Symmetric LDL Block Factorization
Use scalar and two-by-two pivots to factor symmetric matrices of mixed sign.
Free theory articleUse scalar and two-by-two pivots to factor symmetric matrices of mixed sign.
Concept & interpretation
The block LDL factorization decomposes a real symmetric matrix into the product L D LT, where L is a unit lower triangular matrix (or block triangular) and D is a block diagonal matrix composed of 1x1 or 2x2 blocks. Unlike the Cholesky decomposition which is restricted to positive definite matrices and requires computing square roots, the LDL block factorization is square-root-free and works for any symmetric matrix (indefinite, positive, negative, or singular) by using 1x1 scalar pivots when possible and full 2x2 blocks to pivot past zero or small diagonals.
Experiment
You can interact with the Symmetric LDL Block Factorization module to explore different matrices.
- Select the "Indefinite" matrix preset and verify the L and D factors match the worked example above.
- Change the "Matrix" to "Zero pivot" and observe that the algorithm automatically falls back to a 2x2 pivot block (Largest D block = 2) to handle the 0 on the diagonal.
- Adjust the "Scale" to 0 or -1 and observe how the inertia responds. Congruence transformations preserve the sign of inertia, but scaling by a negative number swaps positive and negative counts.