The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation
Xindong Zhang, Xinlong Feng · International Journal of Nonlinear Sciences and Numerical Simulation · 2017
Abstract: In this paper, we study the nonlinear matrix equation X s ± ∑ i = 1 m A i T X δ i A i = Q $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$ , where A i ( i = 1 , 2 , … , m ) $A_{i}\;(i=1,2,\ldots,m)$ is n × n $n\times n$ nonsingular real matrix and Q $Q$ is n × n $n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.