Introduction to the hyperdeterminant and to the rank of multidimensional matrices

TL;DR

This paper introduces the hyperdeterminant and explores the rank of multidimensional matrices, emphasizing geometric properties, boundary cases, and open problems in the context of Segre varieties and tensor ranks.

Contribution

It provides a geometric proof of the triangle inequality for Segre varieties and discusses the boundary format case and tensor rank, highlighting new insights and open questions.

Findings

Geometric proof of the triangle inequality for Segre varieties

Analysis of the boundary format case and pencil of quadrics

Proposes three open problems in the field

Abstract

This is an introduction to the hyperderminant, according to Gelfand, Kapranov and Zelevinsky. The "triangle inequality", characterizing the Segre varieties such that their dual variety is a hypersurface, is proved in a geometric way (Theorems 3.3 and 6.2). Significant emphasis is given to boundary format case, to pencil of quadrics and to the rank of multidimensional matrices. In the last section we propose three open problems.