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### Principal Axis Theorem

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Date: 12/10/97 at 04:56:03
From: Goga
Subject: Principal axis theorem (lin.alg)

Hi,

Could you give me three application examples of the principal axis
theorem? This way I hope to better understand this concept.

```

```
Date: 12/10/97 at 16:43:28
From: Doctor Tom
Subject: Re: Principal axis theorem (lin.alg)

Well, here's one; perhaps other math doctors can supply others.

Suppose you've got a population of animals that you divide into four
age categories: 0-1, 1-2, 2-3, over 3 years old. We can write down the
structure of the population by giving the proportion of animals in
each age category, say p0, p1, p2, and p3 represent the number in each
of the age classes above.

Further, suppose that every year, none of the 0-1 individuals
reproduce, 40 percent of those in the 1-2 category reproduce, 50
percent of those in the 2-3 reproduce, and 10 percent of those
over 3 reproduce.

Also assume that 30 percent of the 0-1 die in that year, 10 percent of
the next two categories die, and 50 percent of the over 3 individuals
die.

You can write a matrix that you multiply (p0 p1 p2 p3) by to indicate
the population structure after one year:

|  0  1-.3  0      0   |    | 0.0 0.7 0.0 0.0 |
|  .4   0   1-.1   0   | =  | 0.4 0.0 0.9 0.0 |
|  .5   0   0     1-.1 |    | 0.5 0.0 0.0 0.9 |
|  .1   0   0     1-.5 |    | 0.1 0.0 0.0 0.5 |

(To see what's happening, individuals either make some babies, which
will generate stuff in the first column, and if they survive, they
advance to the next group, unless they were already in the oldest
group.)

To figure out the population structure over many generations, simply
multiply the initial vector repeatedly by the matrix. (You will have
to re-normalize to get the proportions to add to 1 if you want them to
be true probabilities, but if you simply think of them as ratios of
numbers of individuals in the categories, you don't need to do that.)

If this matrix has a unique largest eigenvalue (and thus a
corresponding unique principal axis), then the axis represents the
limiting population ratios to which any initial population
distribution will tend. If the initial population already has this
exact distribution, it will be unchanged in every generation.

(All of this, of course, assumes infinite (or very large) populations,
and in reality things will wiggle a bit, but in a theoretically
infinite population, this is exactly what would happen.)

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Linear Algebra
High School Linear Algebra

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