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Edinah K. Gnang; Ahmed Elgammal; Vladimir Retakh
A Spectral Theory for Tensors
Annales de la faculté des sciences de Toulouse Sér. 6, 20 no. 4 (2011), p. 801-841, doi: 10.5802/afst.1325
Article PDF | Reviews MR 2918215 | Zbl 1238.15004

Résumé - Abstract

In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors. Our proposed factorization shows the relationship between the eigen-objects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise the notion of matrix hermicity, matrix transpose, and most importantly the notion of orthogonality. Our proposed factorization for a tensor in terms of lower order tensors can be recursively applied so as to naturally induces a spectral hierarchy for tensors.

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