Cholesky Decomposition Webtool

Verification: Check that A = LL^T

This section computes L·L^T and compares it with your input matrix A, showing both the reconstructed matrix and the element-wise differences.

Formulas used in the algorithm

For a symmetric positive definite matrix A ∈ ℝ^n×n, the Cholesky factorization writes A = LL^T, where L is a lower triangular matrix with positive diagonal entries.

General element-wise formulas

We construct L row by row. For indices in 1-based notation:

• Diagonal entries (for i = 1, 2, …, n):
  L_ii = √( A_ii − Σ_k=1ⁱ⁻¹ L_ik² )
• Sub-diagonal entries (for i = 2, …, n and j = 1, …, i−1):
  L_ij = ( A_ij − Σ_k=1^j−1 L_ikL_jk ) / L_jj

If, for some i, the quantity under the square root becomes non-positive, the matrix is not positive definite and the Cholesky factorization fails.

Algorithmic pseudocode (row-wise construction)

For i = 1 to n for j = 1 to i sum = Σ (from k = 1 to j−1) L[i, k] · L[j, k] if i = j then L[i, j] = sqrt( A[i, i] − sum ) # diagonal else L[i, j] = ( A[i, j] − sum ) / L[j, j] # below diagonal end if end for end for

The webtool above implements exactly this scheme and shows each numerical step used to compute L from your input matrix A.