About

Cholesky decomposition factors a symmetric positive-definite matrix A into the product LL^T, where L is a lower triangular matrix with strictly positive diagonal entries. The decomposition exists if and only if A is symmetric and all leading principal minors are positive. It requires roughly n³3 floating-point operations, making it approximately twice as fast as LU decomposition for eligible matrices. Numerical instability or an incorrect assumption of positive-definiteness leads to imaginary square roots during factorization, producing meaningless results. This tool validates symmetry and definiteness before proceeding.

Applications span Kalman filters, Monte Carlo simulation (generating correlated random variables), solving normal equations in least-squares fitting, and finite element analysis. The algorithm implemented here is the Cholesky - Banachiewicz row-by-row variant. Limitation: this calculator uses IEEE 754 double-precision arithmetic, so matrices with condition numbers above 10¹² may produce inaccurate results due to rounding.

Formulas

The Cholesky - Banachiewicz algorithm computes the lower triangular factor L row by row. For diagonal entries:

l_jj = √a_jj − j−1∑k=0 l_jk²

For off-diagonal entries where i > j:

l_ij = a_ij − j−1∑k=0 l_ik ⋅ l_jkl_jj

The determinant is computed as:

det(A) = (n∏i=1 l_ii)²

Where A is the input symmetric positive-definite matrix of order n, L is the resulting lower triangular matrix, a_ij denotes the element at row i column j of A, and l_ij denotes the corresponding element of L. If the expression under the square root becomes negative or zero, the matrix is not positive-definite and decomposition fails.

Reference Data

Property	Requirement / Value	Notes
Symmetry	A = A^T	Strict requirement; a_ij = a_ji
Positive-Definiteness	x^TAx > 0 for all x ≠ 0	All eigenvalues must be positive
Leading Minors	All > 0	Sylvester’s criterion
Diagonal of L	All l_ii > 0	Uniqueness condition
Determinant	det(A) = (n∏i=1 l_ii)²	Computed from diagonal of L
Complexity	n³3 flops	Half of LU decomposition cost
Storage	n(n + 1)2 elements	Only lower triangle stored
LDL^T Variant	Avoids square roots	Uses diagonal matrix D
Hermitian Extension	A = LL^*	For complex matrices (not covered here)
Band Matrix Optimization	O(nb²)	b = bandwidth
Sparse Optimization	Fill-in minimization via reordering	AMD, Nested Dissection
Condition Number	κ(A) = κ(L)²	Squared relationship
Numerical Stability	Backward stable	No pivoting needed
Inverse via Cholesky	A⁻¹ = (L⁻¹)^TL⁻¹	Forward + back substitution
Identity Test	L = I	When A = I
Diagonal Matrix	l_ii = √a_ii	Trivial case
2×2 Explicit	l₁₁ = √a₁₁, l₂₁ = a₂₁l₁₁	Closed-form for small matrices

Frequently Asked Questions

The algorithm will encounter a non-positive value under the square root when computing a diagonal element l_jj. This calculator detects this condition and reports exactly which diagonal step failed. A matrix can be symmetric with all positive diagonal entries yet still fail positive-definiteness. Sylvester's criterion requires all leading principal minors to be positive, not just the diagonal.

IEEE 754 double-precision provides roughly 15-16 significant decimal digits. For matrices with condition number κ(A) near 10¹² or above, the last several digits of L become unreliable. The verification step (LL^T reconstruction) will show residual errors exceeding 10⁻⁶ in such cases. For practical engineering matrices (condition numbers below 10⁶), expect accuracy to at least 8 decimal places.

Yes. Once you have L, solve Ly = b by forward substitution, then L^Tx = y by back substitution. Each step costs O(n²) operations. This calculator provides L; you perform the substitution manually or use the output in another tool.

A valid input must satisfy a_ij = a_ji. Rather than requiring you to type each value twice and risk inconsistency, the tool auto-mirrors the lower triangle to the upper triangle. This enforces symmetry by construction and halves the number of inputs.

LU works on any non-singular matrix but requires pivoting and costs 2n³3 operations. Cholesky exploits symmetry and positive-definiteness, needing only n³3 operations and no pivoting. The result is also inherently numerically stable. If your matrix qualifies, Cholesky is always preferred.

This calculator automatically computes LL^T and displays it alongside the original matrix A. The Frobenius norm of the residual ‖A − LL^T‖_F is shown. Values below 10⁻¹⁰ indicate excellent accuracy.