LA 101
M05 · L03
Module 5: Orthogonality

Orthogonal Matrices

When a matrix has orthonormal columns, its transpose is its inverse — and it preserves every length and angle in space.

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LA 101
M05 · L03
The Definition

QTQ = I

A square matrix Q is orthogonal if its columns are orthonormal. The condition QTQ = I encodes two facts: unit lengths on the diagonal, zero dot products off the diagonal.

Defining Property
Q^TQ = I \quad\Longleftrightarrow\quad Q^{-1}=Q^T
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LA 101
M05 · L03
The Free Inverse

Q−1 = QT

Inverting a general matrix costs O(n³). For orthogonal Q, the inverse is just the transpose — an O(n²) read. Solving Qx = b is simply x = QTb.

O(n³)
general inverse
vs
 
O(n²)
QT inverse
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LA 101
M05 · L03
Geometry Preserved

Rigid Motions

Orthogonal matrices preserve lengths and angles. Multiplying by Q rotates or reflects space — but never stretches or squishes it.

  • ‖Qx‖ = ‖x‖ — length preserved
  • (Qx)·(Qy) = x·y — dot product preserved
  • Angle between Qx and Qy = angle between x and y
  • det(Q) = ±1 always
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LA 101
M05 · L03
Canonical Example

Rotation by θ

2D Rotation Matrix
R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}
Key check
R(θ)ᵀ = R(−θ) = R(θ)⁻¹ · det = +1 (proper rotation)
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LA 101
M05 · L03
Reflections

Householder Reflection

Reflect across the hyperplane perpendicular to unit vector u: H = I − 2uuᵀ.

Householder
H = I - 2\mathbf{u}\mathbf{u}^T,\quad\|\mathbf{u}\|=1

H is symmetric, H² = I, det(H) = −1. Used in numerically stable QR factorization.

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LA 101
M05 · L03
Two Families

det = +1 or −1

det(Q)² = 1 because det(QᵀQ) = det(I) = 1. Two families emerge:

det(Q) = +1
Proper rotations — orientation-preserving (SO(n))
det(Q) = −1
Improper rotations — includes a reflection, orientation-reversing
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LA 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

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LA 101
M05 · L03
Signal Processing

Orthogonal Transforms

  • DFT matrix — unitary (complex orthogonal), F⁻¹ = F* free
  • DCT (JPEG) — orthogonal, energy-preserving image transform
  • MIMO precoding — Q rotates signal into channel eigenvectors
  • Beamforming — QᵀQy = y exact matched filtering
  • Householder QR — numerically stable, used in LAPACK
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LA 101
Up Next
Lesson 3 Complete

M5-L4: Least Squares

You've mastered orthogonal matrices — the rigid transformations that preserve shape. Next: what to do when a system has no exact solution at all — project onto the column space and solve the normal equations.

Coming Up
M5-L4: Least Squares →
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