A Solution of the Bilinear Matrix Equation $AY + YB = - Q$

A conceptually very simple method for the solution of bilinear matrix equations ${\bf A}{\bf Y} + {\bf Y}{\bf B} = - {\bf Q}$ is given. The method involves a transformation of ${\bf B}_{N \times N}$ (or ${\bf A}_{M \times M}^T$) to Hessenberg form, an $M \times M$ (or $N \times N$) matrix inversion and a recursive algorithm. Using the recursive algorithm in an iteration scheme, high accuracy results can be obtained, even for high order matrix equations.