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 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* 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 and uses the eigencircles for easy demonstrations of relations between the eigenvectors, sign of the determinant, etc.

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?