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A semidefinite programming characterization of the Crawford number

Shmuel Friedland, Cynthia Vinzant

Abstract

We give a semidefinite programming characterization of the Crawford number. We show that the computation of the Crawford number within $\varepsilon$ precision is computable in polynomial time in the data and $|\log \varepsilon |$.

A semidefinite programming characterization of the Crawford number

Abstract

We give a semidefinite programming characterization of the Crawford number. We show that the computation of the Crawford number within precision is computable in polynomial time in the data and .
Paper Structure (6 sections, 4 theorems, 29 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. SDP characterization and polynomial computability of $\kappa(C)$
  4. Proof of Theorem \ref{['SDPcr']}
  5. Reformulation as a real symmetric SDP
  6. Complexity results for the Crawford number

Key Result

Theorem 1.1

Let $C = A+\mathbbm{i} B\in\mathbb{C}^{n\times n}$ where $A$, $B$ are Hermitian. Then Here $X$ is an $n\times n$ Hermitian matrix variable and $u,v,w$ are variable real numbers.

Figures (1)

  • Figure 1: The numerical range from Example \ref{['ex:n=2']} and $z\in \mathcal{W}(C)$ minimizing $|z|$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['SDPcr']}.
  • Theorem 2.2
  • proof
  • Example 2.3: $n=2$
  • Theorem 2.4
  • proof