Schur complements for tensors and multilinear commutative rank

TL;DR

This paper establishes the equivalence of three different rank notions for matrices of multilinear forms, generalizing classical results and addressing open questions about tensor ranks.

Contribution

It proves the equivalence of three rank notions for multilinear matrices, extending classical results and resolving a specific open question.

Findings

Three rank notions for multilinear matrices are equivalent.

The result generalizes Flanders' classical theorem.

It addresses a question on the relation between analytic and slice ranks.

Abstract

We show that three notions of rank for matrices of multilinear forms are equivalent. This result generalizes a classical result of Flanders, corrects a minor hole in work of Fortin and Reutenauer, answers a question of Lampert on the relation between the analytic and slice ranks of trilinear forms, and establishes a special case of the conjecture that the analytic and partition ranks of a tensor are equivalent.