On approximate diagonalization of third order symmetric tensors by orthogonal transformations

TL;DR

This paper investigates methods for approximately diagonalizing third order symmetric tensors using orthogonal transformations, exploring classes of such tensors, their relationships, and convergence properties of relevant algorithms.

Contribution

It introduces new classes of approximately diagonal tensors, analyzes their relationships, and proves convergence of the cyclic Jacobi algorithm for this problem.

Findings

Defined classes of approximately diagonal tensors

Analyzed relationships between tensor classes and eigenproperties

Proved convergence of the cyclic Jacobi algorithm

Abstract

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.