Decompose a symmetric positive-definite matrix into a lower triangular matrix.
Last updated: 2026-05-24T22:58:31.700Z | By ForgeCalc Engineering
Lower Triangular Matrix (L):
Verification: A = L·L^T
Product of L and its transpose reconstructs the original matrix
Enter numeric matrix entries; matrix must be symmetric and positive definite.
Cholesky decomposition is a matrix factorization technique that decomposes a symmetric positive-definite matrix A into the product of a lower triangular matrix L and its transpose L^T, written as A = L·L^T. This decomposition is useful in solving systems of linear equations, computing determinants, and numerical simulations. It's more efficient than general LU decomposition and is particularly important in computational statistics, physics simulations, and machine learning algorithms.
Matrix Equation:
Computation Algorithm:
For i = 1 to n:
L[i,i] = √(A[i,i] - Σ(L[i,k]²))
For j = i+1 to n:
L[j,i] = (A[j,i] - Σ(L[j,k]·L[i,k])) / L[i,i]
Decompose the 2×2 matrix: A = [[4, 2], [2, 3]]
A symmetric matrix is positive definite if all eigenvalues are positive, or equivalently, x^T·A·x > 0 for all non-zero vectors x. This ensures the Cholesky decomposition exists.
The Cholesky decomposition assumes A = L·L^T. For this to hold, A must equal its own transpose (symmetric). Non-symmetric matrices require different factorization methods.
Applications include solving linear systems Ax = b efficiently, computing matrix determinants, sampling from multivariate Gaussian distributions, and numerical optimization.
Cholesky is about twice as fast as LU for symmetric positive-definite matrices and requires less memory. However, LU decomposes more general matrices.
The algorithm will fail when taking the square root of a negative number. You can use modified Cholesky (adding perturbations) or find the nearest positive-definite matrix.
No, Cholesky decomposition requires square matrices. For rectangular matrices, use QR or SVD decomposition instead.
Cholesky decomposition has O(n³/3) floating-point operations, making it efficient for large systems compared to LU decomposition which is O(2n³/3).
Compute L·L^T and check if it equals the original matrix A. Due to floating-point errors, allow for small numerical differences (usually < 1e-10).
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