Cholesky Decomposition Calculator - Factor Positive Definite Matrices

Cholesky decomposition is one of the most important matrix factorizations in numerical linear algebra. Given a symmetric positive definite matrix A, the decomposition produces a unique lower triangular matrix L with strictly positive diagonal entries such that A = L·Lᵀ. The algorithm, attributed to French artillery officer André-Louis Cholesky (1875–1918), was published posthumously in 1924 and has since become a cornerstone of scientific computing. The Cholesky-Banachiewicz algorithm used in this calculator proceeds column by column. For each diagonal entry, it subtracts the sum of squares of all previous entries in the same row from the corresponding diagonal entry of A, then takes the square root. For off-diagonal entries below the diagonal, it subtracts a dot product of previously computed entries and divides by the current diagonal. The computational cost is approximately n³/6 multiplications for an n×n matrix, making it roughly twice as efficient as general LU decomposition for symmetric positive definite inputs. The most important prerequisite is that the matrix must be both symmetric and positive definite. Symmetry means A[i][j] = A[j][i] for all pairs (i, j). Positive definiteness means xᵀAx > 0 for every non-zero real vector x, which is equivalent to requiring all eigenvalues to be strictly positive. In the Cholesky algorithm, failure of positive definiteness manifests as an attempt to take the square root of a non-positive number, which is the check this calculator performs. In statistics, covariance matrices are always symmetric and positive semi-definite. When no two variables are perfectly collinear, they are strictly positive definite, which means Cholesky decomposition applies directly. The decomposition is used to generate multivariate normal random samples: if z is a vector of independent standard normal variables, then L·z has the covariance matrix A. This technique underpins Monte Carlo simulation of correlated financial assets, climate model ensembles, and structural reliability analyses. In machine learning, Gaussian process regression and Bayesian neural networks rely heavily on Cholesky decomposition to invert or log-determinant kernel matrices efficiently. The log-determinant of A equals twice the sum of the logarithms of the diagonal entries of L, which avoids the numerical instability of computing the determinant directly. Kalman filter implementations use the so-called square-root Kalman filter, which propagates the Cholesky factor of the covariance matrix rather than the covariance itself, dramatically improving numerical stability in long-running estimation problems. This calculator supports 2×2, 3×3, and 4×4 matrices — the sizes most commonly encountered in classroom exercises, small-scale numerical analysis problems, and prototyping of numerical algorithms. For larger matrices, the same algorithm applies and can be implemented efficiently using BLAS level-3 operations on modern hardware. Whether you are checking homework, verifying a hand-computed decomposition, or exploring the properties of positive definite matrices, this tool provides immediate, accurate Cholesky factors.