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
*A***x**, formed by left-multiplying **x** by *A, i.e*
applying the matrix A on the blue line. The
length of *A***x** 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*A***x**=a**x**, where a is a number, the eigenvalue, i.e. the ratio of*A***x**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 |*A***x**| = |**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