The degree of the E-characteristic polynomial of an even order tensor

Qufu Normal University, Küfow, Shandong Sheng, China
Journal of Mathematical Analysis and Applications (Impact Factor: 1.12). 05/2007; 329(2):1218-1229. DOI: 10.1016/j.jmaa.2006.07.064

ABSTRACT The E-characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form. In this paper, for an even order tensor, we first establish the formula of its E-characteristic polynomial by using the classical Macaulay formula of resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this bound is attainable in some low order and dimensional cases. (c) 2006 Elsevier Inc. All rights reserved.

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    • "Tensors arise in diverse fields such as signal and image processing, nonlinear continuum mechanics, higher-order statistics, as well as independent component analysis; see [4] [5] [6] [8] [11] [18] [19] [20] [24] [31]. In particular, eigenvalues of higher order tensors have become an important topic of study in numerical multilinear algebra, and they have a wide range of practical applications; see [12] [15] [17] [21] [22] [23] [24] [25] [26] [29]. "
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    ABSTRACT: Lower bounds and upper bounds for the spectral radius of a nonnegative tensor are provided. And it is proved that these bounds are better than the corresponding bounds in [Y. Yang, Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl. 31 (2010), 2517-2530].
    Journal of Industrial and Management Optimization 05/2015; DOI:10.3934/jimo.2016.12.975 · 0.84 Impact Factor
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    • "[19] [31] [26]. In the aforementioned papers [22] [20] [8] on eigenvalues of complex tensors, the associated complex polynomials however are not real-valued. The aim of this paper is different. "
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    ABSTRACT: In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomials to be real-valued. As applications of our results, we discuss the relation between nonnegative polynomials and sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for complex tensors are introduced, extending properties from the Hermitian matrices. Finally, we discuss an important property for symmetric tensors, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value; the result is known as Banach's theorem. We show that a similar result holds in the complex case as well.
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    • "In fact, the authors of [22] [23] presented a method to verify if an mth order symmetric space tensor is positive semi-definite or not. This involves to solve an equation of a onedimensional polynomial of degree no more than m 2 − m − 1 [18] to find all stationary points of (4) and solve the minimum eigenvalue problem exhaustedly. Such a polynomial may be approximately solved to any given error bound in polynomial time of m or n = 1 2 (m + 1)(m + 2). "
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    ABSTRACT: Space tensors appear in physics and mechanics. Mathematically, they are tensors in the three-dimensional Euclidean space. In the research area of diffusion magnetic resonance imaging, convex optimization problems are formed where higher order positive semi-definite space tensors are involved. In this short paper, we investigate these problems from the viewpoint of conic linear programming (CLP). We characterize the dual cone of the positive semi-definite space tensor cone, and study the CLP formulation and the duality of positive semi-definite space tensor conic programming.
    Computational Optimization and Applications 10/2014; 59(1-2). DOI:10.1007/s10589-013-9577-0 · 1.32 Impact Factor
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