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.

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