Tools such as Eigenizer
are only as good as their ability to help students interact with and form their own conjectures concerning linear algebra content. After students interacted with Eigenizer
, we asked them in the cognitively-guided activity
to put their observations about eigenvalues and eigenvectors into their own words. The responses included reflections such as these:
- "An eigenvalue when multiplied by an eigenvector yields the same result as when matrix A is multipled [sic] by the eigenvector. Thus matrix A acts like the scalar eigenvalue".
- "An eigenvalue is a value that lets the matrix act like a scalar. An eigenvector is a nonzero vector that corresponds to the eigenvalue, if Ax = the eigenvalue * x".
- "The eigenvector is multiplied by A to make A act like a scalar. The eigenvalue of the vector is the lambda that A acts as when the vector is multiplied by it".
- "An eigenvalue of a matrix A is a scalar that when multiplied with a vector x yields the same resultant vector as A*x. An eigenvector of a matrix A is a vector that when multiplied with a scalar lambda yields the same resultant vector as when multiplied with the matrix A".
- "In our words, an eigenvalue is a way of representing a matrix as a scalar. This will allow the investigation of a matrix of transformation (T) on a vector (T(x)) without complicated calculations. The eigenvector provides a relationship between the values of x for which the lines are co-linear [sic]".
- "An eigenvalue, with it's eigenvector, 'mimic' A when multiplied together. Meaning, A acts like a scalar when multiplied with the eigenvector. Since lambda is derived by solving det(A - lambda In)x = 0, it 'unravels' to Ax = lambda x, which is what I described above".
- "The eigenvalues are related to A in that when multiplied by the In matrix and subtracted from A, you can row reduce to find a basis to the corresponding R space. The corresponding eigenvectors form the basis that spans the R space".
Clearly, some of the students were able to make observations that were generally consistent with mathematical definitions of eigenvalue and eigenvector and others made remarks that were slightly askew from such definitions.
However, being able to explore the geometry associated with eigenvectors and eigenvalues allowed some students to identify that MATLAB provided anomalous results when they encountered the following problem (2c in the accompanying activity):
- Using Eigenizer, determine how many times the vectors x and Ax are collinear? When the vectors x and Ax are collinear, what is the significance of the value of the eigenvalue in relation to the direction of an eigenvector of A, x, and its image T(x)?
- How many eigenvalues does A have? Are they real or nonexistent? Are they distinct or multiple? How many are positive? negative? zero? Is A invertible? Explain.
- Obtain a rough estimate of an eigenvector for each eigenvalue and then reveal the rest of the eigenvalue equation by pressing one of the "Show lambda # equation" buttons. Is the eigenvalue equation a truth within reasonable error? Move the vector x so it is no longer an eigenvector. Is the new equation true? Is the set of eigenvectors linearly independent? Explain.
- Compute by hand or by using Matlab the characteristic polynomial, the exact eigenvalues, and exact eigenvectors. Compare your results to your estimates in part 3. If you use Matlab,
- v = poly(A) gives the coefficients of the characteristic polynomial of matrix A, starting with the highest-degree term.
- k = roots(v) gives the roots of the characteristic polynomial of A, or use eig(A) to accomplish the same thing.
- nulbasis(A-k(1)*eye(2)) and nulbasis(A-k(2)*eye(2)) will give the eigenvectors, where each k(i) is a particular eigenvalue of A.
- If a matrix produced non-existent eigenvalues, what did you determine about the eigenvalues from part 4 when you computed them by hand or via Matlab?
Here are some sample solutions provided by students for the 2c matrix:
Our other samples are in a separate page because of their width.
Better understanding of both the computational and geometric roles of eigenvalues and eigenvectors, when combined with enhanced understandings of change of bases and linear transformations, sets the stage for investigations into diagonalization and potentially Singular Value Decompositions. One of our goals in having students explore the geometric perspective associated with eigenvalues and eigenvectors is to get students to begin to recognize that computations cannot be taken at face value but need to be examined within context for validity.
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