Math Explained to Programmers — Cholesky Decomposition

Refactor Half, Auto-Generate the Rest — If Your Code Isn’t a Total Mess

Understanding Cholesky Decomposition with Developer Language

Imagine you have a symmetric, positive definite matrix A.

“Symmetrical” means it mirrors across the diagonal, like when client and backend share the same data models.

“Positive definite” means it’s stable and well-behaved (which doesn’t crash the moment you breathe near it).

A typical 3×3 symmetric, positive definite matrix looks like:

A = [a₁₁ a₁₂ a₁₃]
    [a₁₂ a₂₂ a₂₃]
    [a₁₃ a₂₃ a₃₃]

Symmetry in developer work — when your frontend and backend share the same data models.

Why LLᵀ?

Cholesky decomposition rewrites A as:

A = L · Lᵀ

Where L is a lower triangular matrix (everything below the diagonal), and It Lᵀ is its transpose.

L = [l₁₁  0   0 ]        Lᵀ = [l₁₁ l₂₁ l₃₁]
    [l₂₁ l₂₂  0 ]              [ 0  l₂₂ l₃₂]
    [l₃₁ l₃₂ l₃₃]              [ 0   0  l₃₃]

Think of it like this:

• L = your backend core logic
• Lᵀ = the client code generated from contracts
• Together, they rebuild the full system A