#diagonalisation

[tf-tumbler]

Diagonalisation of a 3x3 matrix

[rgplus]

Variant No. 1: "The Diagonalization Argument"

A multiplayer variant of my original horizontal writing method, inspired by the method that Georg Cantor published in 1891 for his proof of the existence of infinite sets that cannot be put into 1-to-1 correspondence with the set of natural numbers .

This writing method takes the form of a 2-dimensional exquisite-corpse game for 2 or more writers.  The rules are extremely simple:

Diagonalisation of Horizontals: For 2-to-N players, sequentially assign each player the set of cells along each successive diagonal in the spreadsheet template (see above image).  

1st Move: Player 1 writes one paragraph in the upper-left cell of the spreadsheet to begin the first chapter of the novel.  

2nd Move & Onward: Player 2 may now write either the paragraph that's one cell to the right of the first paragraph (the 1st paragraph of Ch. 2) or the paragraph one cell below (the 2nd paragraph of Ch. 1).  

No Illegal Moves: Players may fill only those empty cells that are adjacent to completed cells—either directly below a completed cell (to continue an existing chapter) or to the right of a completed cell in the first row (to start a new chapter).  

No "Turns" / Simultaneity Encouraged: "Turns" exist only as assigned squares in the spreadsheet.  Players are encouraged to add content simultaneously if spreadsheet software (e.g. Google Drive) allows for real-time collaboration.  (The number of cells available for each player will increase steadily as the game progresses horizontally.)  

No Backediting: Players may edit their completed cells until another player in the game begins to write a neighboring cell.  Once a cell has been "buried", it may not be altered until the draft is complete (exception:  unanimous agreement among all players.  

2+ Paragraphs/Cell is OK: A "paragraph" in the game is defined as the contents of single cell, and not by any grammatical construct.  If players wish to write more than one paragraph in a cell, a suitably unique line-break designator must be used (e.g. "[[BR]]") to ensure that proper line breaks may be inserted once the novel is transferred to a linear format (i.e. plain text rather than spreadsheet document) for reading and editing.  

The Only Competition = Race to Complete Available Assigned Cells:  This writing game was designed for cooperative/collaborative writing—not to produce a nightmarishly paced, unreadable work of fiction (though such an outcome remains a distinct possibility for undisciplined players).  Before the game begins, standards (and penalties) must be established to combat subversion and sabotage w/r/t individual players' plot ideas—else, the project will descend quickly into formless chaos.    

No Intentional Misdirection: Intentionally misleading one's collaborators is forbidden implicitly w/r/t the game's collaborative-creative goals.  

Why So Serious?: Either (or both) of the two prior rules (8 &/or 9) may be ignored if all players agree at start of game.  

Write Now, Edit Later: Once draft is completed, players may edit as a group, divide up chapters to edit individually, designate one player to serve as editor, or delete the file and start again.

    ProTip: Add cleverbot as a player.

[whatilearninmath]

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!

[themodernicide]

In which I dream of the after-écrits...

Les after-écrits, baptised as such by an over enthusiastic Ludivine, is the post-written concours period... a mythical stretch of time between mid-may and mid-june, when the results of the written exams aren't out yet and you're prepping for the interviews.

And so on this dreary Thursday afternoon I'm dreaming of the after-écrits instead of assimilating the fact that eigenvectors of stochastic matrices can be hyperplanes making them non-diagonalisable. I was surprised too. I swear I looked at my drafts with wide eyes and gawked for a few seconds before going: "Really? REALLY?" But I guess it does make sense. Anyway... let's not get into that discussion.

So the after-ecrit period is already shaping up, I finish exams on the 15th of May and that very afternoon, the entire year is being sent for a Spa getaway in Belgium for a week. They want us to recharge our batteries I guess. I've been doing a little planning on my own, and I'll be in London and Oxford for a long weekend from the 25th till the 29th of May. I do hope there aren't mock interviews that weekend! *crosses fingers*. We aren't being given the planning of our oral-preparations till after the concours, but I can't wait till then to book my travel tickets so they'll have to do without me.

In the meantime, this period will be a great opportunity to really sort out my things, get rid of old stuff, clear through my clothes. I'm a bit of a clothes hoarder, and though I gave off two huge bin bags of clothes to my local Ding-Fring, I still have loads of stuff I barely touch.

Then there are all the books I want to read, all the films I want to watch, all the DJs I want to see! I also think it's time I gave a makeover to Modernicide, with it's douchey pretentious name, I do think it deserves an equally pretentious layout, especially now that I'll have far more time to dedicate to it... and I'll be doing more interesting things to blog about.

Can you imagine how Modernicide would have looked had I honestly blogged about my day to day life up till now?! Math nerds across the globe would have been putting bullets through their head! No no no, I couldn't possibly let readers know what I ACTUALLY spend my time doing ... it's far to distressing... 

Ahh my break seems to be up...