Cholesky Decomposition Calculator
Factor a symmetric positive-definite matrix into L·Lᵀ with definiteness verification.
Inputs
L (Lower Triangular)
2 0 0 6 1 0 -8 5 3
||L·Lᵀ − A|| (should be ≈ 0)
0.000000000000
Step by step
Matrix A (must be symmetric positive-definite)
= 4 12 -16 12 37 -43 -16 -43 98
L (A = L·Lᵀ)
= 2 0 0 6 1 0 -8 5 3
Residual ||L·Lᵀ − A||
= 0.0000e+0
How it works
Cholesky decomposition factors a symmetric positive-definite matrix A into L·Lᵀ where L is lower triangular with positive diagonal entries. It is twice as efficient as LU decomposition and is widely used in statistics (covariance matrices), optimisation, and numerical simulations. The matrix must be both symmetric and positive-definite; otherwise the decomposition fails.
Formulas
Cholesky Factorisation
A = L·Lᵀ
- A
- Symmetric positive-definite matrix
- L
- Lower triangular with positive diagonal
Diagonal Element
Lⱼⱼ = √(Aⱼⱼ − Σₖ₌₁ʲ⁻¹ Lⱼₖ²)
- Lⱼⱼ
- j-th diagonal of L
Frequently Asked Questions
What is a positive-definite matrix?
A symmetric matrix A is positive-definite if xᵀAx > 0 for all non-zero vectors x. Equivalently, all its eigenvalues are positive.
What happens if the matrix is not positive-definite?
The algorithm encounters a non-positive value under the square root and reports an error. This is a definitive test for positive-definiteness.