An Even Order Symmetric B Tensor is Positive Definite

TL;DR

This paper provides a new decomposition method for symmetric B and B0 tensors, establishing conditions under which even order symmetric B tensors are positive definite and B0 tensors are positive semi-definite.

Contribution

It introduces a novel decomposition approach for symmetric B tensors, extending positive definiteness criteria from matrices to tensors.

Findings

Symmetric B tensors can be decomposed into a diagonally dominated symmetric M tensor plus positive multiples of all-one tensors.

Even order symmetric B tensors are positive definite.

Symmetric B0 tensors are positive semi-definite.

Abstract

It is easily checkable if a given tensor is a B tensor, or a B$_0$ tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B$_0$ tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B$_0$ tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.