On equivalence and explicit solutions of a class of matrix equations

It is shown in this paper that three types of matrix equations AX−XF=BY,AX−EXF=BYAX−XF=BY,AX−EXF=BY and AXI−EXF=BYAXI−EXF=BY which have wide applications in control systems theory, are equivalent to the matrix equation AXℱ−X=ℬYAXℱ−X=ℬY with their coefficient matrices satisfying some relations. Based on right coprime factorization to (sA−I)−1ℬ(sA−I)−1ℬ, explicit solutions to the equation AXℱ−X=ℬYAXℱ−X=ℬY are proposed and thus explicit solutions to the former three types of matrix equations can be immediately established. With the special structure of the proposed solutions, necessary conditions to the nonsingularity of matrix XX are also obtained. The proposed solutions give an ultimate and unified formula for the explicit solutions to these four types of linear matrix equations.