# An eigencircle applet

Graham Farr, Faculty of IT, Monash University
`Graham.Farr@infotech.monash.edu.au`

## Contents

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 .

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 ) -μ λ
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 -μ 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.

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 `java.com` which might help; if you try it, I would be interested to know how much it helps you.

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.

• Matrix. This enables you to type in or modify the matrix entries (whose fields will now be white, and editable). Alternatively, on the graph, you can move the special points on the axes corresponding to the matrix entries. These are shown as small red boxes. (They will only exist if you already have an eigencircle graph displayed. If not, then the only thing you can do is put matrix entries in their fields.) This option is always selected when the window first starts, and indeed is the only option available (so the others are greyed out and cannot be selected) until you have a valid matrix and its eigencircle has been calculated and displayed.
• Eigencircle centre. This enables you to change the coordinates of the eigencircle's centre, either by editing the appropriate fields, labelled f and g (which are now white and editable), or by moving the centre dot (which is now red) on the graph. When you do this, the circle's radius remains fixed, and the matrix entries are recalculated so that the displayed eigencircle is the correct one for the current matrix. The matrix entries are adjusted so as to preserve the size and shape of the special rectangle (but not its location, since its centre is the same as the eigencircle's centre). This is not the only matrix which has the new circle as its eigencircle, but has been chosen as being the one most like the original matrix in some sense.
• Eigencircle radius. You can change the eigencircle's radius either by editing the appropriate field, labelled ρ (now white and editable), or by moving the circle's perimeter (now red) in or out. The circle's centre stays fixed, and the matrix entries are adjusted so as to preserve the shape and centre (but not the size) of the special rectangle. If you make the radius zero, you will no longer be able to change anything except the matrix entries or the eigencircle centre, and this will remain the case until you enter a matrix whose eigencircle radius is nonzero. (This is because a zero-radius eigencircle has no special rectangle, so if you were to increase the radius the applet does not know what special rectangle to use in order to update matrix entries.)
• Vary a matrix element, fix eigencircle. You can now vary any single matrix element, by moving the corresponding small box (now red) on the axes. All the other entries will adjust so that the eigencircle of the matrix remains the same. This enables you to study the family of all matrices with a given eigencircle. Note that this option does not allow you to edit the fields for the matrix elements. Also, for numerical reasons it does not always allow the rectangle to become the degenerate "rectangle" formed by a single horizontal or vertical diameter of the eigencircle.
• Vary matrix/circle, fix eigenvalues You can now vary anything at all: matrix element, eigencircle centre, or eigencircle radius. All the corresponding items in the graph should be red. Varying any of them will cause all the others to be adjusted so as to keep the eigenvalues of the matrix unchanged (even if they are complex), and so as to preserve the shape of the special rectangle. If the radius becomes zero then some items will become unavailable, and a matrix of positive eigencircle will have to be entered in order to make all editing options available again. If elements a and d are equal, then they will not be movable under this edit option, as in this case the special rectangle is degenerate (a single vertical line segment, forming a diameter of the eigencircle), and such a special rectangle can only remain degenerate in this way if a = d. Other items should still be movable in this situation (except the radius, if it is zero).

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:

• Special rectangle. This is the rectangle formed by the four points with λ-coordinates a or d and μ-coordinates b or -c. These points always lie on the eigencircle. This option also displays the lines leading from the coordinate axes to the sides of this rectangle.
• Determinant. The determinant is (in the language of classical Euclidean geometry) the power of the origin O with respect to the eigencircle. Its magnitude is the square of the length of a particular line from the origin to the eigencircle: if the origin is outside the eigencircle, the line is tangent to the eigencircle (and the determinant is positive); otherwise, it is perpendicular to the line from the origin to the eigencircle's centre C (and the determinant is negative). This option shows this line segment, and also the right-angle triangle (including O and C) of which the line forms one side.
• Eigenvalues. This option displays lines whose lengths are the eigenvalues. If the eigenvalues are real, then their values are the λ-intercepts of the eigencircle. In this case, they are shown as thick dark blue lines along the λ-axis from the origin to these λ-intercepts. If the eigenvalues are complex but not real, then their (identical) real parts are shown as, again, a thick dark blue line from the origin to the λ-coordinate of the eigencircle's centre. The magnitude of the imaginary part is shown as a light blue line, tangent to the eigencircle and touching the end of the real part.
• Eigenvectors. These are only displayed if the eigenvalues are real. In that case, they are shown as grey vectors from the point (λ,μ) = (d,-c) to the λ-intercepts of the eigencircle. Of course, eigenvectors can be any scalar multiple of either of these vectors. Furthermore, eigenvectors actually sit in the x,y-plane, rather than the λ,μ-plane, so strictly speaking we have to imagine the former plane superimposed on the latter. If the eigenvalues are not real, then no eigenvectors are displayed, but this display option will remain available in its current state (checked or unchecked) and that state can be changed by the user. Although checking or unchecking this checkbox has no immediate effect if the eigenvalues are not real, there is always the possibility that the user, in changing the matrix or circle using one of the edit options, may produce a matrix with real eigenvalues. In that case, the current state of this checkbox will be used to determine whether or not eigenvectors are displayed.
• Scales on axes. Scales on axes are provided by default, but sometimes can clutter the graph a bit so you have the option of removing them. The axes themselves cannot be removed, though.
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:

• Calculate. This calculates the eigencircle, and all the various items of matrix information to be presented in the information fields, from the current 2x2 matrix. It also ensures that the eigencircle and axes are nicely positioned in, and framed by, the display area used for the graph.
• Reframe. This just reframes the current eigencircle, without recalculating it or any of the matrix information. The aim is to ensure that the circle and axes are well positioned and framed, as in the final step of the Calculate button described above. It is useful in cases where the eigencircle has been partly moved out of the display area by some action of the user.
• Previous matrix. Pressing this takes you back to the previous matrix, which is the matrix that was being used before the last time the user released the mouse button on the graph, or pressed Enter, or clicked Calculate. This button is only available when there is a previous matrix to go back to, so will be unavailable (and greyed out) initially, and even after the very first matrix has been successfully entered. Only after a second valid matrix has been entered (and its eigencircle graphed) does this button become available.
• Quit. This just makes the eigencircle window quit. But you can always start another window by clicking the launching button again. To stop the applet completely, you need to leave this web page or close the browser window that the web page is displayed in.

## References

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

Created 15 November 2007;
Last updated 1 December 2007.