The Common Hermitian Solutions of the matrix equations B 2 X B 2 = A 2 and B 3 X B 3 = A 3 subject to inequality restrictions

• Published in: Quaestiones mathematicae Vol. 48; no. 8; pp. 1137 - 1152
• Main Authors: Wang, Zhanshan, Zhang, Huiting, Zou, Honglin
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 03.08.2025
• Subjects: generalized singular value decomposition; Moore-Penrose generalized inverse; Variable substitution
• ISSN: 1607-3606; 1727-933X
• DOI: 10.2989/16073606.2025.2466808

Problem CHS: Given , find such that X ≥ 0 (> 0), where is given by In 2014, Tian pointed out in [16] (Y. Tian: Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions, Banach Journal of Mathematical Analysis, 8 (2014) 148-178) that giving the closed-form of the common Hermitian solutions (CHSs) to and subject to X ≥ 0 (> 0) is a challenging problem. Although Zhang and Cheng [22, 23, 24] provided an expression for the general solution, it is presented in an implicit form. It also appears difficult to derive the expressions for the general solution solely by virtue of the ranks of matrices (see [16]), due to the discontinuity and non-convexity inherent in the ranks of matrices. In this paper, we will present a completely explicit expression for the general solution of Problem CHS by utilizing the Moore-Penrose generalized inverses, orthogonal projectors, and some matrix decompositions. The solvability conditions are also proposed. Finally, a numerical example is given to validate the accuracy of the obtained results.