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Topics in Quaternion Linear Algebra$
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Leiba Rodman

Print publication date: 2014

Print ISBN-13: 9780691161853

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161853.001.0001

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PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 19 October 2019

The algebra of quaternions

The algebra of quaternions

Chapter:
(p.9) Chapter Two The algebra of quaternions
Source:
Topics in Quaternion Linear Algebra
Author(s):

Leiba Rodman

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161853.003.0002

This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as axb = c and axxb = c. It studies the Sylvester equation axxb = y; x,y ∈ H; and the corresponding real linear transformation Sa,b(x) = axxb; x ∈ H. Descriptions of all automorphisms and antiautomoprhisms of quaternions are then given. The chapter also considers quadratic maps of the form x ↦ φ‎(x)α‎x, where α‎ ∈ H∖{0} is such that either φ‎(α‎) = α‎ or φ‎(α‎) = −α‎ for a fixed involution φ‎. The chapter also introduces representations of quaternions in terms of 2 × 2 complex matrices and 4 × 4 real matrices.

Keywords:   scalar quaternions, quaternion algebra, Sylvester equation, real linear transformations, automorphisms, antiautomorphisms, quadratic maps, real matrices, complex matrices

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