An eigencircle applet

Graham Farr, Faculty of IT, Monash University


1. Introduction to eigencircles
2. The applet
3. Guide to the eigencircle applet
   3.1. Getting started
   3.2. Edit options
   3.3. Display options
   3.4. Matrix information
   3.5. Buttons

The eigencircle of a 2x2 matrix is a special circle based on the matrix that generalises the set of eigenvalues of the matrix. It can be used to give simple geometric illustrations of many properties of the matrix. It was introduced by Michael Englefield and the author in [1].

The main purpose of this web page is to present an applet to help you explore eigencircles. We also give a very brief introduction to the topic. This introduction just gives you the minimum to get started. We assume you already know about eigenvalues. For further information on eigencircles, see the references.

If you already know about eigencircles, you will probably just want to go straight to the applet. You can do so here if you like. For help on using the applet, consult the guide in Section 3 below.

1. Introduction to eigencircles

Recall that an eigenvalue of a 2x2 matrix
A = ( a b )
c d
is a number λ such that
( a b ) ( x ) = λ ( x )
c d y y
for some x, y not both 0.

Let us rewrite the above equation a little:
( a b ) ( x ) = ( λ 0 ) ( x )
c d y 0 λ y
where, again, x, y are not both 0.

We have just used the correspondence
λ  ←→  ( λ 0 )
0 λ
which represents any real number as a real 2x2 matrix. Real addition and multiplication correspond to matrix addition and multiplication: it is a field isomorphism, between the real numbers and real multiples of the identity matrix.

Our generalisation is inspired by a well-known way of representing complex numbers as real 2x2 matrices. Let λ + μ i be complex (where λ and μ are real). Then the correspondence
λ + μ i  ←→  ( λ μ )
is a field isomorphism that represents any complex number by a real 2x2 matrix of appropriate form. Let us see what happens if we use matrices of this form, instead of just multiples of the identity matrix, in the eigenvalue equation above. We have:
( a b ) ( x ) = ( λ μ ) ( x )
c d y λ y
with, as usual, x, y not both 0. We call a pair of real numbers (λ,μ) that satisfy this an eigenpair. Note that this equation is not the same as just allowing complex eigenvalues in the ordinary eigenvalue equation. It is routine to show that the set of eigenpairs describes a circle in the real λ,μ-plane, and it is this circle which we call the eigencircle.

The equation just given is an example of a multiparameter eigenvalue problem.

2. The applet

Press the following button to launch an eigencircle window.

For this to work, you must have Java installed. If you are not sure whether or not your web browser can run Java applets, click here to test it. If you have Java and want to see what version of Java your web browser is using, click here to find out. To run the applet, you will probably need at least Java 1.2.

The above button should say, "Launch eigencircle window". If all you see is a rectangle with a red cross in it, or some kind of error message, then you have a problem. Try the above links to see whether you have Java and, if so, what version it is. If you do not have Java, you will need to install it for the eigencircle applet to work on your machine: one place to start is here. If you do have Java, but the eigencircle applet still does not work, please email me with "eigencircle" in the subject line, and tell me what version of Java you have, what web browser you are using (including version number), what operating system and computer you are using, what the above button looks like when you first load the web page, and what happens when you click it. There is some error message info at which might help; if you try it, I would be interested to know how much it helps you.

From now on, we assume you have Java and can run applets from your web browser.

Each click of the above button will launch a new window. All such windows are identical as far as the user interface is concerned, but they are independent in that they can each contain their own separate matrix and eigencircle. The Guide below concentrates on the behaviour of a single one of these eigencircle windows.

Note: If you have just arrived at this page and not pressed this button before, and it does not respond to your first press, then you may need to press it a second time: it may take an initial click just to "activate" the button so that it can respond to clicks. After that, a single press should suffice to launch a new eigencircle window promptly.

3. Guide to the eigencircle applet

3.1. Getting started

The applet initially appears as just a button sitting in this web page. The purpose of this button is to launch eigencircle windows. Each such window allows you to enter a 2x2 matrix, view its eigencircle, and experiment interactively.

When an eigencircle window first starts, you will see a large blank area on the left and various buttons and fields on the right. The large blank area is where the graph of the eigencircle will be drawn, but there is nothing there yet as no matrix has been entered. So the first thing to do is to manually enter a matrix (indeed, the "Message Box" along the top tells you to do just this, in dark blue text). You can do this by typing matrix entries in the four fields labelled a, b, c, d (and arranged as a 2x2 matrix) at upper right.

Once you have typed in all matrix entries, press the Enter key, or click on the Calculate button, and the eigencircle will be calculated and graphed at left. Also, various fields will be filled in, in the right side of the display, and the equation of the eigencircle is given in the very bottom field. If you press Enter or click Calculate when some of the fields a, b, c, d are still blank, or when one or more of them contains text that does not represent a number, an error message will appear in the Message Box (this time in red). You just need to complete or correct the field(s) concerned, and then press Enter or click Calculate again.

Note that the fields you can currently edit are white, while all the others, which just display information and cannot be edited, are the same colour as the surrounding area of the window (light blue on my machine, and we'll assume it is this colour from now on). This convention is used throughout the running of the applet.

Once the matrix entries are all valid and the window has displayed the eigencircle (and all the other information) for the first time, you will get another message in dark blue in the Message Box. This tells you that you now have a choice as to what to edit.

3.2. Edit options

You choose what to edit by clicking one of the five radio buttons in the top right of the window.

Note that, if you move (click-and-drag) one of the red items on the graph, then the graph itself updates continuously while you move the item. However the various fields, giving information on the matrix and its eigencircle, will only update once the mouse button is released.

An example of the kind of exploration you can do is to study all matrices with given eigenvalues. To do this: (1) enter some matrix of interest (which has the eigenvalues that interest you, if you have particular eigenvalues in mind), using the Matrix edit option, and get the program to display its eigencircle as explained above; (2) select the last edit option: Vary a matrix element, fix eigencircle; (3) vary the eigencircle by moving any of the movable items, say its centre or radius; (4) select the second-last edit option: Vary a matrix element, fix eigencircle; (5) vary any of the matrix elements, using the small red boxes on the axes, in order to get any desired matrix with that eigencircle. The matrices obtainable in this way are precisely those that have the given eigenvalues (at least in principle; of course, any program is subject to numerical limitations).

3.3. Display options

At first, the graph shows the eigencircle with a certain special inscribed rectangle defined by the matrix entries, and the axes of the graph have scales on them. You have a number of choices as to what is, or is not, displayed in the graph. These choices are made by clicking the checkboxes under the heading DISPLAY OPTIONS in the middle of the right side of the window. Initially, the Special Rectangle and Scales on Axes boxes are checked, so the graph will show these. If you wish, you can uncheck either of these by clicking it, and that feature will no longer be displayed (unless you later click the checkbox again). The display options are:

Initially, the Special Rectangle and Scales on Axes options are selected and the others are not selected.

3.4. Matrix information

Further down on the right there are fields that display information about the matrix: its determinant, trace, eigenvalues (real or complex, as the case may be), and eigenvectors (which are only given if the eigenvalues are real, since only in that case can they be shown on the eigencircle graph). Finally, in the long field underneath the graph, the equation of the eigencircle is given. All these fields are for display of information only, and can never be edited.

3.5. Buttons

There are four buttons at the very right edge of the window:


[1] M J Englefield and G E Farr, Eigencircles of 2x2 matrices, Mathematics Magazine 79 (October 2006) 281--289.
[2] M J Englefield and G E Farr, Eigencircles and associated surfaces, submitted, 2009.

Created 15 November 2007;
Last updated 1 December 2007.