$\text{Matrix and Lie Group}$ $\text{Rotation matrix}$ $A= \begin{bmatrix} \cos(\beta) & -\sin(\beta)\\ \sin(\beta) & \cos(\beta) \end{bmatrix}$

$A \text{ is orthonormal matrix, it means } \langle \vec{v_{i}}, \vec{u_{j}} \rangle = 0 \quad i \neq j ,\quad \rvert \vec{v_{i}} \rvert=1 \quad \forall i$ $\text{The inverse of A is } A^{-1} = \begin{bmatrix} \cos(-\beta) & -\sin(-\beta)\\ \sin(-\beta) & \cos(-\beta) \end{bmatrix} = \begin{bmatrix} \cos(\beta) & \sin(\beta)\\ -\sin(\beta) & \cos(\beta) \end{bmatrix}$

$AA^{-1} = \begin{bmatrix} \cos(\beta) & -\sin(\beta)\\ \sin(\beta) & \cos(\beta) \end{bmatrix} \begin{bmatrix} \cos(\beta) & \sin(\beta)\\ -\sin(\beta) & \cos(\beta) \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

\begin{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} 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{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}

$\text{Derive 2 x 2 rotation matrix}$ $x = \begin{bmatrix} 1\\ 0 \end{bmatrix} \quad \implies \quad x' = \left[ \begin{array}{c} \cos\beta\\ \sin\beta \end{array} \right]$ $y = \begin{bmatrix} 0\\ 1 \end{bmatrix} \quad \implies \quad y' = \left[ \begin{array}{c} -\sin\beta\\ \cos\beta \end{array} \right]$ \begin{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} $\text{Derive 2 x 2 projection matrix}$

\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} $\text{Derive 2 x 2 reflection matrix}$

$\text{let angle between } l \text{ and } \vec{x} =\left[ \begin{array}{c} 1\\ 0 \end{array} \right] \text{ is }\beta$ $\text{ so the reflection of }\vec{x} \text{ is } \vec{x'} =\left[ \begin{array}{c} \cos2\beta\\ \sin2\beta \end{array} \right]$ $\text{angle between } l \text{ and } \vec{y} =\left[ \begin{array}{c} 0\\ 1 \end{array} \right] \text{ is } \pi/2 - \beta$ $\text{ so the reflection of } \vec{y} \text{ is }$ $\vec{y'} =\left[ \begin{array}{c} \sin2\beta\\ -\cos2\beta \end{array} \right]$ $\text{ Thereforce the reflection matrix is given by }$ $H = \begin{bmatrix} \cos2\beta & \sin2\beta\\ \sin2\beta & -\cos2\beta \end{bmatrix}$ $\text{ Rotation and reflection matrix form a group }$ \begin{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} \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} $\langle R(\beta) , H(\alpha) \rangle \text{ generates a group}$ \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} $\text{ Exponential map } \mbox{ maps } \mathfrak{so}(3) \mbox{ to } SO(3)$ $exp(\mathbf{A}) = \sum_{k=0}^{n} \frac{A^{k}}{k!}$ $\text{ If A is n by n matrix, then prove the series is converage}$ \begin{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} $\left[ \begin{array}{c} x(\beta) \\ y(\beta) \end{array} \right] = \begin{bmatrix} \cos\beta & -\sin\beta\\ \sin\beta & \cos\beta \end{bmatrix} \left[ \begin{array}{c} x \\ y \end{array} \right]$ $\frac{d}{d \beta} \left[ \begin{array}{c} x(\beta) \\ y(\beta) \end{array} \right] = \begin{bmatrix} -\sin\beta & -\cos\beta\\ \cos\beta & -\sin\beta \end{bmatrix} \left[ \begin{array}{c} x \\ y \end{array} \right]$ $\frac{d}{d \beta} \left[ \begin{array}{c} x(\beta) \\ y(\beta) \end{array} \right] = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \left[ \begin{array}{c} x \\ y \end{array} \right] \quad \beta = 0$ $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \quad \text{Infinitesimal Generator}$ \begin{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} \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}

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{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} $\left\langle G_x = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{bmatrix} \,, G_y = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ -1 & 0 & 0 \end{bmatrix} \,, G_z = \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \right\rangle \\$ \begin{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} 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{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} 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{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}