Cholesky Decomposition

Factor a symmetric positive definite1 matrix as \(A = LL^T\).
Efficient method for solving linear systems and optimization problems.2

The Cholesky decomposition factors a symmetric positive definite matrix \(A\) as:

\[ A = LL^T \]

where \(L\) is a lower triangular matrix with positive diagonal entries.

Requirements:

  • \(A\) must be symmetric: \(A = A^T\)
  • \(A\) must be positive definite: \(\mathbf{x}^T A \mathbf{x} > 0\) for all \(\mathbf{x} \neq \mathbf{0}\)

Applications:

  • Solving linear systems: \(A\mathbf{x} = \mathbf{b}\) becomes \(L(L^T\mathbf{x}) = \mathbf{b}\)
  • Computing determinants: \(\det(A) = \bigl(\prod_i L_{ii}\bigr)^2\)
  • Generating correlated random variables
  • Numerical optimization (Newton's method)

This calculator computes \(L\) entry-by-entry using the standard Cholesky algorithm4, and displays the result in decimal or surd/fraction form.

Set matrix dimension

Enter the matrix size n and click Generate Matrix to create an n×n input grid.

Output form:
1 — Set Up
Loads a pre-filled 3×3 symmetric positive-definite matrix — ready to decompose.
2 — Compute
Decomposes the symmetric positive-definite matrix as A = LLᵀ with step-by-step detail.
Resets all dimensions, matrix entries, and results.