\[ \text{Matrix and Lie Group} \]
\[ \text{Rotation matrix} \]
\[
A= \begin{bmatrix}
\cos(\beta) & -\sin(\beta)\\
\sin(\beta) & \cos(\beta)
\end{bmatrix}
\]
\[ A \text{ is orthonormal matrix, it means } \langle \vec{v_{i}}, \vec{u_{j}} \rangle = 0 \quad i \neq j ,\quad \rvert \vec{v_{i}} \rvert=1 \quad \forall i\]
\[ \text{The inverse of A is }
A^{-1} =
\begin{bmatrix}
\cos(-\beta) & -\sin(-\beta)\\
\sin(-\beta) & \cos(-\beta)
\end{bmatrix}
=
\begin{bmatrix}
\cos(\beta) & \sin(\beta)\\
-\sin(\beta) & \cos(\beta)
\end{bmatrix}
\]
\[
AA^{-1} =
\begin{bmatrix}
\cos(\beta) & -\sin(\beta)\\
\sin(\beta) & \cos(\beta)
\end{bmatrix}
\begin{bmatrix}
\cos(\beta) & \sin(\beta)\\
-\sin(\beta) & \cos(\beta)
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
\]
\begin{equation}
\begin{aligned}
& A_{\beta=0}= \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix} \quad
& A_{\beta=\pi/2}= \begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}\\
& A_{\beta=\pi}= \begin{bmatrix}
-1 & 0\\
0 & -1
\end{bmatrix}\quad
& A_{\beta=\pi\frac{3}{2}}= \begin{bmatrix}
0 & 1\\
-1 & 0
\end{bmatrix} \nonumber
\end{aligned}
\end{equation}
Derive Rotation, Projection and Reflection matrix
Suppose A is the standard matrix for a linear transformation $T: \mathbb{R} \rightarrow \mathbb{R}$ that maps three linear indepedent vectors in $\mathbb{R}$, $x$, $y$, and $z$ as follows:
\begin{equation}
\begin{aligned}
& T:x \rightarrow x' \text{ or } & Ax = x'\\
& T:y \rightarrow y' \text{ or } & Ay = y'\\
& T:z \rightarrow z' \text{ or } & Az = z'\\
& \implies A[x, y, z] = [x', y', z']\\
& \text{let } X = [x, y, z] \text{ and } X' = [x', y', z'] \\
& \implies AX = X' \\
& \implies A = X'X^{-1} \\
& \text{let } X = I \\
& \implies A = X'
\end{aligned}
\end{equation}
\[ \text{Derive 2 x 2 rotation matrix}\]
\[
x = \begin{bmatrix}
1\\
0
\end{bmatrix} \quad \implies \quad
x' = \left[
\begin{array}{c}
\cos\beta\\
\sin\beta
\end{array}
\right]
\]
\[
y = \begin{bmatrix}
0\\
1
\end{bmatrix} \quad \implies \quad
y' = \left[
\begin{array}{c}
-\sin\beta\\
\cos\beta
\end{array}
\right]
\]
\begin{equation}
\begin{aligned}
& X = \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix} \implies
X' = \begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix}\\
& \implies A =
\begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix}
\end{aligned}
\end{equation}
\[ \text{Derive 2 x 2 projection matrix}\]
\begin{equation}
\begin{aligned}
& X' =\left[
\begin{array}{c}
l\cos\beta\\
l\sin\beta
\end{array}
\right]
\quad \text{ and }l = \cos\beta\\
& \implies X' =\left[
\begin{array}{c}
\cos^{2}\beta\\
\cos\beta\sin\beta
\end{array}
\right]\\
& Y' =\left[
\begin{array}{c}
l\cos\beta\\
l\sin\beta
\end{array}
\right]
\quad \text{ and }l = \sin\beta\\
& \implies Y' =\left[
\begin{array}{c}
\sin\beta \cos\beta\\
\sin^{2}\beta
\end{array}
\right]\\
& \implies A =
\begin{bmatrix}
\cos^{2}\beta & \sin\beta \cos\beta\\
\cos\beta \sin\beta & \sin^{2}\beta
\end{bmatrix}
\end{aligned}
\end{equation}
\[ \text{Derive 2 x 2 reflection matrix}\]
\[
\text{let angle between } l \text{ and }
\vec{x} =\left[
\begin{array}{c}
1\\
0
\end{array}
\right]
\text{ is }\beta
\]
\[
\text{ so the reflection of }\vec{x}
\text{ is }
\vec{x'} =\left[
