Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors
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(2019) 38:183
Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors Chaoqian Li1 · Qilong Liu2 · Yimin Wei3 Received: 31 May 2019 / Revised: 28 July 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we present the pseudospectrum for generalized tensor eigenvalues, and a set to locate this pseudospectrum. By the relations between H-eigenvalues (Z-eigenvalues) of tensors and generalized tensor eigenvalues, a pseudospectral localization for H-eigenvalues (Z-eigenvalues, respectively) is given to seek positive definite tensors surrounding a positive definite tensor. Keywords Pseudospectral localization · Generalized tensor eigenvalues · H-eigenvalues · Z-eigenvalues · Positive definiteness Mathematics Subject Classification 15A18 · 15A69 · 65F15 · 65F10
1 Introduction Generalized eigenvalues of matrices play a significant role in matrix theory and numerical linear algebra (Kosti´c et al. 2009; Liu and Wu 2006). The concepts of generalized eigenvalues are naturally extended to the high-order tensors case by Chang et al. (2008, 2009) and attract much attention in the practical fields such as the higher order Markov chain (Ching et al. 2006; Li and Ng 2014), multi-label learning (Sun et al. 2008; Yan et al. 2007), and the
Communicated by Jinyun Yuan.
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Qilong Liu [email protected] Chaoqian Li [email protected] Yimin Wei [email protected]; [email protected]
1
School of Mathematics and Statistics, Yunnan University, Yunnan 650091, People’s Republic of China
2
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, People’s Republic of China
3
School of Mathematics Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
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stability of nonlinear dynamical systems (Ding and Wei 2015; Ding et al. 2016). The application we focus on here is the positive definiteness identification problem of a multivariate homogeneous polynomial because generalized tensor eigenvalues encapsulate the definitions of H-eigenvalues and Z-eigenvalues of tensors (Kolda and Mayo 2014) which are all closely related to the positive definiteness of a multivariate homogeneous polynomial. Testing the positive definiteness of a multivariate homogeneous polynomial arises in various applications, such as automatic control (Bose and Kamat 1974; Bose and Modarressi 1976; Deng et al. 2018; Fu 1998; Hasan and Hasan 1996; Ku 1965; Li et al. 2014b; Wang and Qi 2005), magnetic resonance imaging (Chen et al. 2013; Hu et al. 2012; Qi et al. 2010a), and spectral hypergraph theory (Hu and Qi 2012, 2014; Hu et al. 2013). The positive definiteness of a multivariate homogeneous polynomial receives much attention in automatic control. When the degree of the polynomial is not greater than three, the positive definiteness of a multivariate homogeneous polynomial can be checked by a method based
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