The expected number of Z-eigenvalues of a real gaussian tensor

TL;DR

This paper calculates the expected number and asymptotic behavior of Z-eigenvalues for real Gaussian tensors, providing insights into their spectral properties in high-dimensional random tensor models.

Contribution

It introduces a method to compute the expected count of Z-eigenvalues for Gaussian tensors and analyzes their asymptotic behavior as tensor dimensions grow.

Findings

Expected number of Z-eigenvalues computed

Asymptotic behavior characterized

Results applicable to high-dimensional tensor analysis

Abstract

A real number $\lambda$ is called a Z-eigenvalue of a tensor $A$, if $\lambda$ is an eigenvalue of $A$ and the corresponding eigenvector $v$ is real and satisfies $v^Tv=1$. In this paper we compute the expected number of Z-eigenvalues of a real gaussian tensor and its asymptotic behaviour. Here we call a tensor $A=(A_{i_0,\ldots,i_{d}}) \in\mathbb{R}^{n^{d+1}}$ gaussian, when the $A_{i_0,\ldots,i_{d}}$ are centered gaussian random variables with variance $\sigma^2= 1$.