Orthogonal Transformations & Rotations

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# Orthogonal Transformations & Rotations ## 🎯 Learning Objectives After this topic you will be able to: - Identify orthogonal matrices ($Q^T Q = I$) - Prove that orthogonal transformations preserve lengths and dot products - Recognise rotation and reflection matrices - Compute with orthogonal matrices including $Q^...

Orthogonal Transformations & Rotations

🎯 Learning Objectives

After this topic you will be able to:

Identify orthogonal matrices ( $Q^T Q = I$ )
Prove that orthogonal transformations preserve lengths and dot products
Recognise rotation and reflection matrices
Compute with orthogonal matrices including $Q^{-1} = Q^T$
Understand the determinant of orthogonal matrices ( $\pm 1$ )

📋 Prerequisites

Orthogonality (Week 8) — orthonormal bases
Matrix Multiplication (Week 1) — matrix operations
Orthogonal transformations are the rigid motions of linear algebra

1. Intuition: Rigid Motions

An orthogonal transformation is a linear map that preserves lengths and angles. In

\mathbb{R}^2

and

\mathbb{R}^3

, these are exactly rotations and reflections — the transformations that don't stretch or squish space. Imagine a rigid object (like a book) in space. You can rotate it or flip it over, but you can't stretch it. Those operations are orthogonal transformations.

2. Orthogonal Matrices

2.1 Definition

Definition (Orthogonal Matrix). A square $n \times n$ matrix $Q$ is orthogonal if:

> Q^T Q = Q Q^T = I_n >

Equivalently, $Q^{-1} = Q^T$ . Key properties:

Columns of $Q$ form an orthonormal basis of $\mathbb{R}^n$
Rows of $Q$ also form an orthonormal basis
$\det(Q) = \pm 1$
$\|Q\mathbf{x}\| = \|\mathbf{x}\|$ for all $\mathbf{x}$ (preserves length)
$\langle Q\mathbf{x}, Q\mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{y} \rangle$ (preserves dot product)

Example 1: 2×2 Rotation Matrix

> Q = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} >

Check:

> Q^T Q = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} >

✓ $\det(Q) = \cos^2\theta + \sin^2\theta = 1$ Example 2: 2×2 Reflection Matrix

Reflection across $x$ -axis:

> Q = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} >

$Q^T Q = I$ ✓, $\det(Q) = -1$

2.2 Determinant = ±1

Q

is orthogonal,

Q^T Q = I

, so:

1 = \det(I) = \det(Q^T Q) = \det(Q^T)\det(Q) = (\det(Q))^2

Therefore

\det(Q) = \pm 1

$\det(Q) = 1$ : Proper rotation (orientation-preserving)
$\det(Q) = -1$ : Reflection or rotation + reflection (orientation-reversing)

3. Geometry of Orthogonal Transformations

3.1 Rotations in $\mathbb{R}^2$

R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Rotates vectors counterclockwise by

\theta

3.2 Rotations in $\mathbb{R}^3$

Rotation around

x

-axis by

\theta

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

Rotation around

y

-axis by

\theta

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

Rotation around

z

-axis by

\theta

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

3.3 Reflections

Reflection across a line through origin with unit direction

\mathbf{u}

H = I - 2\mathbf{u}\mathbf{u}^T

This is called a Householder reflection.

Example 3: Reflection across line y = x

$\mathbf{u} = \left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$ (unit normal to line $y=x$ ).

> H = I - 2\mathbf{u}\mathbf{u}^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - 2\begin{bmatrix} 1/2 & -1/2 \\ -1/2 & 1/2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} >

.

Check: $H(1,2) = (2,1)$ ✓ (swaps coordinates, reflecting across $y=x$ ).

4. Properties

Property	Formula
Inverse	$Q^{-1} = Q^T$
Preserves norm	$\
Preserves dot product	$\langle Q\mathbf{x}, Q\mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{y} \rangle$
Preserves angles	Angle between $Q\mathbf{x}$ and $Q\mathbf{y}$ = angle between $\mathbf{x}$ and $\mathbf{y}$
Determinant	$\det(Q) = \pm 1$
Eigenvalues	$
Product	Product of orthogonal matrices is orthogonal

5. Edge Cases & Gotchas

Situation	What Happens
$Q = I$	Orthogonal (identity is a rotation by 0°)
$Q = -I$	Orthogonal (rotation by 180° in $\mathbb{R}^2$ )
Permutation matrices	Orthogonal — they just reorder coordinates
Non-square matrices with orthonormal columns	Called semi-orthogonal; $Q^T Q = I$ but $Q Q^T \neq I$

6. Common Pitfalls

❌ Pitfall 1: Thinking "orthogonal" means entries are orthogonal

An orthogonal matrix has orthonormal columns, not necessarily orthogonal entries.

❌ Pitfall 2: Confusing orthogonal with diagonalisable

Orthogonal matrices are diagonalisable over

\mathbb{C}

(eigenvalues have

|\lambda|=1

) but may not be diagonalisable over

\mathbb{R}

(rotation matrices have complex eigenvalues).

❌ Pitfall 3: Assuming $Q^T = Q^{-1}$ for non-square matrices

For non-square matrices with orthonormal columns,

Q^T Q = I

but

Q Q^T \neq I

, so

Q^T \neq Q^{-1}

7. Formula Summary Table

Concept	Formula
Definition	$Q^T Q = I$
Inverse	$Q^{-1} = Q^T$
Norm preservation	$\
Rotation in $\mathbb{R}^2$	$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
Householder reflection	$H = I - 2\mathbf{u}\mathbf{u}^T$
Determinant	$\det(Q) = \pm 1$

8. 📝 Practice Questions

Q1: Orthogonal check

Is

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

orthogonal?

