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: 1.05). 01/2007; DOI: 10.1016/j.jmaa.2006.07.064 - [Show abstract] [Hide abstract]

**ABSTRACT:**The quantum eigenvalue problem arises in the study of the geometric measure of the quantum entanglement. In this paper, we convert the quantum eigenvalue problem to the Z-eigenvalue problem of a real symmetric tensor. In this way, the theory and algorithms for Z-eigenvalues can be applied to the quantum eigenvalue problem. In particular, this gives an upper bound for the number of quantum eigenvalues. We show that the quantum eigenvalues appear in pairs, i.e., if a real number $\lambda$ is a quantum eigenvalue of a square symmetric tensor $\Psi$, then $-\lambda$ is also a quantum eigenvalue of $\Psi$. When $\Psi$ is real, we show that the entanglement eigenvalue of $\Psi$ is always greater than or equal to the Z-spectral radius of $\Psi$, and that in several cases the equality holds. We also show that the ratio between the entanglement eigenvalue and the Z-spectral radius of a real symmetric tensor is bounded above in a real symmetric tensor space of fixed order and dimension.05/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we show that the coefficients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. When the dimension is 2, some simplified formulas of the E-characteristic polynomial are presented. A re- sultant formula for the constant term of the E-characteristic polynomial is given. We then study the set of tensors with infinitely many eigenpairs and the set of irregular tensors, and prove both the sets have codimension 2 as subvarieties in the projective space of tensors. This makes our perturbation method workable. By using the perturbation method and exploring the difference between E-eigenvalues and eigenpair equivalence classes, we present a simple formula for the coefficient of the leading term of the E-characteristic polynomial, when the dimension is 2.08/2012; -
##### Article: The number of singular vector tuples and uniqueness of best rank one approximation of tensors

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**ABSTRACT:**In this paper we discuss the notion of singular value tuples of a complex valued d-mode tensor of dimension m_1 x ... x m_d. We show that a general tensor has a finite number of singular value tuples, viewed as points in a corresponding Segre variety. We give the formula for the number of singular value tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e. m_1=...=m_d. We show that best rank one approximations for general tensors are unique. Similarly, for general partially symmetric tensors the best rank one approximation is unique and partially symmetric. We show that a best rank-(r_1,...,r_d) approximation for a general $d$-mode tensor is unique.10/2012;

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