Small matrices

There is a somewhat similar algebra applicable to $2\times 2$ real matrices, which are of frequent occurence in the theory of second order linear differential equations and elsewhere. However, a quaternion basis is not strictly appropriate because the squares of two of the candidfates for basis vectors are $+{\bf 1}$ rather than the $-{\bf 1}$ characteristic of quaternions. The result is that all the theory resembles that of the rotation group except for the fact that it works with a Minkowski metric rather than the accustomed Euclidean metric, which in turn means that there is an effect similar to Lorentz contraction when working with geometric concepts.

Another representation of rotations considers them to be the product of two reflections. The reflecting planes should intersect along the axis of rotation, which is invariant under each of the two reflections and consequently their composite. The dihedral angle separating the planes should be half the angle of rotation, because under the reflection points end up just as far above the plane as they originated below it. Any pair of planes with the same axis and angular aperture will produce the same rotation, inviting the selection of a common intermediate plane, whose presence disappears by squaring when composing two individual rotations, always leaving two reflections to define a rotation.

Reflections aren't a normal part of quaternion lore, but both representations are a part of the common knowledge about rotations. Even so, it is not so easy to track down the historical origins of the representation, or even to be sure that it has not arisen spontaneously with different investigators in altogether different locations. For example, knowing that the exponential of an antisymmetric matrix is orthogonal usually sufficies to deduce Rodrigues' formula, the one which figured in our investigation of Putzer's method in Section 3. Going from that point to the vector representation is harder, making a historical analysis more significant if one were actually interested in origins.

Insofar as my own involvement in the representation is concerned, the question of multiplying $2\times 2$ matrices to solve cascaded transmission line problems and later on to solve differential equations goes back to my graduate school days. Involvement with the rotation group goes almost as far back, both from an interest in group theory, and trying to express a certain practical problem in terms of spherical harmonics. Articles in the physical literature, particularly in American Journal of Physics, frequently spoke of Pauli matrices, quaternions, and other topics.

Before long, differences between the rotation group and the unimodular matrices emerge. The group SL(2,R), as it is known, is locally compact but not compact, so it has infinite dimensional irreducible representations. These were studied by E. P. Wigner in the late 1930's [98], V. Bargmann a decade later [97], and are the principal focus of Serge Lang's book [96] of 1975. But is is harder to find treatments in the the group itself, rather than its representations. Over the years I have looked for a graphical representation equivalent to that of the rotation group. Although such a thing exists and seems to be relatively simple, it is just recently that a clear picture has emerged.

First we repeat some of the background details, taken from lecture notes and various places in the literature.