# Eigenvalues

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 Instructions Please enter a matrix whose values are not too big (less than 5) or too small (greater than .1) Otherwise you will have a difficult time interpreting the results. Note that the changes you have made do not register until you hit enter. The color blue will indicate that the value has not been fully entered. Use your mouse to select the vector on which you want the matrix to operate. You can drag your mouse as well as click, but if you're not on a very fast computer, please drag slowly. If you're experiencing technical difficulties using the applet please let us know. In your message please indicate your operating system as well as the version of your browser.

 Supporting Material The mouse makes the vector x move. At the same time the graph shows Ax, in color and also moving. The green circle appearing is the unit circle and the red oval is its image under the action of the matrix. The are traced as you vary the vector x. Possibly Ax is ahead of x. Possibly Ax is behind x. Sometimes Ax is parallel to x. At that parallel moment, Ax=x and x is an eigenvector. The eigenvalue comes from the length and direction of Ax. Depending on your choices of the matrix A, the applet will demonstrate various possibilities. 1. There are no (real) eigenvectors. The directions of x and Ax never meet. The eigenvalues and eigenvectors are complex. 2. There is only one line of eigenvectors. The moving directions of x and Ax meet but don't cross. 3. There are eigenvectors in two independent directions. This is typical! Ax crosses x at the first eigenvector, and it crosses back at the second eigenvector. Suppose A is singular (rank one). Its column space is a line. The vector Ax can't move around, it has to stay on that line. One eigenvector x is along the line. Another eigenvector appears when Ax = 0. Zero is an eigenvalue of a singular matrix. You can follow x and Ax for these matrices. How many eigenvectors and where? When does Ax go clockwise instead of counterclockwise? A=([0, -1], [1, 0]) A=([3, 0], [0, 3]) A=([1, 3], [1, 0]) (defective) A=([1, 2], [2, 1])