A Spectral Theory for Tensors

Details A Spectral Theory for Tensors A Spectral Theory for Tensors

2010-08-17

arxiv.org/abs/1008.2923v4

Annales de la facult\'e des sciences de Toulouse S\'er. 6, 20 no. 4 (2011), p. 801-841

Mathematics

Spectral Theory

The paper is an updated version of an earlier version

Scientific paper

10.5802/afst.1325

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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