Geometrically, the equation [math]A\vec{x}=\lambda\vec{x}[/math] means that the vectors [math]\vec{x}[/math] and [math]A\vec{x}[/math] lie on the same line. In the applet below, notice there are two vectors on the screen, the [color=#ff0000]red vector[/color] is [math]\vec{x}[/math] and the [color=#0000ff]blue vector[/color] is [math]A\vec{x}[/math]. You can interact with the applet by clicking and dragging the [color=#ff0000]red arrow[/color] around, which will result in the [color=#0000ff]blue arrow[/color] moving (since changing [math]\vec{x}[/math] causes [math]A\vec{x}[/math] to move as well). See if you can find eigenvectors corresponding to different matrices [math]A[/math] by dragging around [math]\vec{x}[/math], keeping in mind the geometric interpretation mentioned above. Can you also approximate the associated eigenvalues to each eigenvector, simply geometrically?