Matrices and Society: Matrix Algebra and Its Applications in the Social Sciences

by Ian Bradley, Ronald L. Meek

Overview

Matrices offer some of the most powerful techniques in modem mathematics. In the social sciences they provide fresh insights into an astonishing variety of topics. Dominance matrices can show how power struggles in offices or committees develop; Markov chains predict how fast news or gossip will spread in a village; permutation matrices illuminate kinship structures in tribal societies. All these invaluable techniques and many more are explained clearly and simply in this wide-ranging book.

Originally published in 1986.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691610207
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Princeton Legacy Library , #501
Pages:	240
Product dimensions:	6.10(w) x 9.00(h) x 0.50(d)

Read an Excerpt

Matrices and Society

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By Ian Bradley, Ronald L. Meek

PRINCETON UNIVERSITY PRESS

Copyright © 1986 Ian Bradley and Ronald L Meek
All rights reserved.
ISBN: 978-0-691-08454-1

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CHAPTER 1

Matrices and How to Manipulate Them

What Is a Matrix?

You are at home in the evening; there is nothing good on television; and you are at a loose end. There are three possibilities open to you: to go out to the pub, to go out to the theatre, or to stay at home and invite some friends round for a game of cards. In order to weigh up the comparative advantages and disadvantages of these three alternatives, you decide to put certain basic facts about each of them down on paper. And, being an orderly, methodical type you put them down in the form of a table, like this:

Motoring Admission Liquor Crisps
(miles) charge (£s) (pints) (packets)

Go to pub:     3
1     4
3
Go to theatre:     2
3     1
1
Cards at home:     0
0     12
9

If you went to the pub, you would have to take your car out and drive three miles. It would cost you £1 to get in, since there is a special entertainment on there tonight, and you would also have to pay for the four pints of liquor and three packets of crisps which you calculate that you would consume on the premises. If you went to the theatre, it would cost you £3 to get in, but as compared with the pub you would save a little on motoring costs and quite a lot on liquor and crisps. If you stayed at home and asked some friends round for cards, you would have to provide a comparatively large quantity of liquor and crisps, but to compensate for this you would not have to take the car out and there would not of course be any admission charge.

Suppose now that just for fun you extracted the array of numbers from this table and put a pair of large square brackets around them, like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

No doubt much to your surprise, you would then have succeeded in constructing a matrix, which is simply a rectangular array of numbers. The individual numbers in a matrix are called its components or elements. This particular matrix, since it has three rows and four columns, is said to be a 3 × 4 matrix. If you had considered only two alternative courses of action – omitting, say, the 'Go to theatre' possibility – the matrix you constructed would have looked like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This matrix has two rows and four columns, and is therefore said to be a 2 × 4 matrix. If on the other hand you had retained all three rows but omitted one of the columns, your matrix would have been a square one with three rows and three columns – that is, a 3 × 3 matrix.

Suppose now that you had seriously considered only one possible course of action – going to the pub, say. Your table would then have consisted of a 1 × 4 matrix – that is, a single row of four numbers:

[3 1 4 3].

An ordered collection of numbers written in a single row like this is a special (and very important) kind of matrix which is called a row vector.

Suppose finally that you had been interested only in the different quantities of liquor consumption associated with the three alternative courses of action. Your table would then have consisted of a 3 × 1 matrix – a single column of three numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An ordered collection of numbers written in a single column like this is another special (and equally important) kind of matrix which is called a column vector.

Matrices in the Social Sciences

The next question, of course, is why matrices are important in the social sciences, and why so many textbooks spend so much time instructing you how to play around with them. What is it all about?

A short answer is that the social sciences are often concerned with unravelling complex interrelationships of various kinds, and that it is often extremely convenient and illuminating to put these interrelationships down on paper in matrix form.

In economics, for example, we may be interested in the implications of the fact that some of the things which an industry produces (that is, its 'output') may be used as ingredients (that is, as 'inputs') in the production of other things – or even in their own production. A large part of the electricity produced in this country, for example, is not consumed directly by you and me, but is used as an input in the production of things like corn, machines, clothes and so on – and of electricity itself. Imagine, then, a very simple economy where there are only three industries, which we shall imaginatively call A, B and C. Industry A produces 300 units of its particular product – tons of steel, kilowatt hours of electricity, or whatever we suppose it to be – every year. It sells 50 of these 300 units to itself (as it were) for use as an input in its own production process; it sells 100 units to industry B and 50 to industry C for use as inputs in their production processes; and the remaining 100 units are sold to final consumers like you and me. Industry B produces 150 units, 70 of which go to itself, 25 to A, 5 to C, and the remaining 50 to final consumers. Industry C produces 180 units, 60 of which go to itself, 30 to A, 10 to B, and 80 to final consumers.

