2D Matrix Transformations, Eigenvectors and Eigenvalues

This applet investigates transformations of the plane under 2x2 matrices, and shows eigenvectors and eigenvalues and their significance to the transformations

Author and programmer: Ron Barrow

UK KS5 Year 13 Further Mathematics (Edexcel FP3, AQA FP4, MEI FP2)
US - Grades 11, 12 - Algebra II

   
           

Instructions below

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How to Use this Applet

The screen is designed to help you understand and visualize 2D matrix transformations, by looking at mappings of simple shapes. The screen starts with the 2×2 identity matrix and a square.
The white square is actually a transformation of a green square, but because the initial transformation matrix is the identity matrix, the white square is identical with the green square, and hides it. As soon as you change the matrix you'll see the two shapes separately.
The green shape (the object) can be changed and moved around by clicking and dragging on the vertices, where there are some small circles drawn. You'll see the effect on the white shape (the image).
The matrix can be changed by clicking the 8 small buttons at the bottom, which change the transformation, so that now a new image appears.

Dragging around the object shape will cause corresponding changes in the image. Play around - you should be able to work out what a, b, c and d represent.
You can zoom in or out (you will probably have to!). Clicking the "Reset Matrix" button resets the matrix to the 2×2 identity matrix. Clicking the "Reset Shape" button will set you back to the start shape.
Eigenvalues (if they are real) are given in cyan (light green/blue) and yellow. If there are corresponding eigenvector directions, then these are drawn on the axes in the matching colours. You should see how the eigenvector directions relate to the transformation of the shape.
Play around and have fun, but to get the most out of this applet, a good deal of thought is required.