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Eigenpictures: Picturing the Eigenvector ProblemAuthor(s): Steven SchonefeldSource: The College Mathematics Journal, Vol. 26, No. 4 (Sep., 1995), pp. 316-319Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2687037 .
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COMPUTER
CORNER
EDITOR
Richard F. Johnsonbaugh School of Computer Science,
Telecommunication Systems, and Information Systems
DePaul University Chicago, IL 60604-2287 [email protected]. edu
In this cotumn, readers are encouraged to share their expertise and experiences with computers as they relate to college-level mathematics. Articles that illustrate how computers can be used to enhance pedagogy, solve problems, and model real-life situations are especially welcome.
Classroom Computer Capsules feature new examples of using the computer to enhance teaching. These short articles demonstrate the use of readily available computing resources to present or elucidate familiar topics in ways that can have an immediate and beneficial effect in the classroom.
Send submissions for both columns to Richard Johnsonbaugh.
Classroom Computer Capsules
Eigenpictures: Picturing the Eigenvector Problem
Steven Schonefeld, Tri-State University, Angola, IN 46703
One picture is worth many words. Yet if you look through books that treat
eigenvalue and eigenvector problems, few pictures appear. Perhaps the pictures described here can help remedy this lack of visual images?at least for real 2 by 2
matrices. My inspiration came from Professor Gilbert Strang's talk at the meeting
of the Indiana section of the MAA at DePauw University on March 19, 1994, in
which Strang suggested a moving picture of a 2 by 2 matrix A. His moving picture would simultaneously display both u and A ? u as the unit vector u moves around
the unit circle. The "stroboscopic" pictures described here I call eigenpictures for
want of a better name. An eigenpictures capture the essence of Strang's movie in a
single image. The reader may recall that a nonzero vector v is an eigenvector for matrix A if
the product A-\ is a scalar multiple, A, or v. The corresponding scalar, A, is an
eigenvalue of A In geometric terms, A-\ and v are parallel when v is an
eigenvector of A. Can we find a way to search visually for real eigenvectors of a real 2 by 2 matrix
A! Since any nonzero scalar multiple of an eigenvector is also an eigenvector, we
need only search through the unit vectors in our guest for real eigenvectors. We
will picture a typical unit vector u as a directed line segment having initial point at
the origin, and we will attach the product A ? u to the terminal point of u, as in
Figure 1. The vector u shown in Figure 1 is not an eigenvector since the vectors u
and A ? u do not align.
3-J 5 THE COLLEGE MATHEMATICS JOURNAL
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Figure 1
Technology to the rescue. Since we cannot look examine the real unit vectors, we try a large, representative sample of them: vectors of the form
u = [cos 0,sin0], for 0 = 0.0,0.1,0.2,. ..,6.2.
The following eigenpictures were generated using the computer algebra system Derive. However, with a little work, any computer algebra system should produce similar eigenpictures. The Derive procedures are given at the end of this article.
A typical eigenpicture. For the matrix
1 A = - 13 9
3 7
we get the eigenpicture in Figure 2. The line segments representing the vectors were plotted with large dots at their endpoints. It is easy to pick out the
(approximate) eigenvectors and to approximate the corresponding eigenvalues. For the unit eigenvector u in the first quadrant, the line segment A ? u is approximately twice as long as u, so Xx = 2 is the eigenvalue corresponding to this eigenvector. For the eigenvector in the fourth quadrant, the corresponding eigenvalue is 1/2.
Eigenvectors
Figure 2 Eigenvalues A,i = 2, X2 = 1/2.
You can see the symmetry in the eigenpictures with respect to the origin. This
symmetry illustrates that if u is an eigenvector for matrix A, then -u is also, with the same eigenvalue.
VOL 26, NO. 4, SEPTEMBER 1995 317
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Complex eigenvalues. The matrix
cos
77 sinl ?
-sin
cosi-
has complex eigenvalues:
l + ?
TT IT
Figure 3 shows the eigenpicture for this matrix. Multiplying a (real) vector by A has the effect of rotating the vector through an angle of tt/4 radians?there are no real eigenvectors. If matrix A has only complex eigenvalues, none of the vectors A ? u will be parallel to the vector u.
Figure 3 Complex eigenvalues.
One zero eigenvalue. The matrix
A = 2 -1
-2 1
has eigenvalues A1 =0, A2 = 3. Figure 4 is an eigenpicture for this matrix. As the vector u gets close to an eigenvector with zero eigenvalue, the vector A ? u gets close to the zero vector. In this example, all the nonzero vectors A ? u are parallel ?they are eigenvectors corresponding to the eigenvalue A2 = 3.
A double eigenvalue. The matrix
A ^ -3
2 + ^
has a double eigenvalue: Xx = 1, A2 = 1. In Figure 5 we see an eigenpicture for this matrix. Whereas we expect a 2 by 2 matrix to have two pairs of unit eigenvectors, this matrix has only one pair. Question: In the eigenpictures, it appears that the
318 THE COLLEGE MATHEMATICS JOURNAL
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Figure 4 A zero eigenvalue.
Figure 5 k\ = X2 = 1.
locus of points at the end of vectors u +A ? u is an ellipse. Is this locus an ellipse for any matrix Al
Derive procedures
1: a:=[[13/8f 9/8], [3/8, 7/8]]
2: TEMP(u) := [[0f 0], u, u + a.u]
3: VECTOR(TEMP([COS(t), SIN(t)]), t, 0, 6.2, 0.1)
Here is a brief explanation of the Derive procedures. Line 1 defines the matrix a used for the eigenpicture in Figure 2. Line 2 defines a temporary function of vector u (which also depends on matrix a), resulting in a triple of points: the origin, the terminal point of u, and the terminal point of u + a ? u. We appro [X] imate line 3 to get a vector of triples of points. A connected plot of this vector of triples gives the eigenpicture in Figure 2. In order to generate the eigenpicture for a different
matrix, you need only: redefine matrix a, appro [X] imate line 3, and plot the result.
VOL 26, NO. 4, SEPTEMBER 1995 319
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