Symmertic/Hermitian Matrixes are too fussy about mutating operations

[oxinabox]

We should allow symmetry/conjugate-symmetry preserving operations on Symmertic/Hermitian matrices.
For example the following failures should not be failures.

julia> AS = Symmetric([1 2; 2 4])
2×2 Symmetric{Int64, Matrix{Int64}}:
 1  2
 2  4

julia> AS .+= deepcopy(AS)
ERROR: ArgumentError: Cannot set a non-diagonal index in a symmetric matrix

julia> AS .+= 1
ERROR: ArgumentError: Cannot set a non-diagonal index in a symmetric matrix

julia> AS .*= 2
ERROR: ArgumentError: Cannot set a non-diagonal index in a symmetric matrix

julia> map!(x -> x==1 ? 10 : 20, AS, AS)  # i suspect this one is harder
ERROR: ArgumentError: Cannot set a non-diagonal index in a symmetric matrix

julia> mul!(deepcopy(AS), AS, 2)
ERROR: ArgumentError: Cannot set a non-diagonal index in a symmetric matrix

These are annoying, they make it hard to write code that is generic to the type of the AbstractArray;
even when you know that the operation you are doing will be symmertic if the input is etc.
E.g. because you are implementing a linear operator, (I think as a rule linear operators tend to preserve a structure one cares about for a type that represents a particular vector subspace)

To keep this issue focused, lets avoid talking about what to do for asymmetric (/non-conjugate symmetric) operations that can be turned into symmetric (/conjugate symmetric) ones.
i.e. not talking about making it so that after running AH[2,3] = 100 then it is observatory as AH[2,3] == AH[3,2] == 100.
Just about things that do preserve the symmetry (/conjugate symmetry)

The effect on this on the parent matrix could either be specifically undefined, or define to only set in the given upper/lower, or actually change both.

---

related: #38055

[fredrikekre]

See #19228, specifically #19228 (comment)

Edit: Also #33071

[oxinabox]

See #19228, specifically #19228 (comment)

Thanks for the links.
Yep, this issue basically is about revisiting that decision.
It's been 4 years; things have changed in the language.
We now have the ability to extensively customize broadcast behavour.

So we could bring the behavour of Symmertic/Hermitian closer inline with Digaonal/LowerTriangualar/UpperTriangular, which all do allow the equalivent structure preserving inplace operations.

[oscardssmith]

I'm adding the triage label so we can get a broader discussion here. I'm personally in favor, as these seem like good operations to have.

[mbauman]

"All" that needs to happen here is that our structured broadcast implementation needs to be expanded to include considerations of symmetric/hermitian matrices:

julia/stdlib/LinearAlgebra/src/structuredbroadcast.jl

Lines 7 to 12 in 95a34a9

	struct StructuredMatrixStyle{T} <: Broadcast.AbstractArrayStyle{2} end
	StructuredMatrixStyle{T}(::Val{2}) where {T} = StructuredMatrixStyle{T}()
	StructuredMatrixStyle{T}(::Val{N}) where {T,N} = Broadcast.DefaultArrayStyle{N}()
	const StructuredMatrix = Union{Diagonal,Bidiagonal,SymTridiagonal,Tridiagonal,LowerTriangular,UnitLowerTriangular,UpperTriangular,UnitUpperTriangular}
	Broadcast.BroadcastStyle(::Type{T}) where {T<:StructuredMatrix} = StructuredMatrixStyle{T}()

It'll be a bit of work, but totally possible and shouldn't really be contentious — especially for the cases where we're broadcasting scalars or other symmetric structures.

[bjarthur]

To keep this issue focused, lets avoid talking about what to do for asymmetric (/non-conjugate symmetric) operations that can be turned into symmetric (/conjugate symmetric) ones. i.e. not talking about making it so that after running AH[2,3] = 100 then it is observatory as AH[2,3] == AH[3,2] == 100. Just about things that do preserve the symmetry (/conjugate symmetry)

@oxinabox at the risk of defocusing this issue, is AH[2,3] == AH[3,2] == 100 not desirable to you? it is to me. is it discussed somewhere else other than #19228 (comment) ?

[bjarthur]

to answer my own question: #33071

[ViralBShah]

Apologies for the noise around locking/transferring - I had moved this overenthusiastically to a discussion.