[Series intro] An introduction to geometric algebra.
[Series intro] An introduction to geometric algebra.
What’s in the pipe.
It’s been a while since I did any math or physics writing. This is the first post in a series where I plan to work my way systematically from an introduction of vectors, to the axioms of geometric algebra. I plan to start with an introduction of vectors as directed “arrows”, building on that to discuss coordinates, tuples, and column matrix representations, and representation…
An inner automorphism is three actions: A B A⁻¹
Sometimes the initial action A and its subsequent reversal A⁻¹ change the overall result ("raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain"),
and sometimes they do not ("take off left glove, take off right glove, put on left glove" has the same effect as "take off right glove only").
Wikipedia, edited by me
The most common inner automorphism for statisticians, I think, has to be:
P = X(XTX)−1XT is the projection matrix which we use for regressions.
Quantum SHO ladder operators as a diagonal change of basis for the Heisenberg EOMs
Quantum SHO ladder operators as a diagonal change of basis for the Heisenberg EOMs
[Click here for a PDF of this post with nicer formatting]
Many authors pull the definitions of the raising and lowering (or ladder) operators out of their butt with no attempt at motivation. This is pointed out nicely in [1] by Eli along with one justification based on factoring the Hamiltonian.
In [2] is a small exception to the usual presentation. In that text, these operators are defined as…
transformations on transformations on transformations
part 5 of linear algebra (toc)
stacks on stacks on stacks. racks on racks on racks. cats in hats on knox in box. Hey classy people, we're rocking diagonalisation and markov chains today! (That means numbers. great.) Inherently included is matrix representation and the change of basis matrix mindboggle... I'll briefly touch on those.
First of all, let me just give it to you straight in terms of how matrix representation works. We pick a standard basis. Conventionally, we use e1, e2, e3, ... etc (depending on the dimension of your space) which are just 0 vectors with "1" in the ith spot. Then we form the matrix by depositing T(e1) in the first column, T(e2) in the second column, and so on. Let's call this matrix form of T, the transformation, as A. So A is one way to represent T as a matrix.
Suppose now we decide we didn't like our original basis and we want to use a different (arbitrary) basis, say f1, f2, f3, etc. Then we have to make a "change of basis matrix", which just involves writing e1, e2, e3, etc in terms of f1, f2, f3, etc and using the coefficients as the entries in the ith column. Let's call our change of basis matrix B. Notice B is an isomorphism (yay! I know what that means now!) so we can think about its inverse, B-1. Turns out (quite intuitively) that this matrix is just the change of basis matrix in the reverse direction; ie, it takes vectors in terms of our f basis and changes them back to vectors in terms of our e basis. So now we're in shape to take vf, a vector written with the f basis, and transform it with BAB-1 (notice we read matrix multiplication right to left, so first we're going to take vf, switch it into ve, apply the transformation written in terms of e, then switch the resulting vector back into the f basis). What that basically tells us is that BAB-1 is the matrix form of T in terms of the basis f.
So we're now good to go in terms of matrix representation of a linear transformation, T, in any kind of basis we want. Why is that knowledge useful?
Here's the motivation: think about a system that we can represent with a matrix of values, S. Every time period, the system changes in a predictable way, modelled by the linear transformation T. As time runs on, we get a compounding of these changes (say, like Conway's Game of Life). <insert something about stochastic processes and markov chains here.> We want to know what the state of the system is at some point in the future, like t = 10 or t = 1000 or t = n in general. What we'll get is Tn(S). If S were a huge matrix, like 800x800, then doing matrix multiplication n times would really stink. There has to be a faster way.
Well...if we're lucky, there is. Recall that we can find eigenvectors of T, so that T(vi) = bivi for all eigenvalues bi. What happens if we have dim(T) eigenvectors? Well, since all eigenvectors are linearly independent, we have a basis of eigenvectors! In this case, T in terms of our basis of eigenvectors is just the diagonal matrix with bi in the ith diagonal spot. Let's use this matrix representation of T under the eigenvectors as A. Then, B would be the change of basis matrix from the standard basis to the basis of eigenvectors: which is just a matrix of all the eigenvectors as columns. Finding B-1 just requires a little bit of inversion to obtain. So now, to write T in the standard basis, we just use T = B-1AB.
I know I'm getting long winded so I'll get to the punch. Writing T as a composition of these matrices instead of directly as its standard basis representation is helpful because now compounding T is a snap. Let's try T2 = B-1ABB-1AB. Matrix multiplication, although not commutative, is fortunately associative! Phew! So we just get B-1A2B. In general, Tn = B-1AnB, which is not too bad to see. Hopefully my point is becoming clearer now: A was just a diagonal matrix, so raising it to the exponent n just means applying the exponent to all the elements on the diagonal (everything else is 0). That's way easier than our previous n times matrix multiplication! What a whole lot of hooplah for nothing! After we raise A to the n (with no sweat like a boss, I might add), then we just use the change of basis matrices to get back to the standard basis.
Okay okay, now onto the reservations. In general, we're pretty lucky if we get a matrix with a full eigenspace. Usually we can't find a basis of eigenvectors, because we don't have enough. In these cases, we just have to make do with "Jordan-Canonical form", which is a cheapskate skimpy way of partially diagonalising a matrix. If that ever comes up, just read the chapter in the book again... ... ...because I honestly don't remember much when it comes to good ole Jordon-Canonical.
I think that about wraps up all I wanted to say about linear algebra for the time being. (Oh yeah there was some cool things about inner product spaces and pseudo-inverses...but again, just read the book, future me!) Look forward to some more abstract algebra and complex analysis -related posts... and maybe some topology ones too if I get back to that Choquet book!
