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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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The algebra of quaternions

The algebra of quaternions

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

Leiba Rodman

Princeton University Press

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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