Given a parameterized space of square matrices, the associated set of
eigenvectors forms some kind of a structure over the parameter space. When is
that structure a vector bundle? When is there a vector field of eigenvectors?
We answer those questions in terms of three obstructions, using a Homotopy
Theory approach. We illustrate our obstructions with five examples. One of
those examples gives rise to a 4 by 4 matrix representation of the Complex
Quaternions. This representation shows the relationship of the Biquaternions
with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity
Theory. The eigenstructure of this representation is very interesting, and our
choice of notation produces important mathematical expressions found in those
fields and in Quantum Mechanics. In particular, we show that the Doppler shift
factor is analogous to Berry's Phase.Comment: 22 pages, also found on http://math.purdue.edu/~gottlie