The way in which the total outputs of the three industries are disposed of can be very conveniently set out in the form of a simple matrix like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One of the advantages of setting out the facts in this way is that two interrelated aspects of the overall situation are presented to us at one and the same time: the three rows tell us where each industry's output goes to, and the first three columns tell us where each industry's physical inputs come from.

Another type of matrix which often crops up in the social sciences is one which sets out the gains and losses accruing from some kind of 'game' which two (or more) participants are supposed to be playing. Suppose, for example, that two persons, Tom and Jerry, find themselves in some sort of conflict situation in which they are obliged to choose (independently of one another) between several alternative courses of action, and in which the final outcome – the gain or loss for each 'player' – depends upon the particular combination of choices which they make. Tom and Jerry, let us say, are two rival candidates for political office. At a certain stage in the election campaign a crisis arises, in which Tom has to choose between two possible strategies and Jerry between three. If we could calculate the numbers of votes which would be gained (or lost) by Tom – and therefore we assume, lost (or gained) by Jerry – in the event of each of the six possible outcomes, our calculations would be usefully presented in matrix form. The matrix might appear like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From this we can immediately see, for example, that if Tom adopts his first strategy and Jerry adopts his third strategy, the outcome will be that Tom will lose (and Jerry will therefore gain) 1000 votes. If Tom adopts his second strategy and Jerry adopts his second, then Tom will gain (and Jerry will therefore lose) 2000 votes. And so on. The advantage of this way of presenting the facts is once again that it puts them before our eyes simultaneously from two points of view – the rows, as it were, from the point of view of Tom, and the columns from the point of view of Jerry.

In sociology, again, we might be interested in what is called a dominance situation, in which the pattern of dominance between three individuals (Pip, Squeak and Wilfred, say) can be represented in a matrix, where the entry 1 indicates that the person whose row the entry is in dominates the person whose column it is in; at the same time an entry 0 in a row means that the person whose row it is in does not dominate the person whose column the 0 entry is in.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus the rows of this matrix show us that Pip dominates Wilfred, Squeak dominates Pip, and Wilfred dominates Squeak; and if we look at the columns we can immediately see whom Pip, Squeak and Wilfred respectively are dominated by.

Or, to take a final example, we might be interested in some kind of transition matrix, setting out the probabilities of a person's proceeding from (for instance) one social class to another in some given time period – a generation, say. Take the following matrix, which might represent the probability of the sons of upper-, middle- and lower-class fathers moving into the upper, middle and lower classes respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here the rows show us the probabilities of the son's class when that of the father is known. The son of a middle-class father, for example, has a one in ten chance of moving to the upper class, a seven in ten chance of staying in the middle class, and a one in five chance of moving to the lower class. In this case the interpretation of the columns is not quite so straightforward. Obviously the first column, for instance, does not tell us what the probabilities are of an upper-class son having an upper-, middle- or lower-class father. Such probabilities must depend on the number of fathers there are in each class.

The Manipulation of Matrices

So far, all we have done is to explain what a matrix is, and to establish that the matrix form may sometimes be a neat and convenient way in which to set out some of the interrelationships in which social scientists are interested. But if that was all there was to it, there would hardly be any need for a book like the present one. The point is that it is often useful not merely to set out the interrelationships in matrix form, but also to be able to manipulate the matrices themselves in various ways. What is meant by this?

In ordinary arithmetic, where we deal with individual numbers, we use certain simple techniques, which we all take in with our mother's milk, in order to add, subtract, multiply and divide them. Suppose, however, that you want to deal not with individual numbers but with arrays of numbers in matrix form. Suppose you think that it might be useful to treat each of these matrices as a unit, and to perform operations upon them analogous to those of addition, subtraction, multiplication and division in ordinary arithmetic. How would you go about it?

Essentially, what you would require is a set of conventions establishing what you are going to mean by addition, subtraction, multiplication and division, when you are dealing with matrices rather than with individual numbers. And the conventions you adopted would depend largely upon their convenience in relation to the particular problems which you were hoping to be able to solve with the aid of these operations.

