Quaternions, Norms and Angles

Norms and Angles

Recall that multiplication in the complex plane multiplies norms and adds angles. This is Demoivre's theorem. A similar Transformation takes place in the quaternions. The norm is the norm you know, and sure enough, the norm of the product is the product of the norms. But what is an angle?

Given a quaternion over a field that includes all its square roots, divide through by the square root of the norm, so that the quaternion has norm 1. This is like scaling a vector to produce a unit vector. If the unit quaternion is a+bi+cj+dk, represent it with the following matrix.

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

Write an arbitrary quaternion as a row vector on the left, and multiply by this matrix. the result is a row vector that represents the product of the two quaternions. Therefore the quaternions with norm 1 can all be represented as 4×4 matrices, such that matrix multiplication corresponds to quaternion multiplication.

Verify that our matrix is orthonormal. It consists of for unit vectors in 4 space, each orthogonal to the other three. Thus each matrix is a rigid rotation in 4 space.

The determinant squared is the determinant of the matrix times its transpose, which is the identity matrix. Therefore the determinant is ±1. When the field is contained within the reals, we can prove the determinant is always 1. Thus each linear transformation is a rotation, not a reflection. Use a continuity argument. Move a and b around their circle of a given radius until a is positive and b is 0. This changes the determinant continuously, but it is always 1 or -1, so it doesn't change at all. Do the same for c and d, and then do the same for a and c. Now we have the identity matrix, and the determinant is 1.

Or you can compute the determinant algebraically. It comes out (a²+b²+c²+d²)², which is 1. Thus we could leave a b c and d unconstrained, and the determinant corresponds to the quaternion norm squared. But then we are no longer separating the "norm and angle" concepts.

The matrix, and hence the rotation, is entirely determined by the top row. This is the new location of the 1 vector in 4 space. If a complex number is a norm and an angle, then a real quaternion is a norm and a location on the unit 3 sphere in 4 space. Multiplication is accomplished by multiplying norms and applying rotations in succession. Unlike the complex plane, the angle group is nonabelian, just as the quaternions are nonabelian.

Realize that this is a subset of the 4 by 4 orthonormals. Place 1,1,-1,-1 down the main diagonal and you have an orthonormal matrix with determinant 1 that does not represent a quaternion.

On the Right

Multiply the aforementioned matrix by a column vector v on the right, where v represents a quaternion, and the new column vector is the product v*(a-bi-cj-dk). Premultiply by another matrix and you multiply on the right by another quaternion. This is a reverse group homomorphism, where f(xy) = f(y)*f(x). As you might imagine, we usually multiply by a row vector on the left, so that the homomorphism runs in the forward direction.

Alternate Representation

The arrangement of rows and columns is somewhat arbitrary. We could, for instance, write the quaternion vectors as a+dk+cj+bi, giving the following matrix.

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

This matrix is conjugate to the one above, swapping the second and fourth rows, and the second and fourth columns. Thus orthogonality and determinant are preserved.