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<   No. 2011   2008-07-29   >

Comic #2011

1 Haken: Good work Erwin.
1 Erwin: {climbing back into the cabin of the truck} I... tried to save him.
1 Haken: Why?
2 Erwin: Er... How will we ever find another rival to match your talents, Herr Kolonel?
3 Haken: My talents are not matchable!
4 Erwin: Sorry, Herr Kolonel. Of course. Die world is fortunate that you are unique...
4 Haken: That is better!

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An Eigen Plot is a story plot that is carefully designed so that every skill of every one of the protagonists, no matter how unlikely or normally useless a skill it is, is needed at some point to overcome an obstacle that would otherwise prevent the heroes from achieving their goal. There are many examples of such plots in fiction. The whole point being, of course, to make sure that each character has something important to do in the story.

The name "Eigen Plot" is taken from a set of concepts in mathematics, namely those of eigenvectors, eigenvalues, and eigenspaces.

Recall from the annotation to strip #1640 that a function is merely a thing that converts one number into another number. But actually, that's not all a function can do. You can extend this definition to say that a function is something which converts any thing into some other thing.

For example, imagine a function that converts a time into a distance. Say I walk at 100 metres per minute. Then in 3 minutes, I can walk 300 metres. To find out how far I can walk in a certain number of minutes, you apply a function that multiplies the number of minutes by 100 and converts the "minutes" to "metres".

That's a pretty simple example, and we could have done pretty much the same thing with a function that simply converts numbers by multiplying them by 100. But let's imagine now that I'm standing in a field with lots of low stone walls running in parallel from east to west. If I walk east or west, I am walking alongside the walls, and never have to climb over them. So I walk at my normal walking speed, and I still walk at 100 metres per minute. But if I walk north or south, then I have to climb over the walls to get anywhere. The result is that I can only walk at 20 metres per minute in a northerly (or southerly) direction.

So, put me in the middle of this field. Now, what function do I need to use to tell me how far I can walk in a given time? You might think that I need two different functions, one multiplying the time by 100, and another multiplying it by 20. But we can combine those into one function by making the function care about the direction of the walking. This is a perfectly good function:

If I am walking east or west, multiply the time by 100 and convert "minutes" to "metres"; if I am walking north or south, multiply the time by 20 and convert "minutes" to "metres"
What happens if I want to walk at an angle, like north-east? If the walls weren't there, I could just walk for a minute to the north-east, and end up 100 metres north-east of where I started. To figure out the equivalent in the walled field, let's break it up into walking east for a bit, and then north for a bit. On the empty field, I would need to walk east 100/sqrt(2) metres (about 71 metres), and then north 100/sqrt(2) metres. In terms of time, the equivalent of walking for 1 minute north-east is walking for 1/sqrt(2) minutes (about 42.4 seconds) east and then 1/sqrt(2) minutes north. It takes me longer (because I'm not walking the shortest distance to my destination), but the end result is the same.

But what happens if I walk for 1/sqrt(2) minutes east and then 1/sqrt(2) minutes north on the field with the walls? I end up 100/sqrt(2) metres east of where I started, but only 20/sqrt(2) metres north of where I started (because the walls slow me down as I walk north). The total distance I've walked is only a little over 72 metres, and my bearing from my starting position, rather than being north-east (or 45° north of east) is only 11.3° north of east. So the function I've described above converts a time of "1 minute north-east" into the distance "72 metres, at a bearing 11.3° north of east". An interesting function!

Here's where we come to the concept of eigenvectors. "Vector" is just a fancy mathematical name for what is essentially an arrow in a particular direction. A vector has both a length and a direction. So a vector of "1 metre north" is different from a vector of "1 metre east". And "1 metre north" is also different from a vector of "1 minute north", by the way. Okay, now we know what a vector is. What's an eigenvector?

Firstly, an eigenvector is a property of a function. Eigenvectors don't exist all by themselves - they only have meaning in the context of a particular function. That's why I went to all the bother of describing a function above. Without a function, there are no eigenvectors. Some functions have eigenvectors, and some functions don't. Okay, ready?

An eigenvector of a function is a vector that does not change direction when you apply the function to it.
An example eigenvector of our function is "1 minute east". The function changes this to "100 metres east". The direction has not changed, therefore "1 minute east" is an eigenvector of our function. Similarly, "1 minute north" is an eigenvector, because the function changes it to "20 metres north". However, "1 minute north-east" gets changed to "72 metres, at a bearing 11.3° north of east" - the direction of this vector is changed, so it is not an eigenvector of our function. In fact, vectors pointing in any direction other than exactly north, south, east, or west are changed in direction by our function, so are not eigenvectors of the function.