So far as addition is concerned, the convention usually adopted is a fairly simple and commonsense one. To add two matrices of the same dimensions, or order as it is often called (that is, having the same number of rows and columns), you simply add the corresponding components. For example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here are two more examples, in the first of which we add two 1 × 3 row vectors, and in the second 2 × 2 square matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But, you may be asking, what about matrices which do not have the same number of rows and the same number of columns? How do you add them? The simple answer is that you don't, and can't. The operation 'addition', when applied to matrices, is defined in terms of the addition of the corresponding components, and is therefore applicable only to matrices of the same dimensions.

The operation subtraction is defined analogously to that of addition, and it is also therefore applicable only to matrices of the same shape. Examples:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is all easy enough, I suppose. But multiplication is defined in a different, less obvious way, and will take a little longer to explain.

First let us get what is called scalar multiplication out of the way. Sometimes we may want to multiply each component of a matrix by a single number, say 2, in which case we just do precisely that, setting out the operation as in the following example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

More often, however, as we shall see later in this book, we will want to multiply the matrix not by a single number but by another matrix, or by itself. What meaning is it most useful for us to give to such an operation?

Let us approach this problem indirectly by having another look at the matrix which we constructed at the beginning of this chapter:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to identify this matrix, let us call it A. Suppose now that you want to take a further step in weighing up the advantages and disadvantages of the three alternatives by calculating the respective total money costs involved in each of them. Suppose also that for some reason best known to yourself you want to separate out the tax element in these money costs, so that you finish up with two separate figures relating to each of the three options – one showing the total money costs of the option, and the other showing the total amount of tax included in these costs.

The motoring cost per mile, reckoned in pence, is, let us say, 20p which includes a tax element of 10p; the admission charge per £1 is (not unnaturally) 100p, which includes a tax element of 25p; the price of liquor per pint is 80p, which includes a tax element of 20p; and the price of crisps per packet is 10p, the tax element here being zero. It will be convenient to put this information in the form of a second matrix – a 4 × 2 one this time – which we will identify as matrix B:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given these two matrices A and B, it is of course simplicity itself to make the calculations you have in mind. To work out the total money costs involved in going to the pub, you multiply each of the components in the first row of A by the corresponding components in the first column of B, and then add up the four products, thus:

Cost (including tax) of going to the pub = (3 × 20) + (1 × 100) + (4 × 80) + (3 × 10) = 510p.

And to work out the tax element included in this total of 510p, you multiply each of the components in the first row of A by the corresponding components in the second column of B, and then add up the four products, thus:

Tax element in cost of going to the pub = (3 × 10) + (1 × 25) + (4 × 20) + (3 × 0) = 135p.

Similarly to work out the total money costs involved in going to the theatre, you multiply each of the components in the second row of A by the corresponding components in the first column of B, and then add up the four products, thus:

Cost (including tax) of going to the theatre = (2 × 20)+ (3 × 100) + (1 × 80)+ (1 × 10) = 430p.

And to work out the tax element included in this total of 430p, you multiply each of the components in the second row of A by the corresponding components in the second column of B, and then add up the four products, thus:

Tax element in cost of going to the theatre = (2 × 10) + (3 × 25) + (1 × 20) + (1 × 0) = 115p.

You should now have no difficulty in making the third and last pair of calculations yourself, or in seeing that the most convenient way of presenting the final results is in the form of a third matrix – a 3 × 2 one which we shall identify as matrix C:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Excerpted from Matrices and Society by Ian Bradley, Ronald L. Meek. Copyright © 1986 Ian Bradley and Ronald L Meek. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

• FrontMatter, pg. 1
• Contents, pg. 5
• Preface, pg. 7
• 1. Matrices and How to Manipulate Them, pg. 13
• 2. Matrix Inversion, pg. 30
• 3. The Ins and Outs of Economic Planning, pg. 42
• 4. Matrices and Matrimony in Tribal Societies, pg. 61
• 5. Dominance in Coops and Courts, pg. 79
• 6. The Simple Mathematics of Markov Chains, pg. 92
• 7. Models of Mobility, pg. 106
• 8. The Mathematics of Absorbing Markov Chains, pg. 122
• 9 ‘Everywhere Man Is in Chains’, pg. 130
• 10. The Seven Ages of Man and Population Problems, pg. 149
• 11. Playing Games in Theory, pg. 173
• 12. Magic, Fishing and Farming — Some Applications of Constant-sum Games Theory, pg. 194
• 13. Conflict or Co-operation, pg. 207
• Epilogue and Further Reading, pg. 231