Quick answer

Cholesky factorization writes A = L Lᵀ with L lower triangular. The site calculator returns L for 2×2, 3×3, and 4×4 inputs.

Formula

  • Factorization: A = L Lᵀ
  • L lower triangular
  • SPD required

Introduction

Whether your syllabus says decomposition or factorization, the Cholesky Decomposition Calculator targets the same L factor. The words differ; the matrix product does not.

Factorization language emphasizes reuse: once L is known, multiple right-hand sides can be processed with forward and backward triangular solves.

Read Cholesky in numerical analysis for stability context, and the entry formulas when you implement the factorization yourself.

Factorization versus decomposition

In linear algebra libraries, factorize often means produce factors, while decompose may mean the same operation. For Cholesky, both refer to A = L Lᵀ on SPD matrices.

The lower factor L is the object you store. The product L Lᵀ reconstructs A if rounding allows.

Some texts introduce LDLᵀ without square roots on the diagonal. That is related but not identical to classical Cholesky; this site implements the sqrt form.

Factorization calculators are useful when homework asks only for L, not for an explicit inverse of A.

Because only half of the symmetric matrix participates in updates, Cholesky factorization is the efficient SPD path compared with generic LU.

Naming aside, always verify matrix requirements before calling a matrix Cholesky-ready.

What is being factorized

  • A = L Lᵀ
  • L invertible when A is SPD
  • det(A) = (det L)² = ( product of L[i,i] )²

The determinant identity follows from L being triangular. It is a quick sanity check on small examples.

Off-diagonal entries of L are not unique if you drop the positive diagonal convention, so stick to the standard unique L.

Reusing L for solves is the practical payoff of factorizing once.

Using factorization on the site

  1. Confirm SPD context. Factorization here assumes real symmetric positive definite A.
  2. Run the calculator. Enter A and read L from the lower grid.
  3. Store L for later solves. In larger software, L persists across many right-hand sides.
  4. Reconstruct if auditing. Multiply L Lᵀ to confirm entries.
  5. Compare with LU when unsure. If Cholesky fails, A may not be SPD.

Factorization check

For A = diag(4, 9), L = diag(2, 3). The factorization rewrites A as a product of two thin triangular pieces.

For a dense 3×3 homework matrix, factorization output matches manual L when symmetry holds.

If you change one off-diagonal without its mirror, factorization stops with a symmetry error instead of silently returning garbage.