Eigenvalues and eigenvectors

This applet is based in the one on Linear mappings, being essentially the same with the option of displaying eigenvectors and eigenvalues.

The black vector represents the input of the linear application, it can be edited by click and drag. The red vector represents the result of the mapping. The mapping matrix can also be edited. Remember that not all linear mappings have real eigenvalues.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Questions:

· What happens if the black vector lies over an eigendirection? Is there any relation with the eigenvalues?

· Introduce the following matrix a = 0, b = 1, c = -1, d = 0. Move the vector and try to guess what the transformation represents. It has no real eigenvectors, why?

· Is it possible for a two dimensional linear mapping to have more than two eigenvectors? Why?

· What happens if we transform the application to a basis of eigenvectors?

Coordinates

This applet aids to visualize the concept of coordinate and the concept of change of basis in linear algebra.

The red vector is editable by click and drag, and so are the basis vectors.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Observe that not only the orientation of the new base vectors is important, but also the module.

Try to insert two linearly dependent vectors as generators of the base B. You'll get an error message. Why?

Linear mappings

This applet shows the behaviour of a linear mapping in two dimensions. The image vector is shown in red, and the black vector can be displaced by click and drag.

The transformation matrix is editable.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Questions:

1. Construct the image of the vector (0,0). What happens?
2. Construct the image of the vector (1,1), and afther that, the image of the vector (2,2). What happens? Why?
3. Can you estimate, just playing with the vector, some eigenvector? Remember that not all linear mappings have real eigenvectors.

Related applets:

Eigenvalues and eigenvectors.