Physlib.Mathematics.SO3.Basic

The set of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$, where $A^\top$ denotes the transpose of $A$ and $I$ is the $3 \times 3$ identity matrix.

The set $SO(3)$, consisting of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$ (where $I$ is the identity matrix and $A^\top$ is the transpose), forms a group. The group operation is defined by matrix multiplication, the identity element is the $3 \times 3$ identity matrix $I$, and the inverse of a matrix $A \in SO(3)$ is its transpose $A^\top$.

The symbol $SO(3)$ is defined as a notation to represent the special orthogonal group in three dimensions, $SO_3$.

The topological space structure on the special orthogonal group $SO(3)$ is defined as the subspace topology (or subtype topology) inherited from the space of $3 \times 3$ real matrices $\mathbb{R}^{3 \times 3}$.

For any element $A$ in the special orthogonal group $SO(3)$, the $3 \times 3$ real matrix representing the group inverse $A^{-1}$ is equal to the matrix inverse of the $3 \times 3$ real matrix representing $A$.

This definition defines the natural group homomorphism from the special orthogonal group $SO(3)$ to the general linear group $GL(3, \mathbb{R})$. It maps each $3 \times 3$ real matrix $A$ in $SO(3)$ to itself, regarded as an invertible matrix in $GL(3, \mathbb{R})$.

The inclusion map that views an element of the special orthogonal group $SO(3)$ as a $3 \times 3$ real matrix in $M_3(\mathbb{R})$ is equal to the composition of the natural group homomorphism $SO(3) \to GL(3, \mathbb{R})$ and the inclusion map from the general linear group $GL(3, \mathbb{R})$ to $M_3(\mathbb{R})$.

The group homomorphism $\iota: SO(3) \to GL(3, \mathbb{R})$ that maps each matrix $A$ in the special orthogonal group $SO(3)$ to itself as an element of the general linear group $GL(3, \mathbb{R})$ is injective. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$.

The group homomorphism $\text{toProd}$ maps an element $A$ of the special orthogonal group $SO(3)$ to the pair $(A, A^\top)$ in the product of the monoid of $3 \times 3$ real matrices $M_3(\mathbb{R})$ and its opposite monoid $M_3(\mathbb{R})^{op}$. Here, $SO(3)$ consists of $3 \times 3$ real matrices $A$ satisfying $AA^\top = I$ and $\det A = 1$, and $A^\top$ denotes the transpose (which is the inverse $A^{-1}$ for elements of $SO(3)$).

For any element $A$ in the special orthogonal group $SO(3)$, the group homomorphism $\text{toProd}$ maps $A$ to the pair $(A, A^\top)$, where $A$ is the underlying $3 \times 3$ real matrix and $A^\top$ denotes its transpose.

The map $\text{toProd} : SO(3) \to M_3(\mathbb{R}) \times M_3(\mathbb{R})^{op}$, which sends an element $A$ of the special orthogonal group $SO(3)$ to the pair $(A, A^\top)$, is injective. Here, $SO(3)$ is the group of $3 \times 3$ real matrices with $\det A = 1$ and $AA^\top = I$, and $A^\top$ denotes the transpose of $A$.

The group homomorphism $\text{toProd} : SO(3) \to M_3(\mathbb{R}) \times M_3(\mathbb{R})^{op}$, which maps an element $A$ of the special orthogonal group $SO(3)$ to the pair $(A, A^\top)$, is continuous. Here, $SO(3)$ is equipped with the subspace topology inherited from the space of $3 \times 3$ real matrices $M_3(\mathbb{R})$.

The map $\text{toProd}: SO(3) \to M_3(\mathbb{R}) \times M_3(\mathbb{R})^{op}$, which maps an element $A$ of the special orthogonal group $SO(3)$ to the pair $(A, A^\top)$, is a topological embedding. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$, and $M_3(\mathbb{R})^{op}$ denotes the opposite monoid of $3 \times 3$ real matrices.

The natural map $\iota: SO(3) \to GL(3, \mathbb{R})$ that sends each matrix $A$ in the special orthogonal group to itself in the general linear group is a topological embedding. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ satisfying $\det A = 1$ and $AA^\top = I$ (where $I$ is the identity matrix and $A^\top$ is the transpose), and $GL(3, \mathbb{R})$ is the group of invertible $3 \times 3$ real matrices.

The special orthogonal group $SO(3)$, consisting of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$ (where $I$ is the identity matrix and $A^\top$ is the transpose), is a topological group. This means that, when $SO(3)$ is equipped with the subspace topology inherited from the space of real $3 \times 3$ matrices $\mathbb{R}^{3 \times 3}$, the group multiplication $(A, B) \mapsto AB$ and the inversion operation $A \mapsto A^{-1} = A^\top$ are continuous functions.

For any matrix $A$ in the special orthogonal group $SO(3)$, the determinant of the matrix $A - I$ is equal to zero, where $I$ denotes the $3 \times 3$ identity matrix.

For any matrix $A$ in the special orthogonal group $SO(3)$, the determinant of the matrix $I - A$ is equal to zero, where $I$ denotes the $3 \times 3$ identity matrix.

For every matrix $A$ in the special orthogonal group $SO(3)$, the real number $1$ is in the spectrum of $A$. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ such that $\det A = 1$ and $AA^\top = I$.

Given an element $A$ of the special orthogonal group $\text{SO}(3)$, this function provides the corresponding linear endomorphism of the 3-dimensional Euclidean space $\mathbb{R}^3$. Specifically, it maps the $3 \times 3$ matrix $A$ to its associated linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ relative to the standard basis.

For every matrix $A$ in the special orthogonal group $SO(3)$, the linear endomorphism of $\mathbb{R}^3$ associated with $A$ has an eigenvalue equal to $1$. Here, $SO(3)$ is the group of $3 \times 3$ real matrices such that $\det A = 1$ and $AA^\top = I$.

For every matrix $A$ in the special orthogonal group $SO(3)$, there exists a unit vector $v \in \mathbb{R}^3$ such that $v$ is stationary under the action of $A$, i.e., $Av = v$. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ satisfying $\det A = 1$ and $AA^\top = I$.

For every matrix $A$ in the special orthogonal group $SO(3)$, there exists an orthonormal basis $\{v_0, v_1, v_2\}$ of the 3-dimensional Euclidean space $\mathbb{R}^3$ such that the first basis vector $v_0$ remains invariant under the action of $A$, i.e., $Av_0 = v_0$. Here, $SO(3)$ is the group of $3 \times 3$ real matrices $A$ satisfying $\det A = 1$ and $AA^\top = I$.