The Complete Positivity of Symmetric Tridiagonal and Pentadiagonal Matrices

TL;DR

This paper characterizes when symmetric tridiagonal and pentadiagonal matrices are completely positive, providing decompositions, proofs, and examples to advance understanding of their structure and properties.

Contribution

It introduces new decompositions and proofs for complete positivity of symmetric tridiagonal and pentadiagonal matrices, extending existing results and clarifying their cp-rank.

Findings

The cp-rank of irreducible tridiagonal doubly stochastic matrices equals their rank.

Provided two decompositions sufficient for complete positivity of symmetric pentadiagonal matrices.

Illustrated constructions with multiple examples.

Abstract

We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix $A$ is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp-rank of any irreducible tridiagonal doubly stochastic matrix is equal to its rank. We then consider symmetric pentadiagonal matrices, proving some analogous results, and providing two different decompositions sufficient for complete positivity. We illustrate our constructions with a number of examples.