The way it should be: “Imagine: a wild pig comes upon a factory farm, and decides to stay?”
This is easily the most wonderful story I’ve heard about farming:
Dan Barber: A surprising parable of foie gras
The most spectacular part starts at 10:10, but the entire thing is amazing.
If is a scalar and is a square matrix, it is very convenient to be able to write . Usually people know immediately what this means, but are uneasy about “abusing” notation, so here’s the detailed justification for why this is perfectly legitimate:
Matrices should be considered first and foremost as linear transformations. You know what a matrix is if you know what it does to vectors. A scalar is also a linear transform on vectors: multiplying a scalar times a vector is a linear operation. Therefore, scalars can also be thought of as linear transformations, and therefore as matrices. It is immediate which matrix the scalar should be: the result of multiplying by a scalar is that all components are scaled by ; the matrix that does this is just .
Here’s another way of saying that. Let be an associative algebra with unity over a field . We can define a homomorphism of algebras via . is an isomorphism since if , implies , a contradiction. is also the unique nontrivial homomorphism from to , since algebra homomorphisms must send to either or . Therefore, there is a unique copy of inside , and we can identify with that copy.
This is a general principle: no one writes when is real and is complex, because the complex numbers are considered a superset of the reals. The same is true of matrices: once we identify scalars with scalar diagonal matrices, matrices are just another superset of the reals.
We’ve decided to give up reading In Search Of Lost Time. Here is a fairly accurate description of why:
He’s not joking: it really is like that. Interesting, yes, but not nearly interesting enough.
Here are some incredibly beautiful images:
http://www.ted.com/index.php/talks/george_smoot_on_the_design_of_the_universe.html
It feels like I didn’t quite realize how large the universe was. I was imagining that it was very large but mostly empty, trailing off out beyond the galaxies. The universe does not look empty in those pictures; it looks full. Somehow a full universe seems much larger than an empty universe.
A week ago someone asked whether the singular values of a general (real) matrix are the magnitudes of its eigenvalues. There are various ways to see that the answer is no, but here’s an amusingly nonconstructive proof:
Consider a random matrix taken from with some smooth distribution. With probability 1 all singular values of will be unique. However, with nonzero probability will be near a rotation matrix and will have a complex conjugate pair of eigenvalues with the same magnitude. Therefore, the singular values of are not always the magnitudes of the eigenvalues.
Latex formula support is provided via itex2MML and Jacques Distler’s excellent WordPress plugin. Frederick Leitner modified these for WordPress 2.0.0/2.0.1. Unfortunately, WordPress 2.6 is different again, so I had to tweak a bit further. Here are instructions:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >→ correctly.Warning: itex2MML produces MathML, which is not readable by a many browsers. Hopefully this situation will improve, but until then this blog will be nonportable.
Basic math support appears to be working, so now I have a blog.
Here’s a test of inline math () and an integral:
Success! I’ll provide details of the mathml/itex setup in a separate post.
