Some matrix equations involving the weighted geometric mean

摘要

In this paper, we consider two matrix equations that involve the weighted geometric mean. We use the fixed point theorem in the cone of positive definite matrices to prove the existence of a unique positive definite solution. In addition, we study the multi-step stationary iterative method for those equations and prove the corresponding convergence. Another equations with a different matrix generalization of the weighted geometric mean for scalars are also discussed.