矩阵方程$X A^*X^{-q}A=Q~(q\geq 1)$的Hermite正定解

Hermitian Positive Definite Solutions of the Matrix Equation $X A^*X^{-q}A=Q~(q\geq 1)$

中文关键词:  非线性矩阵方程  正定解  迭代方法.

英文关键词:nonlinear matrix equations  positive definite solution  iterative method.

中文摘要:

      本文研究非线性矩阵方程$X A^*X^{-q}A=Q~(q\geq 1)$的Hermite正定解. 首先得到了有解的一些新的充分和必要条件, 其次构造了求最小和准最大正定解的迭代方法, 并给出了相应的收敛性定理, 最后用数值例子验证了得出的结论.

英文摘要:

      In this paper, Hermitian positive definite solutions of the nonlinear matrix equation $X A^*X^{-q}A=Q\ (q\geq 1)$ are studied. Some new necessary and sufficient conditions for the existence of solutions are obtained. Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions, and the convergence analysis is also given. The theoretical results are illustrated by numerical examples.