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Third-Order Tensors as Linear Operators on a Space of Matrices

✍ Scribed by Karen Braman

Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
240 KB
Volume
433
Category
Article
ISSN
0024-3795

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✦ Synopsis

A recently proposed tensor-tensor multiplication (M.E. Kilmer, C.D. Martin, L. Perrone, A Third-Order Generalization of the Matrix SVD as a Product of Third-Order Tensors, Tech. Rep. TR-2008-4, Tufts University, October 2008) opens up new avenues to understanding the action of n Γ— n Γ— n tensors on a space of n Γ— n matrices. In particular it emphasizes the need to understand the space of objects upon which tensors act. This paper defines a free module and shows that every linear transformation on that module can be represented by tensor multiplication. In addition, it presents a generalization of ideas of eigenvalue and eigenvector to the space of n Γ— n Γ— n tensors.

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