E-Determinants of Tensors
Abstract
We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.
Keywords
Cite
@article{arxiv.1109.0348,
title = {E-Determinants of Tensors},
author = {Shenglong Hu and Zheng-Hai Huang and Chen Ling and Liqun Qi},
journal= {arXiv preprint arXiv:1109.0348},
year = {2015}
}