*Leiba Rodman*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161853
- eISBN:
- 9781400852741
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161853.003.0002
- Subject:
- Mathematics, Algebra

This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as axb = c and ax − xb = c. It studies the Sylvester equation ax ...
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This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as axb = c and ax − xb = c. It studies the Sylvester equation ax − xb = y; x,y ∈ H; and the corresponding real linear transformation Sa,b(x) = ax − xb; 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.Less

This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as *axb* = *c* and *ax* − *xb* = *c*. It studies the Sylvester equation *ax* − *xb* = *y*; *x*,*y* ∈ H; and the corresponding real linear transformation *S*_{a,b}(*x*) = *ax* − *xb*; *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*)α*x*, 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.