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Schur complements for tensors and multilinear commutative rank

Guy Moshkovitz, Daniel G. Zhu

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.

Schur complements for tensors and multilinear commutative rank

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.
Paper Structure (23 sections, 78 equations)

This paper contains 23 sections, 78 equations.

Table of Contents

  1. Introduction
  2. Linear matrices
  3. Applications to tensors
  4. Proof overview
  5. Outline
  6. Preliminaries
  7. Multi-homogeneous polynomials
  8. Matrices
  9. Tensors
  10. Approximation
  11. Approximating rational functions
  12. Approximating matrices
  13. Schur Complementation
  14. Schur complements over rings
  15. Differential Schur complements
  16. ...and 8 more sections

Theorems & Definitions (16)

  • proof : Proof of \ref{['prop:eltapprox']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 6 more