Article

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

Department of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, PR China; Department of Mathematics, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong; Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong; School of Operations Research and Management Sciences, Qufu Normal University, Rizhao, Shandong 276800, PR China

Journal of Mathematical Analysis and Applications (Impact Factor: 1.12). 05/2007; DOI: 10.1016/j.jmaa.2006.07.064 - [Show abstract] [Hide abstract]

**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.12/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**The adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs, and some fundamental properties for the adjacency tensor and its Z-eigenvalues of a uniform hypergraph are obtained. In particular, some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented.Linear Algebra and its Applications 10/2013; 439(8). · 0.98 Impact Factor - [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.01/2015;

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