## Linear Transformations of the Unit Circle

Suppose
$A$
is a 2 by 2 matrix operating on the unit circle. The transformation that
$A$
represents sends the unit circle onto a curve in the
$xy$
plane.
$\mathbf{x} = \begin{pmatrix}x_1 \\x_2 \end{pmatrix}$
is the position vector of a point on the unit circle. Then
$x_!^2+x_"^2=1$
.
If
$\mathbf{x} \rightarrow A \mathbf{x} =$
then the circle is sent to
$(A \mathbf{x})^T (A \mathbf{x}) =1$
.
Example:
$A= \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)$

$(A \mathbf{x})^T (A \mathbf{x}) = (x_1,x_2) \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}=1 \rightarrow x_1^2 =1$

Hence
$x_1 = -1, \: 1$
.
Example:
$A= \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$

$(A \mathbf{x})^T (A \mathbf{x}) = (x_1,x_2) \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}=1 \rightarrow x_1^2 +x_2^2=1$

The unibt circle is sent to itself, rotated by a right angle.
Example:
$A= \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)$

$(A \mathbf{x})^T (A \mathbf{x}) = (x_1,x_2) \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}=1 \rightarrow x_1^2 +2x_1x_2 +2x_2^2=1$

This is the equation of an ellipse.