User:Bart vanderbeke/sandbox

Eigencircle of a $2\times 2$ matrix

Eigencircles support a geometric interpretation of linear transformations.

This article only shows the basic properties of an eigencircle.

The referred articles derive more properties

An eigenvalue of a square $2\times 2$ matrix $A$ is a number $\lambda$ such that $AX=\lambda X$ for an $X\neq 0$.

Fig. 1: Basic properties of an eigencircle

Fig. 2: Characteristic points on an Eigencircle

Fig. 3: How to read eigenvectors from an eigencircle

Fig. 4: Eigencircle and rotation and scaling of a vector

To determine the eigenvalues $\lambda$ of a matrix $A$ a solution for $\det(A-\lambda I)=0$ is to be found.

The collection of eigenvalues $EV$ can be written as:

$EV=\left\{\lambda \ |\ \exists {\vec {x}}\left[{\begin{matrix}x\\y\\\end{matrix}}\right]and\ \left[{\begin{matrix}\lambda &0\\0&\lambda \\\end{matrix}}\right]\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=A\left[{\begin{matrix}x\\y\\\end{matrix}}\right]\right\}$

We now extend the concept of eigenvalues by looking for the elements of the set  $EC$ below:

$EC=\left\{\left(\lambda ,\mu \right)\ |\ \exists \left[{\begin{matrix}x\\y\\\end{matrix}}\right]and\ \left[{\begin{matrix}\lambda &+\mu \\-\mu &\lambda \\\end{matrix}}\right]\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=A\left[{\begin{matrix}x\\y\\\end{matrix}}\right]\right\}$

Using this broader concept, for every $\left[{\begin{matrix}x\\y\\\end{matrix}}\right]$ a $(\lambda ,\mu )$ can be found such that:

$L_{\lambda \mu }\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=\left[{\begin{matrix}\lambda &+\mu \\-\mu &\lambda \\\end{matrix}}\right]\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=A\left[{\begin{matrix}x\\y\\\end{matrix}}\right]$

$\left(\lambda ,\mu \right)$ is called a $\left(\lambda ,\mu \right)-eigenpair$ and $\left[{\begin{matrix}x\\y\\\end{matrix}}\right]$ is the corresponding $\left(\lambda ,\mu \right)-eigenvector$.

The condition for $\left(\lambda ,\mu \right)-eigenpairs$ to exist can be written as below:

${EC}_{=}\left\{\left(\lambda ,\mu \right)\ |\ \exists {\vec {x}}\left[{\begin{matrix}x\\y\\\end{matrix}}\right]and\ \left[{\begin{matrix}\lambda &+\mu \\-\mu &\lambda \\\end{matrix}}\right]\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=A\left[{\begin{matrix}x\\y\\\end{matrix}}\right]\right\}\neq \left\{\right\}$

$det\left(A-L_{\lambda \mu }\right)=\left|{\begin{matrix}a-\lambda &b+\mu \\c-\mu &d-\lambda \\\end{matrix}}\right|=0$

$\lambda ^{2}-\left(a+d\right)\lambda +\det {\left(A\right)}-\left(b-c\right)\mu +\mu ^{2}=0$

Using the following identities, the equation can be simplified:

$f={\frac {\left(a+d\right)}{2}}$

$g={\frac {\left(b-c\right)}{2}}$

$r^{2}=f^{2}+g^{2}$

$\det {\left(A\right)}=r^{2}-\rho ^{2}$

$\rho ^{2}=\left({\frac {a-d}{2}}\right)^{2}+\left({\frac {b+c}{2}}\right)^{2}$

$\left(\lambda -f\right)^{2}+\left(\mu -g\right)^{2}-\rho ^{2}=0$

The set $EC$ containing all $\left(\lambda ,\mu \right)-eigenvalues$ is a circle on the $\left(\lambda ,\mu \right)-plane$ with center $C(f,g)$ and radius $\rho$.

This circle is called the eigencircle of $A$.

This is illustrated in Figure 1.

$EC=\left\{\left(\lambda ,\mu \right)\ |\ \exists {\vec {x}}\left[{\begin{matrix}x\\y\\\end{matrix}}\right]and\ {\mathcal {t}}\left({\vec {x}}\right)=\left[{\begin{matrix}\lambda &+\mu \\-\mu &\lambda \\\end{matrix}}\right]\left[{\begin{matrix}x\\y\\\end{matrix}}\right]=A\left[{\begin{matrix}x\\y\\\end{matrix}}\right]\right\}$

$EC=\left\{\left(\lambda ,\mu \right)\ |\ \left(\lambda -f\right)^{2}+\left(\mu -g\right)^{2}-\rho ^{2}=0\right\}$

Every eigencircle contains four characteristic $\left(\lambda ,\mu \right)-eigenpairs$, this is illustrated in Figure 2.

The construction in Figure 3 shows how the eigenvectors of the matrix $A$ can be read from the eigencircle.

Figure 4 illustrates how a the existence of a $\left(\lambda ,\mu \right)-eigenpair$ $\left(\lambda _{1},\mu _{1}\right)$ can be interpreted:

If a $\left(\lambda _{1},\mu _{1}\right)_{cartesian}=\left(s_{1},-\theta _{1}\right)_{polar}$ exists on the eigencircle, a vector $x_{1}$, with $\|x1\|=1$, must exist such that $\angle \left(x_{1},Ax_{1}\right)=\theta _{1}$ and $\|Ax1\|=s1$.