The blue line represents the vector x. It will always appear as a line from the center to a point on the unit circle. You can change the vector x by moving the mouse around while holding down the button.

The red line represents the vector Ax, formed by left-multiplying x by A, i.e applying the matrix A on the blue line. The length of Ax changes as x changes.

The gray circle in the center of the applet represents the unit circle, and you can see the pink image of this circle by clicking the option Draw image of unit circle. You can see the orange eigenvectors by clicking the option Show Eigenvectors.

Some mathematical aspects to note:

• Eigenvectors are parallel to input vectors (def: x is EV if Ax=ax, where a is a number, the eigenvalue, i.e. the ratio of Ax and x)
• The eigenvectors are orthogonal if (and only if) A is symmetrical: this is what states the spectral theorem
• The image of the unit circle is an ellipse, that means that the matrix A only stretches and rotates, i.e. it is linear
• if (and only if) A is orthogonal (e.g. [[0,-1], [1, 0]]) the unit circle is mapped on itself (def: A is orthogonal if | Ax | = | x |)
• Matrices of the type [[cos a, sin a],[-sin a, cos a]] (e.g. [[0,-1], [1, 0]]) really rotate the input vector
• The range of matrices A with det A = 0 (e.g. [[2, 1], [1, 0.5]]) is a straight line, i.e. a space of rang 1