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?