EigenExplorer is part of LAVA (The Linear Algebra Visualization Assistant), a larger site that is still under construction. In this applet, the user is able to input the entries of a 2 by 2 matrix, say A. The user is allowed, by using the mouse, to select a unit vector x. The vector Ax is displayed in another color. In addition, the coordinates of x and Ax are shown as well as the length of Ax. Further, it is possible to trace the circle which contains all possible endpoints of x and the ellipse which contains all possible values of Ax.
Type of Material:
This applet should run on any Java-enabled browser.
Identify Major Learning Goals:
To provide geometric insight into the nature of eigenvectors and eigenvalues. The user will also acquire visual exposure to the affect of a linear transformation.
Target Student Population:
This material could be effectively used in a linear algebra course. Students in more advanced courses could use it as supplemental material.
Prerequisite Knowledge or Skills:
The ability to multiply matrices would be sufficient.
This very simple applet effectively illustrates eigenvalues and eigenvectors. EigenExplorer is part of LAVA (The Linear Algebra Visualization Assistant), a larger site that is still under construction. As a part of LAVA there is a webpage which gives several 2 by 2 matrices and asks the user to find the eigenvalues with EigenExplorer. This page could easily be used as the first part of a laboratory session. LAVA contains other interesting applets which illustrate other concepts from linear algebra.
Potential Effectiveness as a Teaching Tool
Of course, this applet can be used to graphically find eigenvectors and eigenvalues. However, it can also be used to examine a vast array of other conjectures/theorems concerning 2 by 2 matrices. These include, but are not limited to, the following: Does a 2 by 2 matrix with negative determinant always have two eigenvalues? Does a real symmetric 2 by 2 matrix have two real eigenvalues? Does a 2 by 2 matrix map the unit circle onto an ellipse? Are the eigenvectors of the matrix always/never/sometimes parallel to the major/minor axes of the ellipse? Under what conditions are the eigenvectors of a matrix perpendicular? What is the relationship between the orientation-reversing nature of a linear map and its determinant? What is the relationship between the orientation-reversing/preserving nature of a linear map and the number of eigenvalues of the matrix which defines it? (It is graphically clear that an orientation reversing map defined by a 2 by 2 matrix has two eigenvalues). What conditions must a matrix satisfy if every vector is an eigenvector? This applet will give students an understanding of the nature of eigenvectors and eigenvalues that they would not get from merely finding them using the characteristic equation and matrix methods.
I see none other than that a student who uses EigenExplorer will only get insight into very low dimensional linear transformations. However, this concern should be addressed by the three dimensional version of EigenExplorer which is under construction.
Ease of Use for Both Students and Faculty
This is a very straightforward applet to use. The ability to see numerical information is a great benefit.
It is somewhat difficult to navigate around the LAVA site at present.
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