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.

Matrix size \(n\):

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

Output form: Decimal Surd / Fraction

1 — Compute

Decomposes the symmetric positive-definite matrix as A = LLᵀ with step-by-step detail.

Resets all dimensions, matrix entries, and results.

Footnotes

A matrix \(A\) is symmetric positive definite if \(A = A^T\) and all eigenvalues are strictly positive, equivalently \(\mathbf{x}^T A \mathbf{x} > 0\) for every nonzero vector \(\mathbf{x}\). ↑
Because \(L\) is triangular, solving \(A\mathbf{x}=\mathbf{b}\) reduces to two triangular solves (forward and back substitution), each costing \(O(n^2)\) operations after the \(O(n^3/3)\) factorization. ↑
Since \(A = LL^T\), we have \(\det(A) = \det(L)\det(L^T) = (\det L)^2 = \bigl(\prod_{i=1}^n \ell_{ii}\bigr)^2\). ↑
The Cholesky algorithm computes: \(\ell_{jj} = \sqrt{a_{jj} - \sum_{k=1}^{j-1} \ell_{jk}^2}\) for diagonal entries, and \(\ell_{ij} = \dfrac{1}{\ell_{jj}}\bigl(a_{ij} - \sum_{k=1}^{j-1} \ell_{ik}\ell_{jk}\bigr)\) for off-diagonal entries \(i > j\). ↑

Cite this tool

DOI: https://doi.org/10.5281/zenodo.20981167

Kapita, S. (2026). Cholesky Decomposition. Math Tools. https://doi.org/10.5281/zenodo.20981167

Kapita, Shelvean. "Cholesky Decomposition." Math Tools, 2026, doi.org/10.5281/zenodo.20981167.

@online{kapita2026choleskyfactors,
  author       = {Shelvean Kapita},
  title        = {{Cholesky Decomposition}},
  year         = {2026},
  organization = {Math Tools},
  doi          = {10.5281/zenodo.20981167},
  url          = {https://doi.org/10.5281/zenodo.20981167}
}