Utilize our free Cholesky Decomposition Calculator to factorize any symmetric positive-definite matrix A into a lower triangular matrix L and its conjugate transpose Lᵀ (A = LLᵀ). Essential for numerical analysis, solving linear equations, and Monte Carlo simulations. Quickly perform matrix factorization online!
Formula:
The Cholesky decomposition expresses a symmetric positive-definite matrix A as the product of a lower triangular matrix L and its conjugate transpose Lᵀ (or just transpose Lᵀ for real matrices):
A = L Lᵀ
Where:
- A is the input symmetric positive-definite matrix (e.g., a 3x3 matrix).
- L is the resulting lower triangular matrix.
- Lᵀ is the transpose of L.
For a 3x3 matrix, the structure is:
A = [a11 a12 a13
a21 a22 a23
a31 a32 a33]
L = [l11 0 0
l21 l22 0
l31 l32 l33]