Investigation of the existence and uniqueness of extremal and positive definite solutions of nonlinear matrix equations

Abstract

We consider two nonlinear matrix equations $X^{r} \pm \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} = I$ , where $- 1 < \delta_{i} < 0$ , and r, m are positive integers. For the first equation (plus case), we prove the existence of positive definite solutions and extremal solutions. Two algorithms and proofs of their convergence to the extremal positive definite solutions are constructed. For the second equation (negative case), we prove the existence and the uniqueness of a positive definite solution. Moreover, the algorithm given in (Duan et al. in Linear Algebra Appl. 429:110-121, 2008) (actually, in (Shi et al. in Linear Multilinear Algebra 52:1-15, 2004)) for $r = 1$ is proved to be valid for any r. Numerical examples are given to illustrate the performance and effectiveness of all the constructed algorithms. In Appendix, we analyze the ordering on the positive cone $\overline{P(n)}$ .