Reducible solution to a quaternion tensor equation
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Reducible solution to a quaternion tensor equation Mengyan XIE,
Qing-Wen WANG
Department of Mathematics, Shanghai University, Shanghai 200444, China
c Higher Education Press 2020
Abstract We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation A ∗N X ∗N B = C via Einstein product using Moore-Penrose inverse, and present an expression of the reducible solution to the equation when it is solvable. Moreover, to have a general solution, we give the solvability conditions for the quaternion tensor equation A1 ∗N X1 ∗M B1 +A1 ∗N X2 ∗M B2 +A2 ∗N X3 ∗M B2 = C , which plays a key role in investigating the reducible solution to A ∗N X ∗N B = C . The expression of such a solution is also presented when the consistency conditions are met. In addition, we show a numerical example to illustrate this result. Keywords Quaternion tensor, quaternion tensor equation, Einstein product, Moore-Penrose inverse, general solution, reducible solution MSC2020 11R52, 15A09, 15A24, 15A69 1
Introduction
The classical matrix equation AXB = C
(1.1)
was studied by Ben-Israel and Greville [4], who gave necessary and sufficient conditions for the existence of a solution of (1.1). Since then, there have been many papers that presented various types of solutions to equation (1.1), such as centro-symmetric solution, skew-symmetric solution, and optimal approximate solution [21,36,40]. To our knowledge, there have been few research on a reducible solution to equation (1.1). A matrix X ∈ Cn×n is called reducible, if there exists a permutation matrix K such that h i X1 X2 X=K K −1 , 0 X3 Received July 27, 2020; accepted September 23, 2020 Corresponding author: Qing-Wen WANG, E-mail: [email protected]
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Mengyan XIE, Qing-Wen WANG
where X1 and X3 are square matrices of order at least 1 over C. When the order of X3 is k (1 6 k < n), X is called k-reducible with respect to the permutation matrix K. For an arbitrary but fixed permutation matrix K, o n X1 X2 −1 (n−k)×(n−k) k×k K 1 6 k < n, X ∈ C , X ∈ C Cn×n = X = K 1 3 k 0 X3 is not empty since the identity matrix I ∈ Cn×n . Reducible matrices are k closely associated with the connection of directed graphs and can be used in compartmental analysis, continuous-time positive systems, stochastic processes, biology, and others [1,10,22,25,34,42]. Therefore, it is interesting and applicable to investigate the reducible solution to equation (1.1) . In recent years, tensor theory has been a hot topic drawing a lot of attentions [2,3,8,11,12,14–16,18,19,23,27–31,37–39,43,44,51,52,54,57]. We find that tensor equations can model the problems of material behaviour in engineering science, and also can apply to continuum mechanics [24]. There have been some papers related to the solution of tensor equation over R or C [7,9,45,46]. The quaternions have become increasingly useful both in theory and application since Hamilton introduced the notion of quaternion [17]. Quaternions are often used in engineering, quantum mechanics, computer sc
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