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

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 $\sigma^2= 1$.