Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness of size-2 positive semidefinite matrix factorizations

Kristen Dawson, Serkan Hoşten, Kaie Kubjas, Lilja Metsälampi

Abstract

We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions.

Uniqueness of size-2 positive semidefinite matrix factorizations

Abstract

We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions.

Paper Structure

This paper contains 15 sections, 37 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. $s$-Infinitesimal and $s$-trivial motions of psd factorizations
  3. $s$-Infinitesimal motions
  4. $s$-Trivial motions and $s$-infinitesimal rigidity
  5. 1-Trivial motions
  6. $k$-Trivial motions
  7. Further preliminary results
  8. $s$-Infinitesimal rigidity in $\mathcal{M}_{3,2}^{p\times q}$
  9. No orthogonal pairs
  10. One orthogonal pair
  11. Two orthogonal pairs
  12. Rigidity and boundaries
  13. Rigidity for positive matrices
  14. Rigidity for matrices with one zero
  15. Non-trivial boundaries

Key Result

Theorem 1

Let $M\in \mathbb{R}_{+}^{p \times q}$ be a matrix of rank $3$ and psd rank $2$, and consider a size-$2$ psd factorization of $M$ given by the factors $A^{(1)}$, $\dots$, $A^{(p)}$, $B^{(1)}$, $\dots$, $B^{(q)} \in \mathcal{S}^2_+$. Let $A^{(1)}$, $\dots$, $A^{(\bar{p})}$, $B^{(1)}$, $\dots$, $B^{(\

Theorems & Definitions (91)

  • Theorem 1
  • Definition 2: $s$-Infinitesimal motion of a psd factorization
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof
  • Remark 6
  • Example 7
  • Definition 8: $s$-Trivial motion
  • Definition 9: $s$-Infinitesimally rigid factorization
  • ...and 81 more