But the vector "2 minutes north" gets changed to "40 metres north". So "2 minutes north" is also an eigenvector of the function, just as "1 minute north" is. You've probably realised that any amount of time in the directions north, south, east, and west are also eigenvectors of the function. Our function has an infinite number of different eigenvectors.

Now we're ready to talk about eigenvalues. Again, eigenvalues are properties of a function. They don't exist without a function.

An eigenvalue of a function is the number that an eigenvector is multiplied by in order to produce the size of the vector that results when you apply the function to it.
What are the eigenvalues of our function? Well, if I walk east (or west), we need to multiply the number of minutes by 100 to get the number of metres I walk. This is true no matter how many minutes I walk. So the number 100 is an eigenvalue of the function. And if I walk north (or south), I multiply the number of minutes by 20 to get the number of metres. So the number 20 is an eigenvalue of the function.

These are the only two eigenvalues of the function: 100 and 20. I don't have to worry about vectors in any other directions, because none of those vectors are eigenvectors. And now we come to the idea of eigenspaces. Once more, eigenspaces are properties of functions.

An eigenspace of a function is the collection of eigenvectors of a function that share the same eigenvalue.
In our case, all of the eigenvectors east and west have the eigenvalue 100. So the collection of every possible number of minutes east and west forms one of the eigenspaces of the function. Similarly, all of the eigenvectors north and south share the eigenvalue 20, so make up a second eigenspace of our function. Considering the sets of eigenvectors together, the two eigenspaces of our function are a line running east-west, and a line running north-south, intersecting at my starting position.

[As an aside, let's examine briefly the walking function in a field without walls. In that case, no matter what direction I walk, I walk 100 metres per minute. "1 minute north-east" becomes "100 metres north-east". So every time vector ends up producing a distance vector in the same direction, with the number of metres equal to 100 times the number of minutes. In other words, every time vector, in every direction, is an eigenvector, with an eigenvalue of 100. This function has a lot more eigenvectors that the walled-field function, but only one eigenvalue! Which means there is only one eigenspace, but it isn't just a line in one direction, it's the whole field!]

You'll notice that the two eigenspaces of the walled-field walking function (the north-south line and the east-west line) end up aligned very neatly with the structure of the walls in the field. In general, this is true of many interesting functions in real life applications such as physics, engineering, and so on. In many fields, there are functions that describe various physical properties of things in multiple dimensions.

An example is the stretching of fabric. If you pull a fabric with a certain amount of force, it stretches by a certain amount. The amount it stretches is actually different, depending on what angle you pull at the fabric. (Anyone who does knitting or sewing will know this is the case.) You can describe this with a function that depends on the angle at which you apply the force. Once you have this function (and you could get it, for example, by actually measuring the stretching of a piece of fabric with various different forces and angles), you can calculate the eigenvectors, eigenvalues, and eigenspaces of the function. And it turns out that the eigenvectors, eigenvalues, and eigenspaces of the stretching function will be closely related to actual physical properties of the fabric, such as the directions of the weave.

A very similar case happens in three dimensions with the properties of rocks. By using seismic probing, core sampling, and other techniques, geologists can build up a three-dimensional analogue of a "stretching function" for rocks. And by calculating the eigenvectors of this function, they can figure out the directions of various features within the rocks, such as fault lines. (This is rather simplified, but that's one of the basic principles behind experimental geophysics.)

We have some loose ends to tie up. Where does the name "eigen" come from? This is actually a German word, meaning "own", in the adjectival form. So an eigenvector is an "own-vector" of a function - a vector that belongs to the function. As you can see, this is a very fitting description. The prefix "eigen" has come from this usage into (geeky) English to refer to other characteristic properties of things. One example is eigenfaces, which refers to the characteristic facial feature patterns used in various computerised facial recognition software functions.

Another example is the TV Tropes coinage of the trope Eigen Plot, which began this annotation. This is an abstracted usage, playing off the "characteristic property" implication of the "eigen" prefix. The characters in a story have a certain set of skills, so an eigen-plot is a plot that is characteristic of that set of skills (i.e. one that will exercise all of the skills).

This anotation has no relevance to today's strip, other than that Colonel Haken and Erwin are German, like the word "eigen".

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