# Symbolic diagonalization

**URL:** <https://discourse.julialang.org/t/symbolic-diagonalization/60875>\
**Category:** Specific Domains\
**Tags:** linearalgebra, symbolic\
**Created:** [May 10, 2021, 10:18am UTC](https://discourse.julialang.org/t/symbolic-diagonalization/60875 "2021-05-10T10:18:08Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![volkerkarle](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/volkerkarle/32/14143_2.png) [@volkerkarle](https://discourse.julialang.org/u/volkerkarle)\
**Post date:** [May 10, 2021, 10:18am UTC](https://discourse.julialang.org/t/symbolic-diagonalization/60875/1 "2021-05-10T10:18:09Z")

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Dear community,

I was wondering to what extend diagonalization in Symbolics.jl is implemented,

> Symbolic linear algebra (factorizations, inversion, determinants, eigencomputations, etc.)

I see that some factorizations are working, but what about diagonalization of a normal matrix?

Best,

v.

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**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 10, 2021, 10:31am UTC](https://discourse.julialang.org/t/symbolic-diagonalization/60875/2 "2021-05-10T10:31:48Z")

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Won’t that be equivalent to solving the characteristic polynomials and so it won’t be possible for `n>5` by Abel’s Theorem to do symbolically? I haven’t thought about it too much but @YingboMa might have an idea.

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**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [May 10, 2021, 10:49am UTC](https://discourse.julialang.org/t/symbolic-diagonalization/60875/3 "2021-05-10T10:49:28Z")

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See [symbolic diagonalization of a matrix - MathOverflow](https://mathoverflow.net/questions/83608/symbolic-diagonalization-of-a-matrix) for a discussion of the general case. Maybe something more specific can be said about normal matrices. With symbolic matrices, I guess one needs to add some constraints on the terms in order to guarantee that they are normal.

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**Author:** ![volkerkarle](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/volkerkarle/32/14143_2.png) [@volkerkarle](https://discourse.julialang.org/u/volkerkarle)\
**Post date:** [May 10, 2021, 11:01am UTC](https://discourse.julialang.org/t/symbolic-diagonalization/60875/4 "2021-05-10T11:01:02Z")

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Agree, but even for n\<5 that diagonalization can prove to be useful. On the other hand, there are some special matrices which can be solved also exactly, I guess one needs to keep a little database of those guys somewhere.