Triangular Decomposition of Third Order Hermitian Tensors

TL;DR

This paper introduces a triangular decomposition for third order Hermitian tensors, extending classical matrix decompositions to higher-order tensors and establishing new properties and equivalences for these tensor classes.

Contribution

It defines lower triangular and sub-symmetric tensors, introduces third order Hermitian tensors, and proves their triangular decomposability, generalizing Cholesky decomposition to tensors.

Findings

Diagonal entries are eigenvalues of lower triangular tensors

Number of independent entries matches symmetric tensors of same order

Positive semi-definite Hermitian tensors are triangularly decomposable

Abstract

We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower triangular sub-symmetric tensor is the same as that of a symmetric tensor of the same order and dimension. We further introduce third order Hermitian tensors, third order positive semi-definite Hermitian tensors, and third order positive semi-definite symmetric tensors. Third order completely positive tensors are positive semi-definite symmetric tensors. Then we show that a third order positive semi-definite Hermitian tensor is triangularly decomposable. This generalizes the classical result of Cholesky decomposition in matrix analysis.