LDLT

LDLT decomposition, or LDL^T factorization, is a factorization of a real symmetric matrix A into A = L D L^T, where L is a unit lower triangular matrix and D is a diagonal matrix. This form separates the elimination multipliers from the pivots, offering numerical advantages in certain computations and storage.

Existence and variants: For symmetric positive definite matrices, an LDL^T factorization exists without pivoting, with L having unit diagonal and D containing positive pivots. For more general symmetric matrices, pivoting is used to ensure numerical stability, via schemes such as the Bunch-Kaufman or Bunch-Parlett algorithms. When pivoting is used, one often writes P A P^T = L D L^T, where P is a permutation matrix.

Computation and relation to Cholesky: The factorization is obtained through a sequence of Gaussian eliminations, with the multipliers stored in L and the pivot elements stored in D. If D has all positive diagonal entries, Cholesky decomposition can be recovered by combining D into a diagonal factor, giving A = (L sqrt(D)) (L sqrt(D))^T. Thus, LDL^T is a generalization of Cholesky that can handle matrices that are not strictly positive definite or require pivoting for stability.

Applications and properties: LDL^T is used to solve linear systems via forward/back substitution, to compute determinants (the determinant is the product of the diagonal entries of D, up to a sign determined by any pivoting), and to estimate eigenvalues. It is particularly advantageous for indefinite matrices and certain sparse problems, where appropriate ordering reduces fill-in during factorization.

Implementation notes: Many numerical linear algebra libraries implement LDL^T factorization with pivoting. For dense matrices, its computational cost is similar to that of Cholesky, with comparable memory requirements, making it a common choice in engineering, physics, and computational mathematics.