Eigencircles of n x n matrices?

Question

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.

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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?

Answer 1

As I understand the problem it can be generalized. The $\small \lambda,\mu $-matrix can be seen as scaled rotation, and using $ \small r^2 = \lambda^2 + \mu^2 $ we can, because the $\small \lambda,\mu$ - matrix is invertible and is the left factor on the rhs (thus it is not really an eigenvalue analogon in my opinion) we can differently rewrite your original equation as

$\qquad \small \frac1{r^2} \begin{bmatrix} \lambda & -\mu \\\ \mu & \lambda \end{bmatrix} \cdot \begin{bmatrix} a & b \\\ c & d \end{bmatrix} \cdot \begin{bmatrix} x \\\ y \end{bmatrix}=\begin{bmatrix} x \\\ y \end{bmatrix} \cdot 1 $

or $\qquad \small \begin{bmatrix} \cos(\varphi) & -\sin(\varphi) \\\ \sin(\varphi) & \cos(\varphi) \end{bmatrix} \cdot \begin{bmatrix} a & b \\\ c & d \end{bmatrix} \cdot \begin{bmatrix} x \\\ y \end{bmatrix}=\begin{bmatrix} x \\\ y \end{bmatrix} \cdot r $

or
$\qquad \small (T \cdot A) \cdot X = X \cdot r$

where then the eigen-problem is
$\qquad \small \left\vert T \cdot A - r\cdot I \right\vert = 0 $

The left factor T is simply a rotation-matrix and r the eigenvalue of the rotated matrix $\small T\cdot A$.

If this characterization of the $\small \lambda, \mu$-matrix is the interesting one (which I assume) it can then be generalized to more dimensions in the obvious way and the $\small \lambda ,\mu $'s have to be determined as scalings of the cos/sin-parameters by the eigenvalues of the multidimensionally rotated matrix.

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[Update] Here is a picture of the construction according to my interpretation of the problem. I take some example-matrix A $\small \begin{bmatrix} 12&24\\\ 13&83 \end{bmatrix} $ then leftmultiply by the rotationmatrices of angles between 0 and $\small 2 \pi $. Each has then two eigenvalues $\small w_0 , w_1 $. The $\small \lambda,\mu$ -matrix is then $\small T \cdot w_0 $. Note, that the eigenvalues and thus the $\small \lambda,\mu$-matrix may become complex for some rotations. Then I plot the log of the absolute value of the two eigenvalues in different colors. Here is the picture:

^(source)

Answer 2

They are not exactly what you are looking for, but I think you should take a look at two-parameter eigenvalue problems. For instance check http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol18_pp420-437.pdf. Essentially, they are a system of two equations of the kind you are considering with different matrices but with the same unknowns $\lambda,\mu$. Since you have two equations in typically that setting, you get a finite number of solutions $(\lambda,\mu)$.

Answer 3

Adding to Gottfried's answer:
The eigencircle ${EC(\mathfrak{t})}_{cart}$ or ${EC(\mathfrak{t})}_{polar}$ of a linear transformation $\mathfrak{t}$ is a circle defined by all possible tuples $(s,\theta)$
where
$\theta=\angle\left(\vec{x,}\mathfrak{t}\left(\vec{x}\right)\right)$ is the angle between an original vector $\vec{x}$ and its image $\mathfrak{t}\left(\vec{x}\right)$ and
$s=\frac{\|\mathfrak{t}(\vec{x})\|}{\|\vec{x}\|}$ is the scaling $s$ from the original length $\|\vec{x}\|$ to the final length $\|\mathfrak{t}(\vec{x})\|$.

${EC(\mathfrak{t})}_{polar}=\left\{\left(\ s_{\vec{x}},\theta_{\vec{x}}\right)| \exists\vec{x}=\left[\begin{matrix}x\\y\\\end{matrix}\right]\ and\ \mathfrak{t}\left(\vec{x}\right)=\left[\begin{matrix}s_{\vec{x}}&0\\0&s_{\vec{x}}\\\end{matrix}\right]\left[\begin{matrix}\cos{\theta_{\vec{x}}}&-\sin{\theta_{\vec{x}}}\\+\sin{\theta_{\vec{x}}}&\cos{\theta_{\vec{x}}}\\\end{matrix}\right]\left[\begin{matrix}x\\y\\\end{matrix}\right]=A\left[\begin{matrix}x\\y\\\end{matrix}\right]\right\}$

${EC(\mathfrak{t})}_{polar}=\left\{\left(s_{\vec{x}},\theta_{\vec{x}}\right) |\ \exists \vec{x}\ such\ that\ s_{\vec{x}}=\frac{\|\mathfrak{t}(\vec{x})\|}{\|\vec{x}\|} \ and\ \theta_{\vec{x}}=\angle(\vec{x},\mathfrak{t}(\vec{x}))\right\}$

${EC(\mathfrak{t})}_{cart}=\left\{\left(\lambda,\mu\right)|\exists\vec{x}=\left[\begin{matrix}x\\y\\\end{matrix}\right]and\ \mathfrak{t}\left(\vec{x}\right)=\left[\begin{matrix}\lambda &-\mu\\+\mu&\lambda\\\end{matrix}\right]\left[\begin{matrix}x\\y\\\end{matrix}\right]=\left[\begin{matrix}s_{\vec{x}}&0\\0&s_{\vec{x}}\\\end{matrix}\right]\left[\begin{matrix}\cos{\theta_{\vec{x}}}&-\sin{\theta_{\vec{x}}}\\+\sin{\theta_{\vec{x}}}&\cos{\theta_{\vec{x}}}\\\end{matrix}\right]\left[\begin{matrix}x\\y\\\end{matrix}\right]\right\}$

I think extending it to higher dimensions will result in an 'eigenball'. But I suspect this will not be useful: the added value of the eigencircle is the visual interpretation and derivation of properties of a linear transform in the plane.

If you want to know more about eigencircles of 2x2 matrices, visit www.heavisidesdinner.com

Public Geogebra example showing eigencircles

visualization of a point, its image and the corresponding point on the eigencircle