I had not encountered Tom Karzes' problem before, and currently have no intelligent and relevant comment to offer; but since this topic has arisen, I shall seize the opportunity to rabbit on about it being one of those not uncommon (double negative --- hah!) situations wherein resides the temptation to regurgitate some standard piece of college mathematics (here decomposition of a real orthonormal matrix into block diagonal form), then trot merrily away under a comforting delusion of having actually grasped what is going on. Cloaked in invisibility, the resulting lacunae stalk one's intellectual landscape at large, assured of perpetual persistence. So here's a little "something for the weekend, sir?" (long ago, the salutation delivered by a hairdresser on conclusion of his labours, in traditional nauseatingly confidential and lugubriously creepy fashion). What lugubrious features of their corresponding (projective) matrices distinguish the following nauseatingly traditional and creepily continuous transformations in (say) Euclidean 3-space? (A) Rotation by an angle about a coline; (B) Translation by a distance along a vector; (C) Dilation by a ratio with respect to a centre point; and (rather less common) in 4-space, (D) Transport along Clifford parallels (aka Hopf fibration). The answers provide (part of) a rationale explaining exactly how these various types arise in the first place, along with what additional exotica might lurk unintuited in other quadratic spaces. At the same time, it does need to be emphasised that matrices provide a less than ideal framework for such an investigation, compared to my preferred option of Clifford algebra. Fred Lunnon On 3/12/16, Tom Karzes <karzes@sonic.net> wrote:
I agree Dan, the "canonical form" of an orthogonal matrix is a nice result. In three dimensions, it basically just says that any rotation can be defined as rotation about some one-dimensional axis. To bring it into canonical form, one need merely rotate that axis onto, for instance, the Z axis, confining the rotation to the XY plane. In higher dimensions, it's a little more complicated.
I remember you mentioning your work in visualizing N-dimensional data, and your "grand tour" method before. My situation is very similar. I want to specify a sequence of Givens rotations that is capable of supporting universal rotation in N-space, and while it is easy to find such sequences, it is not clear to me whether any such sequence works vs. only some sequences. I don't *need* to know the answer, but I'd like to.
Honestly, when I posted this question I thought I'd immediately get several replies saying "Oh, don't you know that? Didn't you take freshman linear algebra?" But it seems like the answer isn't common knowledge after all. I'm honestly wondering if the answer is even known. Seems hard to imagine that it wouldn't be (or at least, that the question wouldn't be well known).
Tom