# Eigenvalues of a real supersymmetric tensor

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

909 Citations (Scopus)

## Abstract

In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An m th -order n-dimensional supersymmetric tensor where m is even has exactly n(m - 1)n-1eigenvalues, and the number of its E-eigenvalues is strictly less than n(m - 1)n-1when m ≥ 4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m - 1)n-1. The n(m - 1)n-1eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.
Original language English 1302-1324 23 Journal of Symbolic Computation 40 6 https://doi.org/10.1016/j.jsc.2005.05.007 Published - 1 Dec 2005

## Keywords

• Eigenvalue
• Supersymmetric tensor
• Symmetric hyperdeterminant

## ASJC Scopus subject areas

• Algebra and Number Theory
• Computational Mathematics

## Fingerprint

Dive into the research topics of 'Eigenvalues of a real supersymmetric tensor'. Together they form a unique fingerprint.