These notes started as a simple exploration of the meaning of complex angles. However, they quickly blossomed beyond recognition and required separation into at least three parts. Two of these parts will cover the general computation of angles between subspaces of R^n and the realification/complexification of vector spaces. These are nearly complete, but need some tidying up before release. The present set of notes covers, often in gory detail, a variety of topics in basic linear algebra that are often glossed over. We look at bases, orthonormality classes of bases, matrix-preserving basis-changes for a variety of matrix types, unordered bases, permutations of bases, and the detailed transformation rules for many types of objects.
Finally, we reach our main topic: complex angles. We look at a variety of possible definitions, and conclude that — of the obvious candidates — only one definition of angle has the required features. It is also the only part of the Hermitian inner product to survive projectivization. From a number of standpoints, this turns out to be a meaningful choice. It is what is known as the “quantum angle”. Our discussion serves to explain and justify the use of an otherwise eclectic-looking formula.
As usual, these notes are homegrown and probably contain some typos. Hopefully, nothing more substantive needs attention — but this is always a possibility. I would be grateful for any feedback or comments. I’ve provided the notes in four formats: Color/B&W and letter/abook size.
A Place of Sand 