A general product of tensors with applications
By J. Shao. 2012
296 citations recorded by the source corpus.
We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of this product will be given. Using this tensor product and some properties on the resultant of a system of homogeneous equations on $n$ variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this tensor product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.
Subfields: Computational Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/a2acca638ed555358ffdb895dc105f974f59a825
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In Computational Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs (2014, methodology): “1 ([3]): Let A (and B) be an order m ≥ 2 (and order k ≥ 1), dimension n tensor, respectively.”
- Some upper bounds on the eigenvalues of uniform hypergraphs (2015, background): “(See [9,12].”
- On some properties of three different types of triangular blocked tensors (2016, background): “4 ([15]) Let A and B be two order m dimension n tensors.”
- Further results on generalized inverses of tensors via the Einstein product (2016, background): “But the increasing ability of data collection systems to store huge volumes of multidimensional data and in recognition of potential modeling accuracy, matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields, including co”