Cholesky Decomposition Calculator — 2×2 Matrix Factorization
Enter the three independent entries of a 2×2 symmetric matrix (a₁₁, a₁₂, a₂₂) to get the Cholesky factor L — the unique lower-triangular matrix with positive diagonal entries such that L × Lᵀ = A.
L₁₁ = √a₁₁
Compute L₁₁
Compute L₂₁
Compute L₂₂
Result: L such that L·Lᵀ = A
- 1
Determinant det(A) = a₁₁ × a₂₂ − a₁₂²
4 × 3 − 2² = 8Must be positive for the matrix to be positive definite. - 2
L₁₁ = √a₁₁
√4 = 2 - 3
L₂₁ = a₁₂ ÷ L₁₁
2 ÷ 2 = 1 - 4
L₂₂ = √(a₂₂ − L₂₁²)
√(3 − 1²) = 1.414214
How does this calculator work?
For a 2×2 symmetric positive-definite matrix A = [[a₁₁, a₁₂], [a₁₂, a₂₂]], the Cholesky factor L is: L₁₁ = √a₁₁, L₂₁ = a₁₂/L₁₁, L₂₂ = √(a₂₂−L₂₁²). The matrix must satisfy a₁₁ > 0 and a₁₁·a₂₂ > a₁₂². Enter the three entries to get L and verify L·Lᵀ = A.
Formula
How this is calculated
The Cholesky decomposition factors a symmetric positive-definite matrix A into the product L · Lᵀ, where L is a lower-triangular matrix with strictly positive diagonal entries. It is the matrix equivalent of taking a square root and is numerically stable and about twice as fast as LU decomposition for symmetric positive-definite problems.
For a 2×2 matrix A = [[a₁₁, a₁₂], [a₁₂, a₂₂]], the three elements of L are found by expanding L · Lᵀ and matching entries to A: L₁₁ = √a₁₁, L₂₁ = a₁₂ / L₁₁, and L₂₂ = √(a₂₂ − L₂₁²). The decomposition exists if and only if the matrix is positive definite — meaning a₁₁ > 0 and det(A) = a₁₁ · a₂₂ − a₁₂² > 0. If either condition fails the calculator reports the matrix as not positive definite.
Applications include solving linear systems (substitute L·y = b, then Lᵀ·x = y), Monte Carlo simulation (generating correlated random variables), and computing matrix square roots in machine learning covariance problems. This calculator handles the 2×2 case analytically; for larger matrices a recursive or blocked algorithm is applied column by column.
Frequently asked questions
A symmetric matrix A is positive definite if xᵀ·A·x > 0 for every nonzero vector x. For 2×2 matrices this is equivalent to requiring a₁₁ > 0 and det(A) = a₁₁·a₂₂ − a₁₂² > 0. Covariance matrices and many physics/engineering stiffness matrices are always positive definite.
Multiply L by its transpose Lᵀ: the product should equal the original A. For the 2×2 case: (L · Lᵀ)[1,1] = L₁₁² = a₁₁, [1,2] = L₁₁·L₂₁ = a₁₂, [2,2] = L₂₁² + L₂₂² = a₂₂. If any entry disagrees by more than rounding error, check that you entered a symmetric matrix.
This calculator is limited to the 2×2 case for which the formulas are closed-form and displayable step by step. For larger matrices, apply the Cholesky–Banachiewicz algorithm: process each column j from left to right, computing L[j,j] = √(A[j,j] − Σ L[j,k]²) and L[i,j] = (A[i,j] − Σ L[i,k]·L[j,k]) / L[j,j] for rows i > j.
TG we-Calculate Editorial Team. (2026). Cholesky Decomposition Calculator — 2×2 Matrix Factorization [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cholesky-decomposition-calculator
TG we-Calculate Editorial Team. "Cholesky Decomposition Calculator — 2×2 Matrix Factorization." TG we-Calculate. 2026. https://we-calculate.com/calculator/cholesky-decomposition-calculator.
TG we-Calculate Editorial Team, "Cholesky Decomposition Calculator — 2×2 Matrix Factorization," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cholesky-decomposition-calculator
@misc{wecalculate_cholesky_decomposition_calculator, title = {Cholesky Decomposition Calculator — 2×2 Matrix Factorization}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cholesky-decomposition-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?