2x2 Eigenvector-Eigenvalue

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Let M be a 2-by-2 matrix, u a 2-by-1 vector, and λ a scalar, such that:

M u = λ u

Then u is called an eigenvector of M and λ is called an eigenvalue of M.

In a linear algebra course, one learns a number of techniques for discovering the eigenvalues and eigenvectors of a matrix. However, in this activity we take a strictly visual approach. The requirement M u = λ u means that the vector M u is a scalar multiple of the vector u. This means that the vectors M u and u must be parallel; i.e., the vector M u must point in the same or opposite direction as the vector u.

In the activity Eigenvalues.ggb, we locate a unit vector u on the unit circle. Drag the point "Drag me" at the tip of the vector u around the unit circle until the vectors M u and u are parallel (i.e., until they point in the same or opposite directions). At that point, the relation M u = &lambda u holds and u is an eigenvector and λ is an eigenvalue.

In the upper left corner of the worksheet we record the matrix M, the vector u, and the absolute value of &lambda. Note that if M u and u point in the same direction, then λ is positive, but if M u and u point in opposite directions, then λ is negative.

If you'd like to try a different matrix, double-click on the matrix M in the Algebra View, then edit accordingly.

I hope you enjoy this activity. It goes a long way towards understanding the geometrical significance of an eigenvector and eigenvalue.

View the Dynamic Worksheet.