# Solving a Shifted Linear System of 2 Symmetric Positive Semi Definite (SPSD) Matrices

**URL:** <https://discourse.julialang.org/t/solving-a-shifted-linear-system-of-2-symmetric-positive-semi-definite-spsd-matrices/131636>\
**Category:** Optimization (Mathematical)\
**Tags:** linearalgebra, convex-optimization, linear-regression, linearsolve, least-squares\
**Created:** [August 16, 2025, 10:47am UTC](https://discourse.julialang.org/t/solving-a-shifted-linear-system-of-2-symmetric-positive-semi-definite-spsd-matrices/131636 "2025-08-16T10:47:40Z")\
**Posts on this page:** 1\
**Showing post:** 17

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**Author:** ![mstewart](https://avatars.discourse-cdn.com/v4/letter/m/b5a626/32.png) [@mstewart](https://discourse.julialang.org/u/mstewart)\
**Post date:** [August 17, 2025, 8:23pm UTC](https://discourse.julialang.org/t/solving-a-shifted-linear-system-of-2-symmetric-positive-semi-definite-spsd-matrices/131636/17 "2025-08-17T20:23:12Z")

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Edit, mostly deleted what I wrote: I wrote a bunch of nonsense here because I focused on the possibility of D\_z potentially containing some nonzero elements, when the point was that it should presumably be a zero block.

Yes. I do think this works if E\_{zz} is nonsingular. But I do not believe that can be guaranteed. So this is not fully general.

The [Fix-Heiberger reduction](https://cmjiang.cs.ucdavis.edu/publications/FHtemp.pdf) starts in this way but then makes a further rank decision on E\_{zz} and further refines the partition of E and D to deal with the possibility that E\_{zz} is singular. It probably would work somewhat robustly in this case, but it’s a more complicated algorithm and I don’t think it ever made it into LAPACK.

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