Suppose the following constraints:
$$ X^2=x\\ Y^2=y\\ XY+YX=c $$
where $X$ and $Y$ are square matrices, and where $x,y,c$ are elements of the complex. How do I find a matrix representation of $X$ and $Y$?
2026-04-04 00:37:04.1775263024
Finding a set of matrices that satisfy a list of non-commutative constraints?
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Assume that $x,y,c,X,Y$ are $n\times n$ complex matrices and that the unknowns are $X,Y$.
If $x,y,c$ are generic (for example, randomly choose them) given matrices, then your system has no solutions except if you are particularly lucky.
Indeed, the equations $X^2=x,Y^2=y$ generically have $2^n$ solutions. Then the couple $(X,Y)$ can only take a finite number of values ($4^n$) and generically, these values cannot be a solution of the last equation.