angle and axis of rotation 2: the two correct answers

Edit 19 Sept 2010. I strongly recommend using the eigenvector rather than trying to make a unit vector out of the off-diagonal terms. Find “edit” below.

Back in what has turned out to be my most popular post, axis and angle of rotation, I showed how to switch between the angle and axis of rotation and the rotation matrix.

Given the axis of rotation as a unit vector (a, b, c), and the angle of rotation, we construct the matrix N

N = \left(\begin{array}{lll} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right)

and then we construct

A = I + N sin \theta + N^2 \left(1-cos \theta \right)

and that’s a rotation. To be specific, it is an attitude matrix for a rotation of coordinate axes, about the axis (a, b, c) through the angle \theta\ .
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PCA / FA Example 7: Bartholomew et al. Correlation matrix

edit 5 Oct 2008: I had omitted the word “constant”. see edit.

The following example comes from Bartholomew et al. “The Analysis and Interpretation of Multivariate Data for Social Scientists.”

It is an excellent example with which to wrap up PCA / FA. (There’s a lot we haven’t done, but it’s almost time for me to move on.)

The example is “employment in 26 European countries”, “eurojob” for short, from chapter 5 (either 1st or 2nd edition), and data for both editions is available at http://www.cmm.bris.ac.uk/team/amssd.shtml . Please note that I am using the 1st edition of the book, and the 1st edition data.

When I first worked this example, I knew that something interesting happened, but not why; and, there was one thing I didn’t understand at all, back then.
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Happenings – 18 Sept

I may very well start putting posts out here real soon now, but I might as well warm up by talking about where the hell I disappeared to.

Nowhere.

And everything is just fine.

All that happened was that I got distracted by other things. There are always potential distractions, but I don’t let too many of them grab me unless I’ve lost my way in the mathematics. Some things ganged up on me, and some things I welcomed.
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The Euler characteristic (triangulations 2)

I want to talk about some things I saw when I was looking at triangulations. This also continues my reading in Bloch.

The major thing I saw was the Euler characteristic. We usually define it for polyhedra, as the alternating sum / difference of the number of vertices, edges, and faces (0-, 1- and 2- simplices)…

\chi = V - E + F

Then we would define it for a surface by taking the Euler characteristic of any polyhedron which is homeomorphic to that surface.

For that to be well-defined, of course, requires that all polyhedra which are homeomorphic to a surface S have the same Euler characteristic (as they do).

By the same token, we could define the Euler characteristic of a surface from any triangulation of the surface, after we show that all triangulations of a surface have the same Euler characteristic. Oh, and we’d better actually prove that every topological surface (every topological 2-manifold) can be triangulated.

Yes, they can be. Not true for topological 4-manifolds, and I think it’s still wide open for higher dimensions. In contrast, I believe that every differentiable n-manifold supports a unique piecewise linear (PL) structure (which is the generalization, they tell me, of triangulations).

Details.
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Correction to Triangulations of Aug 12

I made a single large edit to the post, trying to clarify the Heawood theorem as opposed to the Heawood conjecture. This is mostly history of mathematics, but the distinction as I originally posted it was unreasonable.