Cholesky Decomposition Calculator
Factor a symmetric positive-definite matrix into L·Lᵀ with definiteness verification.
About Cholesky Decomposition Calculator
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
Worked example
With the default values below, Cholesky Decomposition Calculator returns L (Lower Triangular) of 2 0 0 6 1 0 -8 5 3. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Matrix A (rows separated by newlines or semicolons; values by commas or spaces)
- 4 12 -16 12 37 -43 -16 -43 98
Results
- L (Lower Triangular)
- 2 0 0 6 1 0 -8 5 3
- ||L·Lᵀ − A|| (should be ≈ 0)
- 0.000000000000
Step by step
- 1Matrix A (must be symmetric positive-definite)= 4 12 -16 12 37 -43 -16 -43 98
- 2L (A = L·Lᵀ)= 2 0 0 6 1 0 -8 5 3
- 3Residual ||L·Lᵀ − A||= 0.0000e+0
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 12 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.