# The algebra of quaternions

# The algebra of quaternions

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.

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

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