Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenvalues, singular values, and the O'Shea-Sjamaar Theorem

Paul-Emile Paradan

Abstract

The main focus of this work is the study of several cones relating the eigenvalues or singular values of a matrix to those of its off-diagonal blocks.

Eigenvalues, singular values, and the O'Shea-Sjamaar Theorem

Abstract

The main focus of this work is the study of several cones relating the eigenvalues or singular values of a matrix to those of its off-diagonal blocks.
Paper Structure (36 sections, 21 theorems, 57 equations)

This paper contains 36 sections, 21 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Reminder of some classical results
  3. Singular values
  4. Augmented matrices
  5. Horn inequalities
  6. The cone $\hbox{\rm LR}(U,\widetilde{U})$
  7. Admissible elements
  8. Polarized trace
  9. Schubert calculus
  10. Description of $\hbox{\rm LR}(U,\widetilde{U})$
  11. The cone $\hbox{\rm LR}(m,n)$
  12. Proof of Theorem \ref{['theo:LR-m-n']}
  13. Admissible elements
  14. Cohomological conditions and inequalities
  15. A consequence of the O'Shea-Sjamaar Theorem
  16. ...and 21 more sections

Key Result

Theorem 2.2

The triplet $(x,y,z)\in (\mathbb{R}^n_+)^3$ belongs to $\hbox{\rm Horn}(n)$ if and only if the following conditions hold:

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • Lemma 2.11
  • ...and 17 more