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A Unified Non-Strict Finsler Lemma

T. J. Meijer, K. J. A. Scheres, S. van den Eijnden, T. Holicki, C. W. Scherer, W. P. M. H. Heemels

Abstract

In this paper, we present a unified general non-strict Finsler lemma. This result is general in the sense that it does not impose any restrictions on the involved matrices and, thereby, it encompasses all existing non-strict versions of Finsler's lemma that do impose such restrictions. To further illustrate its usefulness, we showcase applications of the non-strict Finsler's lemma in deriving a structured solution to a special case of the non-strict projection lemma, and we use the unified non-strict Finsler's lemma to prove a more general version of the matrix Finsler's lemma.

A Unified Non-Strict Finsler Lemma

Abstract

In this paper, we present a unified general non-strict Finsler lemma. This result is general in the sense that it does not impose any restrictions on the involved matrices and, thereby, it encompasses all existing non-strict versions of Finsler's lemma that do impose such restrictions. To further illustrate its usefulness, we showcase applications of the non-strict Finsler's lemma in deriving a structured solution to a special case of the non-strict projection lemma, and we use the unified non-strict Finsler's lemma to prove a more general version of the matrix Finsler's lemma.
Paper Structure (17 sections, 10 theorems, 40 equations)

This paper contains 17 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Existing non-strict Finsler's lemmas
  3. The indefinite case
  4. The definite case
  5. Main results
  6. A unified non-strict Finsler lemma
  7. Discussion
  8. Applications
  9. Structured solution to the non-strict projection lemma
  10. A matrix Finsler lemma
  11. Conclusions
  12. Proof of Theorem \ref{['thm:nsfl']}
  13. Proof of Proposition \ref{['prop:special-case-sfl']}
  14. Proof of Proposition \ref{['prop:special-case-indef']}
  15. Proof of Proposition \ref{['prop:equivalence-coupling']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let $M$ and $N$ be symmetric matrices. Then, the following statements are equivalent:

Theorems & Definitions (14)

  • Lemma 1
  • Example 1
  • Lemma 2
  • Lemma 3: Anstreicher2000
  • Theorem 1
  • Example 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • ...and 4 more