\begin{array}{c}
\cos2\beta\\
\sin2\beta
\end{array}
\right]
\]
\[ \text{angle between } l \text{ and }
\vec{y} =\left[
\begin{array}{c}
0\\
1
\end{array}
\right]
\text{ is } \pi/2 - \beta \]
\[ \text{ so the reflection of } \vec{y} \text{ is } \]
\[ \vec{y'} =\left[
\begin{array}{c}
\sin2\beta\\
-\cos2\beta
\end{array}
\right]
\]
\[ \text{ Thereforce the reflection matrix is given by } \]
\[
H = \begin{bmatrix}
\cos2\beta & \sin2\beta\\
\sin2\beta & -\cos2\beta
\end{bmatrix}
\]
\[ \text{ Rotation and reflection matrix form a group } \]
\begin{equation}
\begin{aligned}
R(\beta) = \begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix} \quad
H(\alpha) = \begin{bmatrix}
\cos2\alpha & \sin2\alpha\\
\sin2\alpha & -\cos2\alpha
\end{bmatrix} \nonumber
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R(\beta) \circ R(\alpha) &=
\begin{bmatrix}
\cos\beta \cos\alpha - \sin\beta \sin\alpha & -(\cos\beta \sin\alpha + \sin\beta \cos\alpha)\\
\sin\beta \cos\alpha + \cos\beta \sin\alpha & -\sin\beta \sin\alpha + \cos\beta \cos\alpha
\end{bmatrix} &= R(\beta + \alpha)\\
R(\beta) \circ H(\alpha) &=
\begin{bmatrix}
\cos\beta \cos2\alpha - \sin\beta \sin2\alpha & \cos\beta \sin2\alpha + \sin\beta \cos2\alpha \\
\sin\beta \cos2\alpha + \cos\beta \sin2\alpha & \cos\beta \cos2\alpha -\sin\beta \sin2\alpha
\end{bmatrix} &= H(\alpha + \frac{\beta}{2})\\
H(\alpha) \circ R(\beta) &=
\begin{bmatrix}
\cos2\alpha\cos\beta + \sin2\alpha \sin\beta & \cos 2\alpha \sin\beta + \sin2\alpha\cos\beta \\
\sin2\alpha \cos\beta - \cos2\alpha \sin2\beta & -\cos2\alpha \sin\beta - \cos2\alpha \cos\beta
\end{bmatrix} &= H(\alpha - \frac{\beta}{2})\\
H(\beta) \circ H(\alpha) &=
\begin{bmatrix}
\cos2\beta \cos2\alpha + \sin2\beta \sin2\alpha & \cos2\beta \sin2\alpha - \sin2\beta \cos2\alpha \\
\cos2\beta \sin2\alpha - \cos2\beta \sin2\alpha & \sin2\beta \sin2\alpha + \cos2\beta \cos2\alpha
\end{bmatrix} &= R(2(\alpha - \beta)) \nonumber
\end{aligned}
\end{equation}
\[ \langle R(\beta) , H(\alpha) \rangle \text{ generates a group} \]
\begin{equation}
\begin{aligned}
\text{ The determinant of the rotation } \\
\det(R) &= \cos^{2}\beta + \sin^{2}\beta &= 1 \\
\text{ The determinant of the reflection } \\
\det(H) &= -(\cos^{2}2\alpha+ \sin^{2}2\alpha) &= -1
\end{aligned}
\end{equation}
$\text{ Exponential map } \mbox{ maps } \mathfrak{so}(3) \mbox{ to } SO(3)$
\[ exp(\mathbf{A}) = \sum_{k=0}^{n} \frac{A^{k}}{k!} \]
\[ \text{ If A is n by n matrix, then prove the series is converage} \]
\begin{equation}
\begin{aligned}
\exp(\mathbf{A}) & = \sum_{k=0}^{n} \frac{\mathbf{A}^{k}}{k!} \\
\sin(\mathbf{A}) & = \sum_{k=0}^{n} (-1)^{k}\frac{\mathbf{A}^{2k}}{2k!} \\
\cos(\mathbf{A}) & = \sum_{k=0}^{n} (-1)^{k}\frac{\mathbf{A}^{2k+1}}{(2k+1)!} \nonumber
\end{aligned}
\end{equation}
\[
\left[
\begin{array}{c}
x(\beta) \\
y(\beta)
\end{array}
\right] =
\begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix}
\left[
\begin{array}{c}
x \\
y
\end{array}
\right]
\]
\[
\frac{d}{d \beta}
\left[
\begin{array}{c}
x(\beta) \\
y(\beta)
\end{array}
\right] =
\begin{bmatrix}
-\sin\beta & -\cos\beta\\
\cos\beta & -\sin\beta
\end{bmatrix}
\left[
\begin{array}{c}
x \\
y
\end{array}
\right]
\]
\[
\frac{d}{d \beta}
\left[
\begin{array}{c}
x(\beta) \\
