On the nonlinear matrix equation Xp = A#t(MT (X–1 + B)–1M). Issue 2 (22nd January 2022)

• Record Type: Journal Article
• Authors: Lee, Hosoo; Meng, Jie
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 2(2022)
• Page Start: 250
• Page End: 263
• Publication Date: 2022-01-22
• Subjects: Matrix equation -- symmetric positive definite -- Thompson metric -- geometric mean -- perturbation analysis; 15A24 -- 65F10 -- 65H10; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1720575
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: In this paper, the nonlinear matrix equation X p = A # t ( M T ( X − 1 + B ) − 1 M ), where p ≥ 1 is a positive integer, t ∈ [ 0, 1 ], M is an n × n nonsingular matrix, A is a positive definite matrix and B is a positive semidefinite matrix, is considered. The notation A # t B is the t -weighted geometric mean of the positive definite matrices A and B . Based on the properties of the Thompson metric, we prove that the nonlinear matrix equation always has a unique positive definite solution and we compare it with the unique positive definite solution of the equation X p = A + M T ( X − 1 + B ) − 1 M, which has been studied in Jung et al. [On the solution of the nonlinear matrix equation X n = f ( X ) . Linear Algebra Appl. 2009;430:2042–2052]; Meng and Kim [The positive definite solution of the nonlinear matrix equation X p = A + M ( B + X − 1 ) − 1 M ∗ . J Comput Appl Math. 2017;322:139–147]. A fixed-point iteration method for obtaining the unique positive definite solution and an elegant estimate of the solution are given. Perturbation analysis of the unique positive definite solution is presented.