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Tensor Analysis: Spectral Theory and Special Tensors

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by Liqun Qi and Ziyan Luo


2017 / xiv + 305 pages / Softcover / ISBN 978-1-611974-74-4 List Price $84.00 / SIAM Member Price $58.80 / Order Code: OT151

Keywords: tensors, eigenvalues, multilinear algebra, hypergraphs, spectral theory


Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory—some of which are nontrivial—have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors.

Tensor Analysis: Spectral Theory and Special Tensors is unique in that it is the first book on

  • the spectral theory of tensors;
  • the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and
  • the spectral hypergraph theory via tensors, which is covered in a chapter.

The intended audience is researchers and graduate students.

About the Authors
Liyun Qi is Chair Professor of Applied Mathematics in the Department of Applied Mathematics at The Hong Kong Polytechnic University. Listed as one of the 345 most highly cited mathematicians from 1981 to 2007 by ISI Highly Cited Research, he has published more than 290 papers—including more than 110 papers on tensors—in international journals. He introduced eigenvalues of higher order tensors in 2005; proposed the first algorithm for computing the largest eigenvalue of a nonnegative tensor in 2009; introduced positive semidefinite tensors, copositive tensors, and Laplacian and signless Laplacian tensors; and introduced M-tensors, P-tensors, B-tensors, Hilbert tensors, Cauchy tensors, SOS tensors, essentially nonnegative tensors, completely positive tensors, completely decomposable tensors, and strongly symmetric tensors with coauthors. He organized several international conferences and workshops on tensors.

Ziyan Luo is Associate Professor of System Science at the State Key Laboratory of Rail Traffic Control and Safety at Beijing Jiaotong University. She did her postdoctoral work at Beijing Jiaotong University (2010–2012). She was a research assistant at The Hong Kong Polytechnic University (2010), a visiting scholar at Stanford University (2011–2012), a visiting scholar at National University of Singapore (2015–2016), and a research associate at The Hong Kong Polytechnic University (2015). She has published more than twenty academic papers.


ISBN: 9781611974744

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