Principal Axis TheoremDate: 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/ |
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