y(\beta)
\end{array}
\right] =
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\left[
\begin{array}{c}
x \\
y
\end{array}
\right] \quad \beta = 0
\]
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix} \quad \text{Infinitesimal Generator}
\]
\begin{equation}
\begin{aligned}
& \text{Let } \mathbf{A} =
\begin{bmatrix}
0 & -\beta \\
\beta & 0
\end{bmatrix}\\
& R(\beta) =
\begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix} \\
& R(\beta) = e^{\mathbf{A}} = \sum_{n=0}^{\infty} \frac{1}{n!} \mathbf{A}^{n} =
\begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
& \exp(\mathbf{A}) =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
+
\begin{bmatrix}
0 & -\beta \\
\beta & 0
\end{bmatrix}
+
\frac{1}{2!}
\begin{bmatrix}
-\beta^{2} & 0 \\
0 & \beta^{2}
\end{bmatrix}
+
\frac{1}{3!}
\begin{bmatrix}
0 & \beta^{3} \\
-\beta^{3} & 0
\end{bmatrix}
+
\frac{1}{4!}
\begin{bmatrix}
\beta^{4} & 0 \\
0 & \beta^{4}
\end{bmatrix}
+
\frac{1}{5!}
\begin{bmatrix}
0 & -\beta^{5}\\
\beta^{5} & 0
\end{bmatrix}
+ ...\\
& \exp(\mathbf{A}) =
\begin{bmatrix}
1 + \frac{1}{2!}(-\beta^{2}) + \frac{1}{4!}\beta^{4} + ... & -\beta + \frac{1}{3!}\beta^{3} - \frac{1}{5!}\beta^{5} + ...\\
\beta - \frac{1}{3!}\beta^{3} + \frac{1}{5!}\beta^{5} + ... & 1 + \frac{1}{2!}\beta^{2} + \frac{1}{4!}\beta^{4} + ...
\end{bmatrix} =
\begin{bmatrix}
\cos\beta & -\sin\beta\\
\sin\beta & \cos\beta
\end{bmatrix}
\end{aligned}
\end{equation}
Lie algebra, $\mathfrak{so}(3)$, is the set of skew-symmetric matrices. The generator of $\mathfrak{so}(3)$ conrespond to the derivatives of rotation
around the each of standard axes, evaluated at the identity:
\begin{equation}
\begin{aligned}
M_{z}(\theta) & =\begin{bmatrix}
\cos\theta & -\sin\theta & 0\\
\sin\theta & \cos\theta & 0\\
0 & 0 & 1
\end{bmatrix} & \Rightarrow \left. {\frac{d M_z}{d\theta}} \right|_{\theta=0} & =
\begin{bmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{bmatrix} \\
M_{y}(\theta) & =\begin{bmatrix}
\cos \theta & 0 & \sin\theta \\
0 & 1 & 0 \\
-sin\theta & 0 & \cos\theta
\end{bmatrix} & \Rightarrow \left. {\frac{d M_y}{d\theta}} \right|_{\theta=0} & =
\begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
-1 & 0 & 0
\end{bmatrix} \\
M_{x}(\theta) & =\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta& \cos\theta
\end{bmatrix} & \Rightarrow \left. {\frac{d M_x}{d\theta}} \right|_{\theta=0} & =
\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0
\end{bmatrix}
\end{aligned}
\end{equation}
\[
\left\langle G_x =
\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0
\end{bmatrix}
\,, G_y =
\begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
-1 & 0 & 0
\end{bmatrix}
\,, G_z =
\begin{bmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}
\right\rangle \\
\]
\begin{equation}
\begin{aligned}
\omega =
\left[
\begin{array}{c}
\omega_1 \\
\omega_2 \\
\omega_3
\end{array}
\right] & \in \mathbb{R}^{3} \\
\omega_1 G_x + \omega_2 G_y + \omega_3 G_z & \in \mathfrak{so}(3)
\end{aligned}
\end{equation}
The exponential map is take the skew symmetric matrices to rotation matrices is simply the matrix exponential over a linear combination of the generators:
\begin{equation}
\begin{aligned}
\text{Let } \mathbf{M} & =
\begin{bmatrix}
