A Study on Nonnegative Tubal Matrices

TL;DR

This paper extends the Perron-Frobenius theorem to nonnegative irreducible tubal matrices, defining positivity in the tubal setting and exploring which classical matrix results generalize.

Contribution

It introduces the concept of nonnegative and positive tubal matrices, establishes properties analogous to matrices, and generalizes the Perron-Frobenius theorem to this new setting.

Findings

Perron-Frobenius theorem is valid for nonnegative irreducible tubal matrices.

Some classical PF theorem conclusions do not extend to tubal matrices.

Presence of a strongly positive tubal scalar ensures most PF theorem results hold.

Abstract

Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce the irreducible tubal matrix, and provide two equivalent characterizations. Then, the celebrated Perron-Frobenius theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices can be generalized to the tubal matrix setting, while some are not. One reason is the defined positivity here has a different meaning to its usual sense. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.