ChapterZero

I’ve been reading the first chapter of Bhatia’s “Matrix Analysis” to review (multi)linear algebra, and it’s been a pleasure. In just four pages or so, he proves the existence of QR decompositions and Schur decompositions, gives the Cholesky Decomposition as an exercise, proves the Singular Value Decomposition, and gives the Polar decomposition and its equivalence to the SVD as an exercise. The brevity of the proofs are the consequence of them being the most straightforward arguments I’ve seen for proving these decompositions. The exercises which occur inline with the text are challenging, but doable; as an examples

If is a contraction, show that

All in all, this is a great chapter– I can’t wait to work through the problems at the end.

Spring break started yesterday, and I’ve been doing a little reading on topics that have reared their heads over this term: differential geometry, discrepancy, VC dimension, and pattern analysis. Or browsing rather, since to actually absorb this material at a serious level, I’d have to spend way more time on it than I’m willing to. I’m on break, after all.

I’m going to list all my sources here so I can continue this line of inquiry at my leisure in the future. Maybe others will find them interesting (the last three are available online):

• Introduction to Smooth Manifolds. Lee
• A Course in Differential Geometry. Aubin
• Optimization Algorithms on Matrix Manifolds. Absil, Mahony, Sepulchre
• The Discrepancy Method. Chazelle.
• Kernel Methods for Pattern Analysis. Shawe-Taylor, Cristianini
• A Tutorial on Support Vector Machines for Pattern Recognition. (paper) Burges
• Introduction to Statistical Learning Theory. (paper) Bousquet, Boucheron, Lugosi
• Concentration-of-measure Inequalities. (lecture notes) Lugosi

I’m also looking for a good reference (preferably a survey paper) on spectral partitioning, the Fiedler vector, and all that magical stuff. I’ve found some stuff that mentions them, or uses the results, but no proofs.

Here’s an interesting question:

Let where Is compact?

I won’t say where this problem came from, in the interest of making it a little harder, but I will say that I couldn’t solve it (in my defense, I didn’t try very hard ). However, someone did show me a really elegant solution…

I’m reading a paper by one of the professors I’m interested in working with, on the conditioning of random submatrices. Hopefully tomorrow I’m meeting with him to discuss it. I could definitely see myself working with him, because the area he’s working in– probabilistic numerical algebra, if I had to name it– is at the intersection of several fields. Not only would I be working on problems I find interesting in themselves, but I’d be gaining a foundation that would allow me to diversify my research interests in the future. On the one hand, there’s the obvious and broad route that segues into more traditional probabilistic studies, and on the other, there’s a connection to optimization on matrix manifolds, and through that, optimization theory in itself, or differential geometry.