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On semidefinite programming characterizations of the numerical radius and its dual norm for quaternionic matrices

Shmuel Friedland

TL;DR

It is shown that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and using the short step, primal interior point method.

Abstract

We give a semidefinite programming characterizations of the numerical radius and its dual norm for quaternionic matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and $|\log \varepsilon |$ using the short step, primal interior point method.

On semidefinite programming characterizations of the numerical radius and its dual norm for quaternionic matrices

TL;DR

It is shown that the computation of the numerical radius and its dual norm within precision are polynomially time computable in the data and using the short step, primal interior point method.

Abstract

We give a semidefinite programming characterizations of the numerical radius and its dual norm for quaternionic matrices. We show that the computation of the numerical radius and its dual norm within precision are polynomially time computable in the data and using the short step, primal interior point method.
Paper Structure (15 sections, 20 theorems, 100 equations)

This paper contains 15 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Quaternionic matrices
  3. Quaternions
  4. Inner product and the Gram-Schmidt process
  5. Eigenvalues, eigenvectors and the spectral decomposition of normal matrices
  6. Singular value decomposition
  7. Norms on $\mathbb{H}^{m\times n}$
  8. Semidefinite programming for quaternionic matrices
  9. Complexity results for semidefinite programming
  10. Numerical range
  11. The SDP characterizations of qradius and its dual norm
  12. Characterizations of $r^\vee(\cdot)$
  13. SDP Characterization of $r(\cdot)$
  14. Polynomial computability of $r(\cdot)$ and $r^\vee(\cdot)$
  15. A pseudo-numerical range on $\mathbb{C}^{n\times n}$

Key Result

Proposition 2.2

Let $A\in\mathbb{H}^{m\times n}$ then $H(A):=\in \mathrm{H}_{m+n}(\mathbb{H})$. Its nonzero eigenvalues are $\pm\sigma_1(A),\ldots,\pm \sigma_r(A)$.

Theorems & Definitions (34)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • ...and 24 more