Author(s):

David E. Meel and Thomas A. Hern

Many students claim to have difficulty understanding the concepts of eigenvalues and eigenvectors (Meel, 1999b). One possible explanation for this difficulty could be that students typically associate eigenvalues and eigenvectors with a computational process and do not understand the geometric connections.

So, how do we encourage students to begin to look beyond the computations and consider the geometry? We consider an associated cognitively-guided activity as the means of forcing students to come to grips with both the computational process and the underlying geometric relationship between eigenvalues and eigenvectors. Specifically, the activity integrates the capabilities of MATLAB (MATLAB, 1995) with the **Eigenizer** tool. By blending the 2D graphical examination of eigenvalues and eigenvectors through the **Eigenizer** tool with examination of higher dimensional eigenspaces through MATLAB, we lead students to explore eigenvalues and eigenvectors from multiple perspectives.

This experience stimulated one student to make the following comment:

I liked working with eigenizer and MATLAB simultaneously. That way, I could see what it would look like with l = 0 and l = *a+bi.* Also the eigenizer showed why an eigenvalue was (+) or (-) depending on which way the vectors pointed."

Another student stated

"I learned the most from the eigenizer project. The eigenizer helped you see that the formula was true, but you need to do the work in order to find all of the eigenvalues, not rely solely on the computer. Eigenizer needs to be able to do (3) vectors."

Looking at the **Eigenizer** tool and its cognitively-guided activity as a whole, we see that students are being asked to look beyond the computations and understand the meanings behind the concepts eigenvalue, eigenvector, and eigenspace.

Next or or page: **10.** Eigenizer Tool and Sample Activity

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

Journal of Online Mathematics and its Applications