Cholesky Decomposition Calculator

Cholesky decomposition breaks down a symmetric, positive-definite matrix A into the form A = L L^T, where L is a lower triangular matrix with strictly positive diagonal entries. For a 2x2 matrix, the algorithm computes the elements of L using square roots and division: L_{11} = \sqrt{A_{11}}, L_{21} = A_{21} / L_{11}, and L_{22} = \sqrt{A_{22} - L_{21}^2}. This exact iterative logic extends upward for 3x3 and 4x4 dimensions, requiring the input matrix to pass rigorous preliminary checks for symmetry and positive definiteness.

Let us find the Cholesky decomposition for a 2x2 symmetric, positive-definite matrix A where row 1 is [4, 12] and row 2 is [12, 39]. First, compute the top-left element of the lower triangular matrix L: L_{11} = \sqrt{4} = 2. Next, find the sub-diagonal element: L_{21} = A_{21} / L_{11} = 12 / 2 = 6. Finally, calculate the bottom-right element using the remaining terms: L_{22} = \sqrt{A_{22} - L_{21}^2} = \sqrt{39 - 6^2} = \sqrt{39 - 36} = \sqrt{3} \approx 1.732. Thus, the lower triangular matrix L has row 1 as [2, 0] and row 2 as [6, 1.732].

Always verify that your matrix is fully symmetric before running computations, as asymmetric inputs will immediately fail validation. Ensure all leading principal minors are strictly positive to confirm positive definiteness. When working with floating-point numbers in larger 3x3 or 4x4 systems, round intermediate values carefully to prevent rounding propagation errors in later triangular rows.