Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/12251
Title: The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory
Authors: Li, G
Qi, L 
Yu, G
Keywords: Characteristic tensor
Maximum Z-eigenvalue
Polynomial optimization
Semismoothness
Spectral graph theory
Symmetric tensor
Issue Date: 2013
Publisher: John Wiley & Sons
Source: Numerical linear algebra with applications, 2013, v. 20, no. 6, p. 1001-1029 How to cite?
Journal: Numerical linear algebra with applications 
Abstract: In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Z-eigenvalues of a real symmetric tensor with even order. We first establish that the maximum Z-eigenvalue function is a continuous and convex function on the symmetric tensor space and so provide formulas of the convex conjugate function and ε-subdifferential of the maximum Z-eigenvalue function. Consequently, for an mth-order N-dimensional tensor A, we show that the normalized eigenspace associated with maximum Z-eigenvalue function is ρth-order Hölder stable at A with ρ=1m(3m-3)n-1-1. As a by-product, we also establish that the maximum Z-eigenvalue function is always at least ρth-order semismooth at A. As an application, we introduce the characteristic tensor of a hypergraph and show that the maximum Z-eigenvalue function of the associated characteristic tensor provides a natural link for the combinatorial structure and the analytic structure of the underlying hypergraph. Finally, we establish a variational formula for the second largest Z-eigenvalue for the characteristic tensor of a hypergraph and use it to provide lower bounds for the bipartition width of a hypergraph. Some numerical examples are also provided to show how one can compute the largest/second-largest Z-eigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula.
URI: http://hdl.handle.net/10397/12251
ISSN: 1070-5325
EISSN: 1099-1506
DOI: 10.1002/nla.1877
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