Wednesday, October 24, 2012

Singular Value Decomposition Delights, Part 2

First off, congratulations to math homie and guest blogger Adam on his advancement to candidacy! Now back to our explorations from last week. So the goal of the previous vector bundle problem is to consider a vector bundle over the circle, represented as planes over an interval such that the fibers at one end of the interval are identified with the other end using some linear transformation T. In order to visualize this, we shall adopt the following approach: we put our base space circle in the xy-plane. arrange the fiber planes perpendicularly to circle (planes θ = constant), like so:

and then we choose a reference circle in one of these planes (say, of radius 1). Then as we move around the main circle, we consider the surface swept out as we carry that circle with us and apply the interpolated transformation

A(t) = exp() Σtexp(–)

to that circle as we go. When we come around to the beginning, this circle should be deformed by the original transformation T. We do the experiment with the matrix

0.498502 0.214052
0.0433899 0.60181

which is a rotation of -2.62394 radians (-150.3°), then stretching with singular values 0.69543 and 0.41803, and finally applying a rotation of 2.24233 radians (128.5°). We start with the identity matrix, and as we move around, rotate the appropriate fraction t, times of the first angle, stretch by the second factor to the t power, and finally rotate by the same fraction of the second angle. So at time = 1, a full revolution about the main circle, we get full operator T applied to it. This is the surface swept out:

And if we kept going, it would spiral inward more and more, which when doing a cutaway view (essentially, starting only with, say, 3/4 or 1/2 of the initial circle), we get the original picture shown in Part 1. But actually, it is more interesting to see what happens if we just unfurl it in a straight line (in the spirit of the original definition as an identification space)... and let the mapping continue past t = 1. The result is the title picture, which shows what happens when we proceed to t = 3 (which corresponds to the transformation T3). Forget donuts, let's do pasta (more comprehensively, here)!

Of course, we know that this shows our bundle is trivial—the only way to get a nontrivial bundle this way is to have a transformation with a negative singular value, in which case, since we can't raise negative numbers to real powers and stay in the reals, makes the above construction fail, as it should (since a nontrivial bundle fails to have a continuous global frame). So what good is this, really, besides making things pretty? As it turns out, such oddly behaving, yet topologically trivial bundles, do have some interesting, different geometry. It is encoded in the notion of connection—a connection will reveal nontrivialities in the geometry that topology cannot (in fact, Riemannian geometry starts out by considering a connection specifically on the tangent bundle, the Levi-Civita [LAY-vee CHEE-vee-tah] connection). This is an interesting topic for yet another part (I'm not promising it soon, however!).

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