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The algebra of quaternions
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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
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.
Title: The algebra of quaternions
Description:
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.
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