We study the matrix equation , where is a complex square matrix and is complex symmetric.
Special cases of this equation appear in Green's function calculation in nano research and also in
the vibration analysis of fast trains. In those applications, the existence of a unique
complex symmetric stabilizing solution has been proved using advanced results on linear operators.
The stabilizing solution is the solution of practical interest.
In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption
that is satisfied for the two special applications. Moreover, our new approach here reveals that the unique
complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm.