## What is an eigencircle of a 2x2 matrix?

### Bottom-up definition of an eigencircle

If $$\mathfrak{t}$$ is a linear transformation, the relation between a vector $$\vec{x}$$ and its image $$\mathfrak{t}\left(\vec{x}\right)$$ can be described as

rotating the original vector until it is collinear with $$\mathfrak{t}\left(\vec{x}\right)$$ and

scaling until its length is the same as the length of $$\mathfrak{t}\left(\vec{x}\right)$$.

For each vector $$\vec{x}$$ this effect can be described as a tuple $$(s,\theta)$$ where s is the scaling and θ the rotation.

The set of all possible couples of rotation and scaling $$\left(s,\theta\right)\$$is a circle.

This circle is called the eigencircle of the linear transformation $$\mathfrak{t}$$.

### Top-down definition of an eigencircle

The eigencircle 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 $$\|\mathfrak{t}(\vec{x})\|$$.

### Formal denotation of an eigencircle

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

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

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

## Can I find the eigenvalues and eigenvectors of a matrix in a geometric way?

Using the concept of an eigencircle the eigenvalues and eigenvectors of a 2x2-matrix can be derived.

If the eigencircle of a matrix is constructed using the steps below, the eigenvectors and eigenvalues can be read from the geometric construction.

Construct the circle with center $$C(f,g)$$ and radius $$\rho$$.

 $$f=\frac{\left(a+d\right)}{2}$$ $$g=\frac{\left(c-b\right)}{2}=-\frac{\left(b-c\right)}{2}$$ (The value of g is the negation of the formula in the articles) $$\rho^2=\left(\frac{a-d}{2}\right)^2+\left(\frac{b+c}{2}\right)^2$$
 Eigencircle of A: $$\left(\lambda-f\right)^2+\left(\mu-g\right)^2-\rho^2=0$$

## Who defined the concept 'eigencircle'?

The concept eigencircle is described in two articles by EngleField and Farr.

Englefield, M. J., & Farr, G. E. (2006). Eigencircles of 2 x 2 Matrices. Mathematics Magazine Vol. 79 Oct.,2006, 281-289.

Englefield, M. J., & Farr, G. E. (2010). Eigencircles and associated surfaces. The Mathematical Gazette Vol.94 No. 531 (November 2010), 438-449.