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Working with **Eigenizer**, similar to **Transformer2D**, involves coordinated actions between defining the column vectors of the matrix of transformation. The yellow box controls the column vectors defining the matrix of transformation. In particular, the **green** vector controls the first column vector and the **blue** vector controls the second column vector. By grabbing the ends of these two vectors, you can construct any 2x2 matrix.

Below the yellow box is a box that controls the vector **x**. As you move **x** (the **red** vector) about the domain of the transformation, you can watch both **x** and the image *T*(**x**) (the **magenta** vector) change in the large area to the right of the screen, depicting the codomain of the transformation. The movement of the vector *T*(**x**) depends on the nature of the matrix of transformation.

The large codomain region also displays information concerning the length of the vector **x**, the length of the vector *T*(**x**), the radian measure of the angle between these two vectors, and a lambda approximator. Two buttons at the bottom of this region control the display of the Eigen Equations in a red box above the codomain box.

Open Eigenizer in new window

Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About *JavaSketchpad* web page.

This sample activity provides a guided exploration of eigenvalues and eigenvectors of a particular matrix and includes a set of questions that can be asked for any other matrix, as well as some general questions about the tool and observations made from interacting with the tool.

- Using the vectors contained in the yellow box, move (by click dragging) the
**green vector**to (3,-1) and the**blue vector**to (-2,2). This should construct the following matrix of transformation: . - Move the red vector
**x**(in the domain portion of the tool) and observe the movement of the magenta vector,*T*(**x**). If possible, move the vector**x**so that the vectors*T*(**x**) and**x**are collinear. Note: This can be aided by examining the angle measure at the bottom of the*y*-axis in the range portion of the tool. - Click on the "Show lambda 1 equation" button, and observe whether the equation is true, i.e., do the vectors displayed on the right and left sides of the equation match each other? If they do not match, click on the "Show lambda 2 equation" button, and check if truth is found.
- Move the red vector
**x**(in the domain portion of the tool) so that it and the*T*(**x**) vector are collinear in a different location. - Redo step 3 for this new location.

Given your exploration (and perhaps some additional ones), answer the following questions:

- What are the eigenvalues for this matrix?
- What are corresponding eigenvectors for these eigenvalues?
- What does the lambda approximator do?
- If an eigenvalue is positive, what does this mean concerning the form of collinearity between the vector
**x**and the vector*T*(**x**)? If an eigenvalue is negative, what does this mean concerning the form of collinearlity between the vector*x*and the vector*T*(**x**)? - Can you have a matrix with two positive eigenvalues or two negative eigenvalues? Explain why or why not
- For a given eigenvalue, is there a unique eigenvector or a set of corresponding eigenvectors? If unique, explain why it's unique and if there is a set, explain how to describe the set.
- Define a matrix that does not have any real eigenvalues? In general, what would be the nature of the column vectors of such a matrix?
- Is it possible to define a matrix that has eigenvalues of -3 and 2? If it is possible, state how many such matrices could be constructed, and provide a specific example of at least one.
- Is it possible to define a matrix that has only a single eigenvalue, say -2? Explain why or why not.

The tool allows students to explore specific matrices as well as hypothesize about possible matrices with particular properties. It is with the latter type of explorations that the worlds of geometry and computation can fuse. Students need to think beyond computations that MatLab might be able to perform and ponder the possibilities of "what if?".

Next page:

Next or page: **11.** Student Responses

David E. Meel and Thomas A. Hern, "Tool Building: Web-based Linear Algebra Modules - Eigenizer Tool and Sample Activity," *Convergence* (May 2005)

Journal of Online Mathematics and its Applications