关于非线性矩阵方程X+A~*X^(-q)A=Q的Hermite正定解

On the Hermite Positive Definite Solution of the Nonlinear Matrix Equantion X+A~*X^(-q)A=Q

考虑非线性矩阵方程X+A*X-qA=Q,其中A是n阶非奇异复矩阵,Q是n阶hermite正定阵.考虑q∈(0,1]和q∈[1,∞)两种情况下非线性矩阵方程存在正定解(唯一正定解)的充分条件和必要条件,并在最后给出一个获得矩阵方程正定解的迭代序列.

In this paper, the nonlinear matrix equantion X ＋ A ＊ X^-qA = Q is investigated. Based on fixed point theorem, some necessary conditions and some sufficient conditions for the equation to have a Hermite positive definite solution are derived under two cases q∈（0,1] and q∈[1,∞） respectively. In the end, an effective iterative method to obtain the positive definite solution of the equation is presented.