Principal Axis Theorem

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. 

Thanks greatly for your help.

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/