Solving a Matrix Polynomial by Conjugate Gradient Methods

One of well known and much studied nonlinear matrix equations is the matrix polynomial which has the form G(X)=$A_0X^m+A_1X^{m-1}+{\cdots}+A_m$ where $A_0$, $A_1$, ${\cdots}$, $A_m$ and X are $n{\times}n$ real matrices. We show how the minimization methods can be used to solve the matrix polynomial G(X) and give some numerical experiments. We also compare Polak and Ribi$\acute{e}$re version and Fletcher and Reeves version of conjugate gradient method.