Solution: Compute $Q^T Q$ (should be $I$ ).

> Q^T Q = \frac{1}{9}\begin{bmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{bmatrix}\begin{bmatrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{bmatrix} >

. Entry (1,1): $\frac{1}{9}(4+4+1) = 1$ . (1,2): $\frac{1}{9}(4-2-2)=0$ . (1,3): $\frac{1}{9}(2+4+2)=\frac{8}{9}\neq 0$ .

Not orthogonal. Wait — let me recheck. $2(1) + (-2)(2) + 1(2) = 2 - 4 + 2 = 0$ . So (1,3)=0 ✓. Let me recompute: $2(1)+(-2)(2)+1(2)=2-4+2=0$ ✓. All off-diagonal should be 0. And diagonal entries = 1. Yes, it is orthogonal. $\det(Q) = \frac{1}{27}(2(2-(-4))-2(-4-2)+1(4-1)) = \frac{1}{27}(12+12+3)=\frac{27}{27}=1$ . So $Q$ is orthogonal with $\det=1$ . Q2: Inverse of orthogonal matrix

If

> Q = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} >

, find $Q^{-1}$ .

Solution: For orthogonal matrices,

> Q^{-1} = Q^T = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} >

. Check:

> Q Q^T = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} >

✓. Q3: Product of orthogonal matrices

If $Q_1$ and $Q_2$ are orthogonal, show $Q_1 Q_2$ is orthogonal.

Proof: $(Q_1 Q_2)^T (Q_1 Q_2) = Q_2^T Q_1^T Q_1 Q_2 = Q_2^T I Q_2 = Q_2^T Q_2 = I$ . ✓ Q4: Rotation by 45°

Find the image of $(1,0)$ under rotation by $45^\circ$ .

Solution: $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$ .

> R\left(\frac{\pi}{4}\right) = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix} >

. $R(1,0) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ . Q5: Reflection matrix

Find the matrix that reflects across the line $y = -x$ .

Solution: Unit normal to $y=-x$ : $\mathbf{u} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ .

> H = I - 2\mathbf{u}\mathbf{u}^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - 2\begin{bmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} >

.

Check: $H(1,0) = (0,-1)$ which is $(1,0)$ reflected across $y=-x$ ✓. Q6: Determinant of orthogonal matrix

If $Q$ is $5 \times 5$ orthogonal with $\det(Q) = -1$ , what does this tell us?

Solution: $\det(Q) = -1$ means $Q$ reverses orientation. It's a reflection, or a rotation composed with a reflection (improper rotation). It must have an odd number of eigenvalues equal to $-1$ . Q7: Orthogonal diagonalisation

If $A$ is symmetric, it can be orthogonal diagonalised: $A = Q^T D Q$ where $Q$ is orthogonal and $D$ is diagonal. Why is this useful?

Solution: This is the Spectral Theorem. It means we can find an orthonormal basis of eigenvectors. Computing $A^k = Q^T D^k Q$ is easy. PCA, covariance matrices, and quadratic forms all use this. Q8: Norm preservation proof

Prove $\|Q\mathbf{x}\| = \|\mathbf{x}\|$ for orthogonal $Q$ .

Proof: $\|Q\mathbf{x}\|^2 = \langle Q\mathbf{x}, Q\mathbf{x} \rangle = (Q\mathbf{x})^T (Q\mathbf{x}) = \mathbf{x}^T Q^T Q \mathbf{x} = \mathbf{x}^T I \mathbf{x} = \mathbf{x}^T \mathbf{x} = \|\mathbf{x}\|^2$ . Taking square roots gives the result. ✓ Q9: 3D rotation

Find the matrix for rotation around the $z$ -axis by $90^\circ$ .

Solution: $\cos 90^\circ = 0$ , $\sin 90^\circ = 1$ .

> R_z(90^\circ) = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} >

. Check: $R_z(1,0,0) = (0,1,0)$ ✓. Q10: Orthogonal matrix with det = -1

Find a $3 \times 3$ orthogonal matrix with determinant $-1$ .

Solution: Reflection across $xy$ -plane:

> Q = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} >

. $Q^T Q = I$ ✓. $\det(Q) = -1$ ✓.

🔗 Cross-References

Next topic: Multivariable Functions
Week 12 (Eigenvalues): Orthogonal matrices have eigenvalues with $|\lambda|=1$
BSCS3004 (Deep Learning): Orthogonal weight matrices prevent vanishing gradients
BSCS2004 (ML Foundations): PCA uses orthogonal transformations Join Discord Previous8.2 Gram-Schmidt & QRNext9.1 Multivariable Functions

What this note is really saying

Orthogonal Transformations & Rotations

🎯 Learning Objectives

📋 Prerequisites

1. Intuition: Rigid Motions

2. Orthogonal Matrices

2.1 Definition

2.2 Determinant = ±1

3. Geometry of Orthogonal Transformations

3.1 Rotations in R2\mathbb{R}^2R2

3.2 Rotations in R3\mathbb{R}^3R3

3.3 Reflections

4. Properties

5. Edge Cases & Gotchas

6. Common Pitfalls

❌ Pitfall 1: Thinking "orthogonal" means entries are orthogonal

❌ Pitfall 2: Confusing orthogonal with diagonalisable

❌ Pitfall 3: Assuming QT=Q−1Q^T = Q^{-1}QT=Q−1 for non-square matrices

7. Formula Summary Table

8. 📝 Practice Questions

🔗 Cross-References

3.1 Rotations in $\mathbb{R}^2$

3.2 Rotations in $\mathbb{R}^3$

❌ Pitfall 3: Assuming $Q^T = Q^{-1}$ for non-square matrices