Symmetric matrix polynomial equations

For problems of linear control system synthesis, an apparatus of polynomial equations (for single-variable case) and of matrix polynomial equations (for multivariable case) was successfully developed in recent times, cf. [1]. In connection with quadratic criteria, we are led to equations of special type, containing an operation of conjugation ah-* a* representing a(s) i-> a( — s) for continuous-time systems and a(d) t —>• a(d~) for discrete-time ones, cf. [2], [3]. The key problem is solution of a quadratic polynomial equation x*x = b (b = b*, b > 0 on the boundary of stability, x stable), known also as a spectral factorization problem. The solution can be found by iterating a linear equation a*x + x*a = 2b (a stable), see [4]. Such equations were investigated in [5].