What is the condition for unique solution for the Sylvester equation over the field F_{2} ?

25 February 2019 1 643 Report

I'm looking for a necessary and sufficient condition for unique solutions to the Sylvester matrix equation

AX-XD=B over the field F_{2}.

Does anybody know about such condition that involves the coprimeness of the minimal polynomials of A and D ?

It can be proved easily that if the minimal polynomials of A,D are co-prime over F_{2} then unique solution exists for any choice of B.

The question now is: is this condition is also necessary for the existance of unique solution for any choice of B?

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