Orthogonal Group
The orthogonal group \( O(n, \mathbb{R}) \) encompasses all \( n \times n \) square matrices that are orthogonal. $$ O(n, \mathbb{R}) = \{ A \in M_n(\mathbb{R}) : AA^T = A^TA = I \} $$ A matrix \( A \) is deemed orthogonal if the product of the matrix and its transpose yields the identity matrix \( I \), denoted as \( A^T A = AA^T = I \).
The defining properties of an orthogonal matrix include the formation of orthonormal sets with its columns (and rows). This entails that the constituent vectors are mutually perpendicular (orthogonal) and possess unit length (normal).
Geometrically, an orthogonal matrix represents a transformation in space that conserves both distances and angles, such as rotations or reflections.
Specifically, the orthogonal group has two connected components: the special orthogonal group, \( SO(n, \mathbb{R}) \), comprising matrices with determinant +1 representing rotations, and the component with matrices having determinant -1, which signifies reflections.
When referring to the orthogonal group "leaving the origin fixed," we imply that these transformations maintain the position of the coordinate system's origin intact; instead, they alter other points to preserve distances and angles relative to the origin. In disciplines like physics and engineering, these motions are recognized as rigid movements, and the orthogonal group can be viewed as the group of rigid motions of \( \mathbb{R}^n \) that uphold the origin's position.
Example
Here's an illustrative example of a real \( 2 \times 2 \) orthogonal matrix. Consider a rotation matrix of 45 degrees (or \( \frac{\pi}{4} \) radians):
$$ A = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $$
To verify the orthogonality of this matrix, we conduct multiplication between the two matrices.
The product of these matrices involves multiplying each row of \( A \) by each column of its transpose \( A^T \), followed by summing the products of corresponding elements. Here are the steps:
$$ A^T A = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $$
$$ A^T A = \begin{bmatrix} \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} -\frac{\sqrt{2}}{2} \cdot ( -\frac{\sqrt{2}}{2} ) & \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} -\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} \end{bmatrix} $$
$$ A^T A = \begin{bmatrix} \frac{\sqrt{2} \cdot \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2} \cdot \sqrt{2}}{2 \cdot 2} & 0 \\ 0 & \frac{\sqrt{2} \cdot \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2} \cdot \sqrt{2}}{2 \cdot 2} \end{bmatrix} $$
$$ A^T A = \begin{bmatrix} \frac{1}{2} + \frac{1}{2} & 0 \\ 0 & \frac{1}{2} + \frac{1}{2} \end{bmatrix} $$
$$ A^T A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
This confirms the orthogonality of matrix \( A \). Geometrically, it signifies a counterclockwise rotation of the plane around the origin by 45 degrees, preserving the distance from the origin.
