# Paired symmetric matrix decomposition?

**URL:** <https://discourse.julialang.org/t/paired-symmetric-matrix-decomposition/29092>\
**Category:** Numerics\
**Created:** [September 23, 2019, 8:48pm UTC](https://discourse.julialang.org/t/paired-symmetric-matrix-decomposition/29092 "2019-09-23T20:48:22Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![tim.holy](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tim.holy/32/52_2.png) [@tim.holy](https://discourse.julialang.org/u/tim.holy)\
**Post date:** [September 23, 2019, 8:48pm UTC](https://discourse.julialang.org/t/paired-symmetric-matrix-decomposition/29092/1 "2019-09-23T20:48:22Z")

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Do any of the linear algebra gurus here know the name of or a citation for the following paired matrix decomposition: given symmetric matrices `A` and `B`, and an arbitrary vector `v`, find a decomposition of the form

```julia
A = W DA W'
B = W DB W'

```

for real matrices `W` satisfying `W' v = [1,1,...,1]` and `DA` and `DB` block-diagonal with 1x1 or 2x2 blocks. `W` is not required to be orthonormal.

I can probably figure out how to compute this myself (it seems very close to the generalized eigendecomposition), but if it already exists, I’d rather acknowledge & call it by name.

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<div class="post-metadata">

**Author:** ![tim.holy](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tim.holy/32/52_2.png) [@tim.holy](https://discourse.julialang.org/u/tim.holy)\
**Post date:** [September 24, 2019, 7:42am UTC](https://discourse.julialang.org/t/paired-symmetric-matrix-decomposition/29092/2 "2019-09-24T07:42:36Z")

</div>

To answer my own question (I finally found the right combination of search terms), see [Simultaneous tridiagonalization of two symmetric matrices](https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.733?casa_token=KCTe53Idw6MAAAAA:7-Tn3PRsthrg8c98ikQ8IVvtSVm8HKzqdFMHvoeSuDo-ualmd7CQB_hmhaNLLLSSHsr4P1Dp35IAhw).