A system of real quaternion matrix equations with applications

• Published in: Linear algebra and its applications Vol. 431; no. 12; pp. 2291 - 2303
• Main Authors: Wang, Qing-Wen, van der Woude, J.W., Chang, Hai-Xia
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2009; Elsevier
• Subjects: [formula omitted]-symmetric matrix; Algebra; Exact sciences and technology; Inner inverse of a matrix; Linear and multilinear algebra, matrix theory; Mathematics; Moore–Penrose inverse of a matrix; Quaternion matrix; Reflexive inverse of a matrix; Sciences and techniques of general use; System of matrix equations; Quaternion matrix; Moore–Penrose inverse of a matrix; 15A24; 15A57; 15A33; 15A06; Reflexive inverse of a matrix; Inner inverse of a matrix; 15A09; System of matrix equations; ( P , Q ) -symmetric matrix; Existence of solution; Moore Penrose inverse; Quaternion; Moore-Penrose inverse of a matrix; Generalized inverse; Commutativity; Algebra; Inverse matrix; Matrix equation; Real matrix; Operator equation; (P, Q)-symmetric matrix; Involution; Linear system; Matrix inversion; Symmetric matrix; Ring; Solvability
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.02.010

Let H be the real quaternion algebra and H n × m denote the set of all n × m matrices over H . Let P ∈ H n × n and Q ∈ H m × m be involutions, i.e., P 2 = I , Q 2 = I . A matrix A ∈ H n × m is said to be ( P , Q ) -symmetric if A = PAQ . This paper studies the system of linear real quaternion matrix equations A 1 X 1 = C 1 X 1 B 1 = C 2 A 2 X 2 = C 3 X 2 B 2 = C 4 A 3 X 1 B 3 + A 4 X 2 B 4 = C c . We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied. As applications, we discuss the necessary and sufficient conditions for the system A a X = C a , XB b = C b , A c XB c = C c to have a ( P , Q ) -symmetric solution. We also show an expression of the ( P , Q ) -symmetric solution to the system when the solvability conditions are met. Moreover, we provide an algorithm and a numerical example to illustrate our results. The findings of this paper extend some known results in the literature.