0 & -\omega_3 & \omega_2 \\
\omega_3 & 0 & -\omega_1 \\
-\omega_2 & \omega_1 & 0
\end{bmatrix} \\
\exp(\mathbf{M}) & = exp \left(
\begin{bmatrix}
0 & -\omega_3 & \omega_2 \\
\omega_3 & 0 & -\omega_1 \\
-\omega_2 & \omega_1 & 0
\end{bmatrix}
\right) \\
& = \mathbf{I} + M + \frac{1}{2!}M^2 + \frac{1}{3!}M^3 + ... \\
\text{Writing the term in pair, we have:} \\
\exp(\mathbf{M}) & = \mathbf{I} + \sum_{k=0}^{\infty} \left[ \frac{M^{2k + 1}}{(2k + 1)!} + \frac{M^{2k+2}}{(2k+2)!} \right] \\
\text{Now We take advantage of skew-symmetric matrices:} \\
\mathbf{M}^3 & = -(\omega^T \omega) \mathbf{M} \quad \text{ where }
\omega =
\left[
\begin{array}{c}
\omega_1 \\
\omega_2 \\
\omega_3
\end{array}
\right] \\
\text{First we extend the identity to general case:} \\
\omega^{T} \omega & = \langle \omega \,, \omega \rangle = \| \omega \|^{2} = \theta^{2} \\
\mathbf{M}^{2k+1} & = (-1)^{k} \theta^{2k} \mathbf{M} \\
\mathbf{M}^{2k+2} &= (-1)^{k} \theta^{2k} \mathbf{M}^{2} \\
\exp(\mathbf{M}) & = \mathbf{I} + \sum_{k=0}^{\infty} \left[ \frac{(-1)^{k} \theta^{2k} \mathbf{M}}{(2k+1)!} + \frac{(-1)^{k} \theta^{2k)} \mathbf{M}^{2}}{(2k+2)!} \right] \\
\exp(\mathbf{M}) & = \mathbf{I} + \sum_{k=0}^{\infty} \left[ \frac{(-1)^{k} \theta^{2k} \mathbf{M}}{(2k+1)!} \right] +
\sum_{k=0}^{\infty} \left[ \frac{(-1)^{k} \theta^{2k)} \mathbf{M}^{2}}{(2k+2)!} \right] \\
\exp(\mathbf{M}) & = \mathbf{I} + (\frac{1}{1!} - \frac{\theta^2}{3!} + \frac{\theta^4}{5!} + ...) \mathbf{M} +
(\frac{1}{2!} - \frac{\theta^2}{4!} + \frac{\theta^4}{6!} + ...)\mathbf{M^2} \\
\exp(\mathbf{M}) & = \mathbf{I} + \left( \frac{\sin \theta }{\theta} \right) \mathbf{M} + \left( \frac{1 - \cos\theta }{\theta^2} \right)\mathbf{M^2} \\
\end{aligned}
\end{equation}
This is the Rodrigue formula. The exponential map is yield a rotation by $\theta$ radians around axis given by
\[
\omega =
\left[
\begin{array}{c}
\omega_1 \\
\omega_2 \\
\omega_3
\end{array}
\right]
\]
Symmetric and Skew-Symmetric matrices
\[
\text{Let } \mathbf{C}_{nxn} \text{ be square matrix }. \text{We can write} \\
\mathbf{C} = \frac{1}{2}(\mathbf{C} + \mathbf{C}^t) + \frac{1}{2}(\mathbf{C} - \mathbf{C}^t) = \mathbf{A} + \mathbf{B} \\
\text{Where } \mathbf{A} = \mathbf{A}^t \text{is symmetric and } -\mathbf{B} = \mathbf{B}^t \text{ is skew-symmetric} \\
\text{Examples:} \\
\]
\begin{equation}
\begin{aligned}
\mathbf{C} &= \begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\\
\end{bmatrix} \\
\mathbf{A} &= \mathbf{C} + \mathbf{C}^t \Rightarrow
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\\
\end{bmatrix} +
\begin{bmatrix}
1 & 4 & 7\\
2 & 5 & 8\\
3 & 6 & 9\\
\end{bmatrix} =
\begin{bmatrix}
2 & 6 & 10\\
6 & 10 & 14\\
10 & 14 & 18\\
\end{bmatrix} \\
\mathbf{B} &= \mathbf{C} - \mathbf{C}^t \Rightarrow
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\\
\end{bmatrix} +
\begin{bmatrix}
1 & 4 & 7\\
2 & 5 & 8\\
3 & 6 & 9\\
\end{bmatrix} =
\begin{bmatrix}
0 & -2 & -4\\
2 & 0 & -2\\
4 & 2 & 0\\
\end{bmatrix} \\
\mathbf{C} &= \frac{1}{2}(\mathbf{A} + \mathbf{B})
\end{aligned}
\end{equation}
Suppose A is the standard matrix for a linear transformation $T: \mathbb{R} \rightarrow \mathbb{R}$ that maps three linear indepedent vectors in $\mathbb{R}$, $x$, $y$, and $z$ as follows: