Orthogonal Transformations

Overview

Orthogonal transformations preserve lengths and angles. They include rotations and reflections, and are represented by orthogonal matrices with special properties.

Definition

A linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ is orthogonal if it preserves the inner product:

$T(\mathbf{u}) \cdot T(\mathbf{v}) = \mathbf{u} \cdot \mathbf{v} \quad \text{for all } \mathbf{u}, \mathbf{v}$

Equivalently, $T$ preserves length:

$\|T(\mathbf{u})\| = \|\mathbf{u}\| \quad \text{for all } \mathbf{u}$

Orthogonal Matrices

A matrix $Q$ is orthogonal if:

$Q^T Q = QQ^T = I$

Equivalently: $Q^T = Q^{-1}$

Properties

• Columns are orthonormal (unit length, mutually perpendicular)
• Rows are orthonormal
• $\lvert\det(Q)\rvert = 1$
• Eigenvalues have $\lvert\lambda\rvert = 1$

Examples in 2D

Rotation by $\theta$

$R_\theta = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

Verify: $R_\theta^T R_\theta = I$ ✓

$\det(R_\theta) = \cos^2\theta + \sin^2\theta = 1$ ✓

Reflection Across Line Through Origin

Across x-axis:

$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Across line $y = x$:

$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Across line at angle $\alpha$ to x-axis:

$\begin{bmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{bmatrix}$

Rotations vs Reflections

Type	$\det(Q)$	Geometric Effect
Rotation	$+1$	Preserves orientation
Reflection	$-1$	Reverses orientation

Examples in 3D

Rotation Around z-axis

$R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Rotation Around x-axis

$R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix}$

Rotation Around y-axis

$R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{bmatrix}$

Properties Preserved

Orthogonal transformations preserve:

• Lengths: $\|Q\mathbf{v}\| = \|\mathbf{v}\|$
• Angles: $\angle(Q\mathbf{u}, Q\mathbf{v}) = \angle(\mathbf{u}, \mathbf{v})$
• Dot products: $(Q\mathbf{u}) \cdot (Q\mathbf{v}) = \mathbf{u} \cdot \mathbf{v}$
• Cross products: $Q(\mathbf{u} \times \mathbf{v}) = (Q\mathbf{u}) \times (Q\mathbf{v})$ (if $\det(Q) = 1$)
• Orthogonality: $\mathbf{u} \perp \mathbf{v} \Rightarrow Q\mathbf{u} \perp Q\mathbf{v}$

Orthogonal Group O(n)

The set of all $n \times n$ orthogonal matrices forms a group:

• Identity: $I$ is orthogonal
• Closure: $Q_1 Q_2$ is orthogonal if $Q_1, Q_2$ are
• Inverse: $Q^{-1} = Q^T$ is orthogonal

Special Orthogonal Group SO(n)

Orthogonal matrices with $\det = 1$ (rotations only).

Gram-Schmidt Process

Converts any basis to orthonormal basis:

1. $\mathbf{v}_1' = \mathbf{v}_1 / \|\mathbf{v}_1\|$
2. $\mathbf{u}_2 = \mathbf{v}_2 - (\mathbf{v}_2 \cdot \mathbf{v}_1')\mathbf{v}_1'$, then $\mathbf{v}_2' = \mathbf{u}_2 / \|\mathbf{u}_2\|$
3. Continue for remaining vectors

Example: Verify Orthogonality

$Q = \frac{1}{3}\begin{bmatrix} 2 & 1 & -2 \\ 2 & -2 & 1 \\ 1 & 2 & 2 \end{bmatrix}$

Check $Q^T Q$:

• Each column has length $\sqrt{4+4+1}/3 = 1$ ✓
• Column products: $(2)(1)+(2)(-2)+(1)(2) = 2-4+2 = 0$ ✓

$Q$ is orthogonal.

Applications

• Computer graphics: rotations, reflections
• Physics: rigid body motion
• Signal processing: orthogonal transforms (FFT, DCT)
• Statistics: orthogonal regression
• Quantum mechanics: unitary operators