Math Explained to Programmers — QR Decomposition (Gram-Schmidt)

Clean Code by Orthogonalizing Your Monolith

Imagine you have a matrix A with each column as a vector:

A = [x₁ x₂ x₃ ... xₙ]

Our goal is to decompose A into:

A = QR

Where:

• Q is a matrix with orthonormal columns (clean, independent directions)
• R is an upper-triangular matrix showing how to reconstruct A from Q

This is exactly what the Gram-Schmidt process achieves — step-by-step orthogonalization, just like refactoring a tangled monolith into clean, decoupled modules.

Why Orthogonalize?

The Gram-Schmidt process removes redundancy and overlap among vectors, making each new vector orthogonal to those we’ve already processed.

To developers:

It’s similar to refactoring your (yes, also mine) spaghetti code into clearly defined, independent modules.

Gram-Schmidt, Step-by-Step

Step 1: Start With the First Vector (Your First “Clean Module”)

Define the first orthogonal vector simply as:

u₁ = x₁
q₁ = u₁ / ‖u₁‖