**Lab #9**

- 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.

- 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*.) - For each position you find, what is the scalar that you would
multiply
*v*by in order to get the vector*Av*? - Using the proceeding answers, estimate the eigenvalues and
eigenvectors of the matrix
*A*. - 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

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.

- Estimate the length of the ellipse's semi-major and semi-minor axes.
- What is the ratio of the area of the ellipse to the area of the unit circle?
- 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

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

- 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. - 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.

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.

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Last modified: Jan 16, 1997

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