Rotation of a Plane

ianmcloughlin.github.io linkedin github

Matrix

$\begin{bmatrix} w & x \\ y & z \end{bmatrix}$ $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ $= \begin{bmatrix} \cos A \\ \sin A \end{bmatrix}$

$\begin{bmatrix} w & x \\ y & z \end{bmatrix}$ $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ $= \begin{bmatrix} -\sin A \\ \cos A \end{bmatrix}$

$ \Rightarrow w = \cos A, $ $ y = \sin A,$ $x = -\sin A,$ and $z = \cos A$

$ M = \begin{bmatrix} \cos A & \sin A \\ -\sin A & \cos A \end{bmatrix}$

Eigenvalues

$ \det (\lambda I - A) = \begin{vmatrix} (\lambda - \cos A) & -\sin A \\ \sin A & (\lambda - \cos A) \end{vmatrix} $

$ = (\lambda - \cos A)^2 + \sin^2 A$

$ = \lambda^2 -2 (\cos A) \lambda + \cos^2 A + \sin^2 A$

$ = \lambda^2 -2 (\cos A) \lambda + 1$

$ \det (\lambda I - A) = 0 $

$\rightarrow \lambda^2 -2 (\cos A) \lambda + 1 = 0$

$\rightarrow \lambda = \frac{1}{2}\left(2 \cos A \pm \sqrt{4 \cos^2 A - 4}\right)$

$\rightarrow \lambda = \cos A \pm \sqrt{\cos^2 A - 1}$

$\rightarrow \lambda = \cos A \pm \sqrt{-(1 - \cos^2 A)}$

$\rightarrow \lambda = \cos A \pm \sqrt{-\sin^2 A}$

$\rightarrow \lambda = \cos A \pm (\sqrt{-1}) \sin A$

$\rightarrow \lambda = \cos A \pm i \sin A$