Positivity of GCD tensors and their determinants

TL;DR

This paper investigates the positivity and divisibility properties of GCD tensors constructed from positive integers, establishing their strong complete positivity, infinite divisibility, and providing a formula for their determinants using Euler's totient function.

Contribution

It introduces the concept of GCD tensors' positivity, proves their strong complete positivity and infinite divisibility, and derives a determinant formula involving Euler's totient function.

Findings

GCD tensors are strongly completely positive.

GCD tensors are infinitely divisible.

Determinant of GCD tensors expressed via Euler's totient function.

Abstract

Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $T_{[S]}=(t_{i_{1}i_{2}\ldots i_{m}}),$ where $t_{i_{1}i_{2}\ldots i_{m}}=GCD(s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}),$ the greatest common divisor (GCD) of $s_{i_{1}},s_{i_{2}},\ldots,$ and $s_{i_{m}}$ is called the GCD tensor on $S$. The earliest result on GCD tensors goes back to Smith [Proc. Lond. Math. Soc., 1976], who computed the determinant of GCD matrix on $S=\{1,2,\ldots,n\}$ using the Euler's totient function, followed by Beslin-Ligh [Linear Algebra Appl., 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the $k$-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number $r,$ the tensor $T_{[S]}^{\circ r}=(t^{r}_{i_{1}i_{2}\ldots i_{m}})$ is strongly CP. Finally, we obtain an interesting decomposition of GCD tensors using Euler's totient function $\Phi$. Using this decomposition, we show that the determinant (also called hyperdeterminant) of the $m$th-order GCD tensor $T_{[S]}$ on a factor-closed set $S=\{s_1,\dots,s_n\}$ is $\prod\limits_{i=1}^{n} \Phi(s_{i})^{(m-1)^{(n-1)}}$.