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Complex numbers and Quaternions as Matrices
Interpretation of the complex numbers and quaternion algebra as algebras of matrices

Prerequisites

To understand this, you need to be familiar with complex numbers, such as are often taught in a high school second-year algebra course, and with matrix arithmetic, which is often taught in such a high school course, or in a college linear algebra course.

Introduction

Some of the algebras of the Cayley-Dickson Construction can be modeled as matrix algebras on certain subsets of square matrices.

Matrices are used to represent linear transformations; these representations are useful in studying the effect of quaternion multiplication as a linear transformation.

Two related but distinct algebraic operations may be represented this way. One is the action of multiplying by given hypercomplex number, as a linear transformation of the space of hypercomplex numbers. The other is the multiplicative algebra of hypercomplex numbers amongst themselves.

The modeling process is not obvious however, and furthermore, does not seem to extend to the octonions.

The complex numbers as matrices

The complex numbers can be thought of as ordered pairs of real numbers (a, b) with addition defined in the obvious way, and with a special multiplication defined by

(a, b) (c, d) = (acbd, ad + bc) .

Consider the special set of 2 × 2 matrices of real numbers whose diagonal elements are equal and whose off-diagonal elements are negatives of one another, like this:

a b
b a
 .

Try multiplying two such matrices:

a b
b a
c d
d c
 = 
acbd −(ad + bc)
ad + bc acbd
 .

So the product of two such matrices is again a matrix of the same form (the diagonal elements are identical, and the other elements are negatives of one another). The set of such matrices is therefore said to be closed under multiplication. Note that the entries correspond to the real and complex parts of the product (a, b) (c, d).

In this way, the algebra of this set of matrices models the algebra of the complex numbers. The correspondence between this matrix algebra and that of complex numbers is called a homomorphism; the two algebras are said to be homomorphic to one another.

So, the algebra of the complex numbers is a proper sub-algebra of the algebra of 2 × 2 matrices of real numbers.

Even though the matrices have four entries, the diagonals entries are identical and the off-diagonals are just negatives of one another, so the set forms a 2-dimensional vector space, just as do the complex numbers.

complex multiplication as a linear transformation

Now, multiplication of the points in the plane by a complex number amounts to a linear transformation of the plane. So the next question is: what is the relation between these matrices, and the matrix of that linear transformation?

This is a little tricky: there is more than one way to arrange the signs in the above matrices to reflect complex multiplication. I deliberately arranged the signs in the matrix entries above so that they correspond exactly the matrices of the corresponding linear transformations of the plane. (This is even trickier for the quaternion case.)

This correspondence between complex multiplication and linear transformations, allows a direct reading of linear algebraic properties of the transformation effected by complex multiplication.

a b
b a
 
c
d
 = 
acbd
ad + bc
 .

the geometry of the transformations

These matrices (as well as the complex numbers) can also be viewed as a special class of transformations of the 2-dimensional plane into itself. This class consists precisely of rotations and isotropic scalings of the plane, excluding other transformations such as non-isotropic deformations, skews, and reflections. The image of a triangle under such a transformation is geometrically similar to the original triangle, and its angles appear in the same order.

The transformation effected by multiplication by a complex number of unit norm is a rotation. This can be seen from the matrix form by multiplying the matrix by its transpose, which results in an identity matrix. Thus again, multiplication by a complex number is a rotation of the plane and a scaling.

This is the linear case of a conformal transformation. Here I discuss the connection of conformal transformations to quaternions.

The quaternions as matrices

To model the multiplication of quaternions, form matrices as before, but with the entries now being complex numbers rather than real numbers, and with the bottom row being complex-conjugated:

a b
b* a*
 .

If the entries are real, the extra conjugation of the bottom row makes no difference, so this matrix would be the same as that of a complex number.

This set is closed under multiplication. Consider a general product of two such matrices:

a b
b* a*
c d
d* c*
 = 
acbd* adbc*
a*d* + b*c a*c*b*d
 
 = 
acbd* −(bc* + da)
(bc* + da)* (acbd*)*
 ,

using the commutativity of complex numbers. The matrices are all of the same form with regard to minus signs and conjugation, so as above with the matrices representing the algebra of complex numbers, the set of matrices is closed under multiplication. This matrix formula is just an expanded form of the Cayley-Dickson multiplication formula, (a, b) (c, d) = (acbd*bc* + da), so the algebra of these matrices represents the algebra of the quaternions.

The algebra defined by matrix multiplication on this set is therefore a proper sub-algebra of the algebra of 4 × 4 matrices of real numbers.

They form a 4-dimensional vector space, since there are only two independent entries in each matrix, but each entry is from the two-dimensional set of complex numbers.

quaternion multiplication as a linear transformation

By regarding the quaternions as a 4-dimensional vector space over the reals, the linear transformation effected by multiplying by a quaternion may be represented as a matrix. Again, there are two different transformations for most quaternions, corresponding to left and right multiplication. The multiplication formula for quaternions produces for multiplication on the left by the quaternion (a, b, c, d) the matrix

a b c d
b a d c
c d a b
d c b a

For multiplication on the right by the same quaternion, it produces the matrix:

a b c d
b a d c
c d a b
d c b a

Fortunately these cumbersome representations are rarely necessary in practice; more compact descriptions are more perspicuous. But there is a price to pay.

