Cholesky Decomposition Calculator

Your details

Choose the dimension of the square matrix to decompose.
Element in row 1, column 1 of matrix A.
Element in row 1, column 2 of matrix A. Must equal A[2,1] for symmetry.
Element in row 1, column 3 of matrix A. Must equal A[3,1] for symmetry.
Element in row 2, column 1 of matrix A. Must equal A[1,2] for symmetry.
Element in row 2, column 2 of matrix A.
Element in row 2, column 3 of matrix A. Must equal A[3,2] for symmetry.
Element in row 3, column 1 of matrix A. Must equal A[1,3] for symmetry.
Element in row 3, column 2 of matrix A. Must equal A[2,3] for symmetry.
Element in row 3, column 3 of matrix A.
L[1,1]
2

Diagonal element of L, row 1

L[2,1]1
L[2,2]3
L[3,1]-1
L[3,2]1
L[3,3]1.732051
det(A)108
Symmetric?Yes
Positive definite?Yes
Reconstruction error ||A - LL^T||0
L[1,1]2
L[2,2]3
L[3,3]1.732051

Cholesky decomposition succeeded for your 3x3 matrix.

  • The lower triangular factor L has diagonal entries 2, 3, 1.732051, all positive, confirming positive definiteness.
  • det(A) = 108, computed cheaply as the square of the product of L's diagonal elements.
  • Reconstruction error ||A - LL^T|| is 0.00e+0, showing the factorization is numerically exact.
  • Once L is known, you can solve Ax = b in two back-substitution passes (Ly = b, then L^T x = y), which is about twice as fast as LU decomposition for symmetric systems.

Next stepTo solve a linear system Ax = b, use L to perform forward substitution (Ly = b), then back substitution (L^T x = y).

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