where is incorrect?

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, conjugate(a*c)-b*d = 0,a*d-b*c=sqrt(3)*I],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, a*d-b*c=sqrt(3)*I],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, conjugate(a*c)-b*d = 0],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1],[a,b,c,d]);

Solution: Matrix([[a,b],[c,d]]) should be
Jesus2 := Matrix([[1313319/2 (3439+1145 I sqrt(3))/((-2292+1313317 I sqrt(3)) (-1146+I sqrt(3))),1313319/2 (3439-1145 I sqrt(3))/((-2292-1313317 I sqrt(3)) (-1146-I sqrt(3)))],[-1/2 (-3942243+1311025 I sqrt(3))/(-2292+1313317 I sqrt(3)),-1/2 (-3942243-1311025 I sqrt(3))/(-2292-1313317 I sqrt(3))]]);

Answer 1

The equation is not solvable.

Let us restate the problem. We want to solve the Sylvester equation $AX+XA=0$, where $A$ is the $2\times2$ matrix Jesus7 in your Maple code and the unknown matrix $X=\pmatrix{a&b\\ c&d}$ is subject to the constraints that $ab=cd=1,\ \overline{ac}=bd$ and $\det(X)=ad-bc=i\sqrt{3}$.

Your $A$ is numerically close to $\pmatrix{e^{2i\pi/3}&e^{-2i\pi/3}\\ e^{-2i\pi/3}&e^{2i\pi/3}}$. I guess the difference is just rounding error, but I'll use the precise value of your Jesus7 anyway. In general, the Sylvester equation $AX+XA=0$ has a non-trivial solution $X\ne0$ if and only if some two eigenvalues of $A$ are negatives of each other. The two eigenvalues of your $A$ are approximately $i\sqrt{3}$ and $-1$. Hence the Sylvester equation $AX+XA=0$ alone has only the trivial solution $X=0$. Yet, the additional constraints for $a,b,c,d$ imply that $X$ cannot be zero. Therefore, with these constraints, the equation is not solvable.