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Cholesky Decomposition Calculator.

Factor a symmetric positive-definite 2×2 matrix as L·Lᵀ.

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Set your values

Results update as you type.

L₁₁: 2

L₁₁

2.000000
L₂₁: 0.5

L₂₁

0.500000
L₂₂: 1.322876

L₂₂

1.322876

Results update automatically as you type.

Use Cases

Solving linear systems efficiently

Cholesky decomposition is used to solve systems of linear equations Ax = b for symmetric positive-definite matrices, often faster than general methods.

Example: Given A = [[4, 2], [2, 3]], find L such that A = L·Lᵀ.

Numerical stability in simulations

In Monte Carlo simulations, Cholesky decomposition generates correlated random variables from uncorrelated ones, ensuring numerical stability.

Example: Generate correlated normal variables using the Cholesky factor of the covariance matrix.

Frequently Asked Questions

What is the Cholesky decomposition of a 2×2 matrix?
For a symmetric positive-definite 2×2 matrix A, the Cholesky decomposition factors A as L·Lᵀ, where L is a lower triangular matrix. For a 2×2 matrix, L has the form [[l₁₁, 0], [l₂₁, l₂₂]]. The calculator computes these entries from the input values a₁₁, a₁₂, and a₂₂.
What are the conditions for a 2×2 matrix to have a Cholesky decomposition?
The matrix must be symmetric (a₁₂ = a₂₁) and positive-definite. For a 2×2 matrix, this means a₁₁ > 0 and the determinant a₁₁·a₂₂ - a₁₂² > 0. If these conditions are not met, the decomposition does not exist.
How do I use the Cholesky Decomposition Calculator?
Enter the values for a₁₁, a₁₂ (which equals a₂₁), and a₂₂ in the input fields. The calculator will then compute the lower triangular matrix L such that A = L·Lᵀ. Ensure the matrix is symmetric and positive-definite for valid results.

Tips & Common Mistakes

Tips

  • Check that a₁₁ > 0 and a₁₁·a₂₂ - a₁₂² > 0 to ensure the matrix is positive-definite.
  • Remember that the matrix must be symmetric: the value for a₁₂ must equal a₂₁.
  • Use the decomposition to verify your matrix is positive-definite; if the calculator returns an error, the matrix may not be positive-definite.
  • For a 2×2 matrix, the Cholesky factor L can be computed manually: l₁₁ = √a₁₁, l₂₁ = a₁₂/l₁₁, l₂₂ = √(a₂₂ - l₂₁²).

Common Mistakes to Avoid

  • Entering a₁₂ and a₂₁ as different values, breaking symmetry.
  • Using a matrix that is not positive-definite, such as a₁₁ ≤ 0 or a₁₁·a₂₂ - a₁₂² ≤ 0.
  • Confusing the Cholesky decomposition with LU decomposition; Cholesky is specific to symmetric positive-definite matrices.

Last updated: August 13, 2026