Math Explained to Programmers — LU Decomposition

LU Decomposition If It Were a Git Commit

LU Decomposition Step-by-Step

Imagine you have the following 3x3 matrix:

A = [1 2 3]
    [4 5 6]
    [7 8 9]

Your goal is to factor it into:

A = LU

Where L is a lower triangular matrix, and U is an upper triangular.

Why LU Decomposition?

LU decomposition simplifies matrix operations, particularly solving linear equations.

In developer words, think of LU decomposition as refactoring a messy codebase step-by-step. You’re not rewriting everything at once — instead, you log each transformation carefully. That way, solving linear systems later becomes easier (like debugging or adding features becomes smoother after a clean refactor — a better developer life).

Step 1: Eliminate Element a₂₁

We begin by eliminating the element in the second row, first column:

Operation: Row₂ → Row₂ – 4 × Row₁

Represented by an elementary matrix:

L₁ = [ 1  0  0]
     [-4  1  0]
     [ 0  0  1]

Apply this to A (refactoring):

U₁ = L₁ · A =
[ 1  2  3]
[ 0 -3 -6]
[ 7  8  9]

We also keep track of the inverse of this transformation (git history):