Multiplications and eigenvalues of tensors via linear maps

2018/09/12 by Zhaobing Fan, Chunli Deng, Haifeng Li +1 · 7 citations
Computer Science · Mathematics · #Algebra over a field #Associative property #Cartesian tensor #Eigenvalues and eigenvectors #Exact solutions in general relativity #Geometry #Invariants of tensors #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Physics #Product (mathematics) #Pure mathematics #Tensor (intrinsic definition) #Tensor contraction #Tensor decomposition and applications #Tensor density #Tensor field #Tensor product #Tensor product of Hilbert spaces

paper · doi:10.1080/03081087.2018.1515173

published in Linear and Multilinear Algebra 68(3), 606-621 (Taylor & Francis)

openalex publication_date 2018/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

In this paper, we study multiplications and eigenvalues of tensors by using linear maps. By the composition of linear maps, we naturally obtain two multiplications of tensors, which generalize Shao's product, Einstein product and n-mode product etc. Some properties of product of tensors are obtained directly, which don't need proofs, for example the associative law of Shao's product etc. Moreover, we extend eigenvalues of tensors to tensor representations of quivers. By using tensor representations of quivers, eigenvalues of tensors and properties are given.

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