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.

  • Source
    [Show abstract] [Hide abstract]
    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.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The positive definiteness of an even-degree homogeneous polynomial form f(x) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automatic control, and the positive definiteness of f(x) is equivalent to that of an even-order supersymmetric tensor which defines f(x). In this paper, we provide some criterions for identifying the positive definiteness of an even-order real supersymmetric tensor. Moreover, an iterative algorithm for identifying the positive definiteness of an even-order real supersymmetric tensor is obtained. Numerical examples are given to verify the corresponding results.
    Journal of Computational and Applied Mathematics 01/2014; 255:1-14. DOI:10.1016/ · 1.08 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety $\mathbb S=\{\mathbf x\in\mathbb P^n\;|\;\sum\limits_{i=0}^nx_i^2=0\}$. We show that a generic tensor has no eigenvectors on $\mathbb S$. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in $\mathbb P^n$. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor $\mathcal T$ is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by $\mathcal T$ and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces, which completes Cartwright and Strumfels' formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor $\mathcal T$ as irreducible factors.
    Linear and Multilinear Algebra 03/2013; 62(10). DOI:10.1080/03081087.2013.828721 · 0.70 Impact Factor


1 Download
Available from