`cholesky(::Symmetric{<:Any,<:SMatrix})` doesn't respect `uplo`

StaticArrays' implementation of Cholesky decomposition assumes that uplo == 'U' in Symmetric and Hermitian wrappers. This gives incorrect results when uplo == 'L' and the underlying array is not actually symmetric/hermitian.

MWE:

julia> using LinearAlgebra, StaticArrays

julia> A = Symmetric(SA[1.5 -1; 1 2], :L)
2×2 Symmetric{Float64, SMatrix{2, 2, Float64, 4}}:
 1.5  1.0
 1.0  2.0

julia> C = cholesky(A);

julia> C.L * C.U
2×2 SMatrix{2, 2, Float64, 4} with indices SOneTo(2)×SOneTo(2):
  1.5  -1.0
 -1.0   2.0

julia> C.L * C.U ≈ A
false

I suppose the easiest solution is to replace the following line:

https://github.com/JuliaArrays/StaticArrays.jl/blob/0feac146bb71ff48ca0a8465739f6b442395292c/src/cholesky.jl#L9

with something like

    if A.uplo == 'L'
        return _cholesky(Size(A), A.data', check)
    else
        return _cholesky(Size(A), A.data, check)
    end

However, it would be even nicer if uplo were preserved by the factorization, i.e., if cholesky(A).uplo == A.uplo, the way it currently works in Base. That would require a change slightly deeper in the code, but could use the same basic trick: if uplo == 'L', take the conjugate transpose, carry out the factorization as if uplo == 'U', and finally take the conjugate transpose of the output and use it to instantiate a Cholesky instance with uplo == 'L'.

Start in src/cholesky.jl at the linked implementation and reproduce the issue with the Julia MWE. Check how the Symmetric or Hermitian uplo value reaches the factorization. Done means cholesky(A) produces factors matching the selected lower or upper triangle, with uplo preserved if that behavior is implemented.