Lab #9

You will have one week to complete this lab, professionally write up the answers, and submit it for a grade. The purpose of the lab is
• to discover the geometry of eigenvectors of matrices
• to understand relationships between eigenvalues, eigenvectors and the determinant.
• to discover that the determinant gives information about the way that a linear transformation expands or contracts area.

Background: Eigenvalues and eigenvectors play a vital role in understanding the behavior of differential equations in the neighborhood of an equilibrium. In this lab you will explore the geometrical relationship between eigenvalues, eigenvectors, and the determinant.

To get started, launch Matlab and type in the following command:
eigshow
Matlab will prompt you to type in a matrix. At the prompt type:
[3 3; 1 2]
This command will store the matrix in the Matlab procedure eigshow. Note how to enter matrices in Matlab: type the first row (separating entries either by a space or by a comma), then a semi-colon, then the next row, etc.

A window should appear with a white unit circle in the center. By holding down any mouse button and moving the mouse, you can move the yellow radius around the circle. As you do this a blue vector will appear that traces out an ellipse. Think of the yellow line as a unit vector, v. Matlab is plotting the image of that vector, Av, in blue.

Recall that an eigenvector of a matrix A is any non-zero vector u such that where is a scalar called the eigenvalue. Geometrically, this says u is an eigenvector of a matrix if its image under A is a scalar multiple of itself. Note on your screen that Matlab shows in the lower left hand corner the unit vector v in yellow, and in the lower right hand corner the image of the vector, Av, in blue.

QUESTION 1: Slowly move the unit vector, v, around the unit circle. As you do so, watch the image vector, Av.
1. Approximately find two linearly independent positions for v so that Av is a scalar multiple of v. (Hint: in one position, the image vector will be pointing opposite from v.)
2. For each position you find, what is the scalar that you would multiply v by in order to get the vector Av?
3. Using the proceeding answers, estimate the eigenvalues and eigenvectors of the matrix A.
4. When you write up your report, you will know how to compute the exact eigenvalues and eigenvectors for A. Do so, and compare the exact values to your estimates.

As you move the vector v (in yellow) about the unit circle, its image traces out an ellipse (in blue). In other words, the image of the unit circle under the linear transformation represented by A is an ellipse.

The area of an ellipse is where a is the length of its semi-major axis, and b is the length of its semi-minor axis. For our ellipse, the semi-major axis will be the longest image vector, and the semi-minor axis will be the shortest image vector. In the upper right hand corner of the Figure window, Matlab displays the norm of the vector Av. When this number is largest, you have approximately found a semi-major axis; when it is smallest, you have found a semi-minor axis.

QUESTION 2:
1. Estimate the length of the ellipse's semi-major and semi-minor axes.
2. What is the ratio of the area of the ellipse to the area of the unit circle?
3. Fill out the remaining entries of Table 1 for this matrix.

Now let's change the matrix we've been studying. To do this, type eigshow and hit return in the window from which you started Matlab. When prompted for a new matrix, type [1 3; 3 1]. This will create the matrix .

QUESTION 3: Answer the previous two questions for the new matrix. Formulate a conjecture relating the eigenvalues of a matrix to the way that the matrix expands or contracts area.

Let's change the matrix one more time, this time to .

QUESTION 4:
1. Why can't you use the previous procedure to find the eigenvectors and eigenvalues for this matrix? In a paragraph, explain the geometric relationship between v and its image, Av, as v moves around the unit circle.
2. Even for this matrix, the image of all unit vectors is an ellipse. Compute the length of the semi-major and semi-minor axes, and the ratio of the area of the ellipse to the area of the unit circle. Based on your experience with the previous two matrices, predict the remaining entries of Table 1. Explain how you reached your conclusions.

QUESTION 5: Compute the determinant of each matrix we've encountered in this lab. Formulate a conjecture relating the determinant of a matrix to its eigenvalues and to the way that the matrix transforms area.

 Matrix Eigenvalues a b

This page is brief summary of some Matlab commands for matrices and vectors. Although none of these commands are required to do this lab, you are expected to know them for future labs.

 Matlab Command Translation v = [3; 2] assigns A = [1 6; 5 -2] assigns Solve for the vector x such that Ax = v. l = eig(A) Computes the eigenvalues of matrix A and stores the result in vector l. [V, D] = eig(A) Computes the eigenvalues and eigenvectors for a matrix A, puts the eigenvectors into the columns of matrix V and puts the corresponding eigenvalues into the diagonal entries of matrix D. det(A) Computes the determinant of A. trace(A) Computes the trace of A. null(A) Computes the null space of A. rank(A) Computes the number of linearly independent rows of A.