Quick answer
Cholesky decomposition (Cholesky factorization) writes a real symmetric positive definite matrix A as A = L Lᵀ, with L lower triangular and L[i,i] > 0.
Formula
- A = L Lᵀ
- L lower triangular
- L[i,i] > 0 for all i
Introduction
Cholesky decomposition is one of the first structured factorizations many students meet after LU. It turns a symmetric positive definite matrix into a product of a lower-triangular factor and its transpose. Use the Cholesky Decomposition Calculator on our home page to see numeric factors while you read.
The method is not a trick for any square matrix. It assumes symmetry and positive definiteness, which is why the algorithm only needs the lower triangle of A and why each diagonal square root stays real when the theory applies.
After this overview, study the Cholesky decomposition formula for the entry-by-entry updates, then follow the how to perform Cholesky decomposition checklist when you compute L by hand.
Definition and meaning
Start with a real symmetric matrix A. If A is also positive definite, there exists a unique lower triangular L with positive diagonal entries such that A = L Lᵀ. That equality is the whole definition: you are not finding a generic LU pair, you are finding one structured factor designed for SPD matrices.
Positive definite means xᵀ A x > 0 for every nonzero vector x. In practice this shows up as positive leading principal minors, positive eigenvalues, or a successful Cholesky run without hitting a negative value under a square root.
Cholesky is sometimes called Cholesky factorization. Both names refer to the same object L. The transpose convention L Lᵀ is standard in many engineering texts; some software libraries store an upper factor U with A = Uᵀ U instead.
Compared with a general LU factorization, Cholesky uses roughly half the arithmetic on SPD problems because symmetry is built into the update formulas. That efficiency is why covariance matrices and normal equations often go through Cholesky rather than a full LU path.
When A is only symmetric but not positive definite, the classical real Cholesky algorithm fails at the first nonpositive pivot under the square root. You may still need a decomposition, but it will be a different method, such as LDLᵀ with pivoting or an eigenvalue-based approach.
Students meet Cholesky right after matrix factorizations and before large-scale numerical methods. The definition is the anchor: every other article on this site refers back to A = L Lᵀ and to the SPD assumptions.
Core statement
- A = L Lᵀ
- L lower triangular
- L[i,i] = sqrt( A[i,i] - sum_{k<i} L[i,k]² )
The formula section on the home page lists the full off-diagonal update as well. The diagonal rule is the quick test: if the quantity inside the square root is negative, A is not positive definite in exact arithmetic.
Uniqueness comes from requiring L to have positive diagonals. Without that sign convention, you could multiply columns of L by ±1 and still form L Lᵀ in some cases.
For 2×2 matrices you can memorize a pattern, but for 3×3 and 4×4 sizes an organized table of L entries prevents index mistakes.
How to recognize Cholesky decomposition
- Confirm symmetry. Check A[i,j] = A[j,i]. Cholesky for real A expects a symmetric matrix.
- Confirm positive definiteness. Use a leading-minor test, eigenvalue sign check, or attempt the algorithm and watch each diagonal square root.
- Read A = L Lᵀ as the target. You are solving for L, not for separate L and U factors as in LU.
- Connect to applications. Covariance matrices, normal equations, and many optimization kernels assume this structure.
- Verify with a tool when learning. Compare hand work to the calculator once per example to catch a single wrong subtraction early.
Tiny numeric illustration
Let A = [[4, 2], [2, 3]]. Then L[1,1] = sqrt(4) = 2, L[2,1] = 2/2 = 1, and L[2,2] = sqrt(3 - 1) = sqrt(2).
Multiply L Lᵀ to recover A. If off-diagonal entries do not match, symmetry was violated or an off-diagonal update used the wrong column of L.
Try the same matrix on the home calculator and read L from the lower-triangular grid. Matching one row by hand before trusting automation builds confidence.