On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$

• Main Authors: Mohamed A. Ramadan, Naglaa M. El–Shazly
• Format: Article
• Language: English
• Published: SpringerOpen 2020-01-01
• Series: Journal of the Egyptian Mathematical Society
• Subjects: Nonlinear matrix equation; Maximal positive solution; Iteration; Matrix differentiation; Perturbation bound
• Online Access: https://doi.org/10.1186/s42787-019-0052-7
• ISSN: 2090-9128

Abstract In this paper, we study the perturbation estimate of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ using the differentiation of matrices. We derive the differential bound for this maximal solution. Moreover, we present a perturbation estimate and an error bound for this maximal solution. Finally, a numerical example is given to clarify the reliability of our obtained results.