Click here to open a new window with a Java applet. Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active.

This applet can be used to explore a linear transformation written in the form

where **A** is a two-by-two matrix. The elements of the matrix are displayed at the right hand side of the applet. By clicking or dragging on the graph you can move the vector **x** which is drawn on red. As you move the vector **x**, the vector **y = Ax** also moves. It is drawn in blue. Before reading on, move the vector **x** to get an idea what this particular linear transformation does. For example, you might note that

- The vector
**y**is always the same length as the vector**x**except that it has been rotated counter-clockwise by**pi/2**radians. - If you stretch the vector
**x**then the vector**y**stretches as well.

You can change the entries in the matrix **A** by clicking on the entry you want to change. A dialog box will ask you for the new entry.

This applet provdes a good way for students to literally see some of the ideas involved in linear equations. For example, change the matrix **A** to a singular matrix like

**a _{11} = 0.5 a_{12} = 1.0**

a_{21} = 1.0 a_{22} = 2.0

Notice that you can observe the following things experimentally and visually

- No matter what the value of
**x**the value of**y**always lies along the same line. Thus, the linear transformation**A**is not onto and is not invertible. - There is a line that is mapped into zero -- that is, you can determine the kernel visually.

We begin our study of eigenvalues and eigenvectors by looking at two applications where eigenvalues and eigenvectors come up naturally and add significantly to our understanding.

Application 1:Collegetown has two pizza restaurants and a large number of hungry pizza-loving students and faculty. 5,000 people buy one pizza each week.

Joe's Chicago Style Pizzahas the better pizza and 80% of the people who buy pizza each week atJoe'sreturn the following week.Steve's New York Style Pizzauses lower quality cheese and doesn't have a very good sauce. As a result only 40% of the people who buy pizza atSteve'seach week return the following week. As usual, we can represent this situation by a discrete dynamical systemP_{n + 1}= AP_{n}where

a

_{11}= 0.80 a_{12}= 0.60a

_{21}= 0.20 a_{22}= 0.40Click here to open a Mathematica notebook to investigate this situation. Evaluate the notebook. Notice that if we start with 2500 customers at each pizza restaurant in the first week then after a very few weeks

Joe's Chicago Style Pizzaseems to have three times as many customers each week asSteve's New York Style Pizza.Next enter the matrix above in the Java applet. Play with the applet a bit to see if you notice anything worthy of note.

The two screenshots below show some behavior that is worthy of note.

Notice in the screen shot above that when the first coordinate (Joe's Pizza) of the vector

xis roughly three times its second coordinate (Steve's Pizza) thenAx = x. Notice in the screen shot below that when the first and second coordinates have the same absolute value but opposite signs thatAxlines up withxbut is roughly one-fifth as long.Notice the following lines from the evaluated Mathematica notebook.

Do you notice any connection between this rather cryptic Mathematica output and our observations above? We will return to this point later but first we look at another application.

Application 2:The three figures below show the current population by age and gender and the projected population by age and gender in 25 years for three countries. These figures were obtained from the United States Census Bureau International Data Base. Demographic information like this is extraordinarily important for understanding the vitality, needs, and resources of a country. The three examples given above have very different properties. The United States and japan are highly developed countries with quite different demographics. Compare the fraction of the population that is young in the two countries both currently and, as projected by the U.S. Census Bureau, in 25 years. Notice how different these two countries are from Pakistan.

We will look at a very simplified model of population growth for a hypothetical species and habitat. You can build much more realistic models using data available at the United States Census Bureau. Our simplified model will ignore gender and immigration and emigration and will use only two age groups. These are serious simplifications but the tools we will develop this semester will enable us to look at much more realistic models. This example is just a starting point. Ignoring immigration, for example, ignores one of the most important differences between Japan whose immigration is close to zero and the United States.

Our model has two age groups -- the young (less than one year old) and the old (one or more years old). Thus, each year the population is represented by a two-dimensional vector

P = (A, B)whose first coordinate is the young population and whose second coordinate is the old population. In this example the fertility rate for young people is 30% which means that on the avergae each young individual gives birth to 0.30 new (and hence young) individuals. The fertility rate for old individuals is 80% which means that on the average each old individual gives birth to 0.80 new (and hence young) individuals. In our model the survival rate for young individuals is 90% and for old individuals is 10%. This gives us the model

Enter the matrix

Ainto the Java applet and experiment. Do you notice anything noteworthy?The two screen shots below show two things that are worthy of note. The first one shows a vector that when multiplied by the matrix

Aresults in another vector that is in the same direction but somewhat longer. The second one shows a vector that when multiplied by the matrixAresults in anotrher vector in exactly the opposite direction that is considerably shorter.Click here for another Mathematica notebook. This Mathematica notebook explores this same model. Evaluate the notebook and compare the results with your observations above. Notice that over time the total population is growing at a rate of 5.44% per year and seems to be settling into a pattern with the young population being 51.4% of the total population.