Cholesky Decomposition Calculator

The Cholesky decomposition factors a symmetric positive-definite matrix A into the product A = L × Lᵀ, where L is a lower triangular matrix. It is named after French military officer André-Louis Cholesky and is widely used in numerical linear algebra for solving systems of equations and simulations.

A matrix decomposition (or matrix factorization) expresses a matrix as a product of two or more matrices with special properties. Just as the number 16 can be written as 4×4 or 2×8, a matrix A can be factored into simpler component matrices — making computations like solving linear systems much more efficient.

A matrix must be symmetric (A = Aᵀ) and positive-definite (all eigenvalues are strictly positive) to have a Cholesky decomposition. Equivalently, all leading principal minors (determinants of the top-left sub-matrices) must be positive. If the algorithm encounters a negative value under a square root, the matrix is not positive-definite and decomposition fails.

Start from the top-left element: L[i][j] = (A[i][j] − Σ L[i][k]×L[j][k]) / L[j][j] for i > j, and L[i][i] = √(A[i][i] − Σ L[i][k]²) for diagonal entries. Compute entries column by column, filling in only the lower triangle. All upper-triangle entries of L are zero.

Cholesky decomposition is used in solving systems of linear equations, Monte Carlo simulations (generating correlated random variables), optimization (especially quadratic programming), kalman filters, and finite element analysis. It is roughly twice as fast as LU decomposition for symmetric positive-definite matrices.

It splits a symmetric positive-definite matrix A into A = L × Lᵀ, producing a lower triangular matrix L. This factored form makes it much cheaper to solve Ax = b (solve Ly = b then Lᵀx = y), compute determinants (det(A) = (det(L))²), and generate correlated multivariate random samples.

A symmetric matrix satisfies A = Aᵀ (the element at row i, column j equals the element at row j, column i). It is positive-definite if xᵀAx > 0 for every non-zero vector x — equivalently, all its eigenvalues are strictly positive. These properties together guarantee that the Cholesky decomposition exists and is unique.

LU decomposition works on general square matrices, factoring A into a lower triangular L and upper triangular U. Cholesky decomposition is a specialization for symmetric positive-definite matrices and exploits that symmetry so that U = Lᵀ — requiring roughly half the arithmetic operations and storage, making it more efficient in those cases.