# Cholesky Decomposition of a Sparse Symmetric Positive Semidefinite (SPSD) Singular Matrix

**URL:** <https://discourse.julialang.org/t/cholesky-decomposition-of-a-sparse-symmetric-positive-semidefinite-spsd-singular-matrix/119682>\
**Category:** Numerics\
**Tags:** linearalgebra, numerics, sparse, factorization\
**Created:** [September 21, 2024, 2:57pm UTC](https://discourse.julialang.org/t/cholesky-decomposition-of-a-sparse-symmetric-positive-semidefinite-spsd-singular-matrix/119682 "2024-09-21T14:57:40Z")\
**Posts on this page:** 1\
**Showing post:** 16

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**Author:** ![RoyiAvital](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/royiavital/32/571_2.png) [@RoyiAvital](https://discourse.julialang.org/u/RoyiAvital)\
**Post date:** [September 23, 2024, 12:47pm UTC](https://discourse.julialang.org/t/cholesky-decomposition-of-a-sparse-symmetric-positive-semidefinite-spsd-singular-matrix/119682/16 "2024-09-23T12:47:12Z")

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I see.  
Summarizing the case of solving a linear system with rank deficient SPSD matrix:

1. Dense Case: Use the Bunch Kaufman decomposition (Sometimes called LDLt as well, see [Shouldn’t `bunchkaufman()` Be Named `ldlt()`](https://discourse.julialang.org/t/77641)).
2. Sparse Case: Use the LDLt decomposition.

In case \boldsymbol{b} \in R \left( \boldsymbol{A} \right), one may use Cholesky Decomposition with `check = false`. Yet the decomposition is not guaranteed to work still.

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