An eigenvalue of a 2 x 2 matrix satisfies the equation 

$$  \left(\begin{array}{cc} a & b \\\\ c & d \end{array} \right)\left( \begin{array}{c} x \\\\ y  \end{array}\right) = \lambda \left( \begin{array}{c} x \\\\ y \end{array}\right)   $$

[Graham Farr][1] multiplies by the identity matrix.  It is still defines eigenvalues $(A - \lambda I ) \vec{x} = 0$.

$$  \left(\begin{array}{cc} a & b \\\\ c & d \end{array} \right)\left( \begin{array}{c} x \\\\ y  \end{array}\right) =  \left(\begin{array}{cc} \lambda & 0 \\\\ 0 & \lambda \end{array} \right)\left( \begin{array}{c} x \\\\ y \end{array}\right)   $$

He embeds the complex numbers into the space of 2 x 2 matrices and asks for vectors which are rotated + dilated the matrix.

$$  \left(\begin{array}{cc} a & b \\\\ c & d \end{array} \right)\left( \begin{array}{c} x \\\\ y  \end{array}\right) =  \left(\begin{array}{cc} \lambda & \mu \\\\ -\mu & \lambda \end{array} \right)\left( \begin{array}{c} x \\\\ y \end{array}\right)   $$

The characteristic equation is defines a circle in $\mu,\lambda$.  This *[eigencircle][2]* is not really a fixed circle in the plane, but a collection of "characteristic" pairs of values forming a circle of values $(\mu,\nu) \in \mathbb{R}^2$.

\\[  \left| \begin{array}{cc} a - \lambda & b - \mu \\\\ c + \mu & d - \lambda \end{array} \right| = 0 \\]



Farr made an [applet demonstrating these calculations][3] and uses the eigencircles for easy demonstrations of relations between the eigenvectors, sign of the determinant, etc.
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Can you have eigencircles in more than two dimensions?  I suppose you can ask for two basis vectors $v_1, v_2$ such that $Av_1 = \lambda v_1 + \mu v_2 $ and $Av_2 = - \mu v_1 + \lambda v_2$.  It also seems 2 x 2 matrices are singling out not only two directions, but a whole circle's worth of vectors.

Is there a correct higher-dimensional generalization of this?  Have these objects been studied under a different name?


  [1]: http://www.csse.monash.edu.au/~gfarr/
  [2]: http://www.csse.monash.edu.au/~gfarr/research/slides/Farr-eigentalk.pdf
  [3]: http://www.csse.monash.edu.au/~gfarr/research/eigencircles.html