One more compact form employs the representation of quaternions as pairs of complex numbers. Here, the transformation effected by multiplication from each direction may be represented as a kind of 2 × 2 matrix.

A difficulty arises in representing quaternions as complex matrices that is not present in the representation of complex numbers as real matrices. It will happen that there are some transformations corresponding to complex multiplication that can’t be consistently represented as 2 × 2 matrices of complex numbers. There are a couple of options.

Typically, multiplication by a quaternion does not correspond to the linear transformation described by any 2 × 2 matrix of complex numbers. One might write

a b
b* a*
 
c
d*
 = 
acbd*
(bc* + da)*
 ,

but the conjugation of d on the left hand side is a cheat—it already represents a transformation of the vector representing (c, d) prior to the multiplication by the matrix. Inasmuch as conjugation cannot be achieved by multiplication by a complex number, this is saying that quaternion multiplication does not represent a linear transformation of 2.

Another option, which maintains the notational compactness of the complex pair, is to introduce an operator ~ that is the left equivalent of conjugation:

~z = z*.

Viewed as a 2 × 2 matrix, this is just the one that reverses the sign of the second coordinate. Despite being a simple operation, it is not equivalent to multiplication by any complex number.

Using this ~ operator in 2 × 2 complex matrices is more compact than writing out the full 4 × 4 matrices, but it has a cost: the operator behaves differently with regard to the end complex operands than with intermediate matrix entries. With the end operands it just effects a conjugation; with a complex matrix entry a, the rule is ~a = a*~.

In this notation, multiplication on the left by the quaternions is given by the relation (a, b) is:

 
a b
b* a*
 
I 0
0 I~
 
c
d
 = 
I 0
0 I~
 
acbd*
bc* + da
 ,

that is, multiplication of a quaternion (c, d) on the left by quaternion (a, b) may be represented as

 
a b~
b~ a
 
c
d
 = 
acbd*
bc* + da
 .

A switch of the roles of (a, b) and (c, d) in the above general formula yields a formula for multiplication (c, d) on the right by (a, b):

a b*
b a*
 
c
d
 = 
ca + db*
da* + bc
 .

This means that multiplication on the right by a quaternion is equivalent to a linear transformation of 2, while multiplication on the left is not: it always involves a conjugate of the second coordinate, which is not achievable by complex multiplication. So it may sometimes be notationally convenient use a 2 × 2 matrix to represent the linear transformation of multiplication on the left by a quaternion, the notation confers little algebraic insight.

A block-matrix form is also sometimes useful (see Representations of the Quaternions.) The multiplication of a quaternion ( u, w ) on the left by quaternion ( t, v ), where t and u are real numbers and v and w are 3-vectors, may be represented by

 
t v ·
v t I + v ×
 
u
w
 = 
tuv · w
uv + tw + v × w
 .

Due to the commutativity of the dot product and the anticommutativity of the cross product, the linear operator of multiplication of a quaternion ( t, v ) on the right by quaternion ( u, w ) is easily represented in this notation:

 
u w ·
w u Iw ×
 
t
v
 = 
tuv · w
uv + tw + v × w
 .

One stumbling block with this formulation is that the cross product operator × is itself not associative. That must always be kept in mind when applying more than one of these matrices.

The octonions

In an attempt to represent the algebra of the octonions, one might form matrices as in the previous step, but with the entries p and q to be quaternions rather than complex numbers. This fails as a representation of octonions however.

The set of such matrices is not closed under multiplication. Consider the octonion multiplication whose factors represented as matrices analogous to the quaternion case above. Here the matrix elements of the factors are quaternions, which don’t commute, so the diagonal elements acbd* and a*c*b*d are not conjugates of one another. So the matrix multiplication doesn’t agree with that of the octonions. That is, the multiplication of octonions can’t be represented as the multiplication of 2 × 2 matrices of quaternions.

But moreover, there can be no real 8 × 8 matrix representation of octonion multiplication, because matrix multiplication is associative, while octonion multiplication isn’t.

Although the action of multiplication (on either left or right) by an octonion is a linear transformation of 8 (and so can be represented by an 8 × 8 real matrix), the composition of two such transformations is not typically the action of multiplication by any octonion. That is, no sub-algebra of a matrix algebra is homomorphic to the octonion algebra.

Matrices can still be used to represent the octonions, but only with matrix multiplication rules different from the usual ones. This is discussed in recent papers of Daboul and Delbourgo, and of Tian.

There are several approaches. However, the notation as matrices is little more than a typographic convenience: the advantage of a familiar algorithm for multiplying them is lost.

See also

William Rowan Hamilton, On Quaternions

John Baez, The Octonions

J Daboul, R Delbourgo (1999) “Matrix Representation of Octonions and GeneralizationsJ. Math. Phys., 40

Yongge Tian (2000) “Matrix Representations of Octonions and Their ApplicationsAdvances in App Clifford Algebras 10