In my research, I need to solve a matrix equation very similar to Lyapunov equation but with one extra term.
The equation is X+DXD-WXW=A, where X is the unknown n*n matrix. W, D and A is known. W is a symmetric n*n matrix, A is not symmetric. D is an diagonal matrix, and D and W cannot commute. So DXD is an extra term the original Lyapunov equation.
So may I ask whether this equation or its analogue has been studied in the literature? My difficulty is since D and W cannot commute, I cannot perform a simultanenous triangularization for D and W.
Thank you very much for your help!