Article

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].
    Full-text · Article · May 2015 · Journal of Industrial and Management Optimization
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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.
    Full-text · Article · Jan 2015
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    • "Eigenvalue of a tensor, as a natural generalized notion of the eigenvalue of a square matrix, is proposed in recent years to study tensors independently by Lim[16]and Qi[22]. Among others, the number of eigenval- ues[1,5,15,18,22], Perron-Frobenius theorem for nonnegative tensors[2], and applications to spectral hypergraph theory[3, 11–13, 17, 23]are the well-studied topics. The eigenvalues of a tensor are the roots of the characteristic polynomial, which is a monic polynomial with degree being determined by the order and the dimension of the tensor[10,22]. "
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    ABSTRACT: We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue $\lambda$ of a tensor: algebraic multiplicity $\operatorname{am}(\lambda)$ and geometric multiplicity $\operatorname{gm}(\lambda)$. The former is the multiplicity of the eigenvalue as a root of the characteristic polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors) corresponding to the eigenvalue. We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this action, which is the main difficulty to study their relationships. However, we show that for a generic tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity and geometric multiplicity are one. In general, we suggest for an $m$-th order $n$-dimensional tensor the relationship \[ \operatorname{am}(\lambda)\geq \operatorname{gm}(\lambda)(m-1)^{\operatorname{gm}(\lambda)-1}. \] We show that it is true for serveral cases, especially when the eigenvariety contains a linear subspace of dimension $\operatorname{gm}(\lambda)$ in coordinate form. As both multiplicities are invariants under the orthogonal linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the algebraic mutliplicity is not smaller than the geometric multiplicity.
    Full-text · Article · Dec 2014
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