A constraint system of coupled two-sided Sylvester-like quaternion tensor equations
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A constraint system of coupled two-sided Sylvester-like quaternion tensor equations Qing-Wen Wang1 · Xiao Wang1 · Yushi Zhang1 Received: 30 August 2020 / Revised: 15 October 2020 / Accepted: 24 October 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We first investigate some necessary and sufficient conditions for the solvability to a constraint system of coupled two-sided Sylvester-like quaternion tensor equations. We also construct an expression of the general solution to the system above when it is solvable. As an application of the system, we discuss some solvability conditions and the η-Hermitian solution to some system of Sylvester-like quaternion tensor equations. Keywords Quaternions · Sylvester-like tensor equation · Einstein product · Moore–Penrose inverse · η-Hermitian tensor Mathematics Subject Classification 11R52 · 15A09 · 15A24 · 15A69 · 15B33
1 Introduction The concept of quaternions was first introduced by Hamilton in 1843 (Hamilton 1866). Quaternions and quaternion matrices have been used in many fields such as signal and color image processing, statistics and probability, quantum computing (e.g., Bihan and Mars 2004; Bihan and Sangwine 2003; De Leo and Scolarici 2000; Fernandez and Schneeberge 2004; Jia et al. 2018; Shi and Funt 2007; Took and Mandic 2011; Took et al. 2011; Zhang 1997). Tensors are multidimensional arrays. As higher order generalizations of vectors and matrices, tensors can be used in machine learning, deep learning, data mining, pattern recognition (e.g., Geng et al. 2018; Hou 2017; Ji et al. 2019; Lechuga et al. 2016; Qi and Luo 2017; Qi et al. 2018; Tao et al. 2005; Yuan et al. 2018; Zhou et al. 2013). Tensor equations are mainly applied to model certain problems in continuum physics and engineering, isotropic and anisotropic elasticity (Lai et al. 2009). There have been many researches on tensor equations. (e.g., Chen and Lu 2012; Ebner 1974; He 2019; Li et al. 2020; Rosati 2000; Sanchez et al. 2014; Wang and Xu 2019; Wang et al. 2019; Xie and Jin 2018; Xu and Wang 2019). As described in Li et al. (2012), the following two-sided Sylvester-like tensor equation over
Communicated by Jinyun Yuan.
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Qing-Wen Wang [email protected]; [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China 0123456789().: V,-vol
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the quaternion algebra A ∗N X ∗M B + C ∗N Y ∗M D = E ,
(1.1)
where the operation ∗ N is the Einstein product, is greatly useful in discretization of a linear partial differential equation of high dimension. The solvability conditions and the analytical solution to Eq. (1.1) were considered in He et al. (2017). Wang and Wang (2020) investigated the system of coupled two-sided Sylvester-like quaternion tensor equations A1 ∗ N X ∗ M B1 + C1 ∗ N Z ∗ M D1 = E1 , (1.2) A2 ∗ N Y ∗ M B2 + C2 ∗ N Z ∗ M D2 = E2 , where X , Y , Z are unknown quaternion tensors. Motivated by the work mentioned above and recent developments in quaternion tensor
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