Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computing Left Eigenvalues of Quaternion Matrices

Michael Sebek

TL;DR

Tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.

Abstract

We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.

Computing Left Eigenvalues of Quaternion Matrices

TL;DR

Tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.

Abstract

We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.
Paper Structure (38 sections, 17 theorems, 111 equations)

This paper contains 38 sections, 17 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Quaternions and matrices
  3. The skew field H
  4. Quaternion vectors and matrices
  5. Real embedding
  6. Left eigenvalues
  7. Left eigenvalues: definition and basic facts
  8. Spectrum geometry and cardinality
  9. The zero eigenvalue: singularity and multiplicity
  10. Algebraic reductions, and computational approaches
  11. Residual certificates for left eigenvalues
  12. Contrast with right eigenvalues
  13. Newton iteration for left eigenpairs with a gauge constraint
  14. Gauge constraint and regauging
  15. Quaternion-form Newton correction with gauge constraints
  16. ...and 23 more sections

Key Result

Proposition 1

Let $A\in\mathbb{H}^{n\times n}$ and set Then $\dim\ker(A)=m$.

Theorems & Definitions (63)

  • Proposition 1: Rank--nullity over $\mathbb{H}$
  • Definition 1: Real coordinate map on $\mathbb{H}^n$
  • Definition 2: Real left-multiplication matrix
  • Theorem 1: Real representation of quaternion multiplication
  • Definition 3: Real embedding of quaternion matrices
  • Theorem 2: Intertwining and algebraic rules for $\rho(\cdot)$
  • Theorem 3: Kernel and the factor $4$
  • proof
  • Definition 4: Left eigenvalues
  • Remark 1: Distinct left eigenvalues
  • ...and 53 more