Fix eigvals gradient for symmetric matrices (FluxML/Zygote#1369)

[CarloLucibello]

Fixes the silent wrong gradient reported in FluxML/Zygote#1369.

Problem

rrule(eigen, A::StridedMatrix) reuses the symmetric-manifold cotangent convention whenever A happens to be Hermitian: it pushes the cotangent through _symherm_back and triu!, projecting it onto the stored (upper) triangle and zeroing the other triangle. That is the right convention for a Symmetric/Hermitian wrapper (where only one triangle is a free variable), but wrong for a plain Matrix, whose entries are all independent.

The result is a gradient that disagrees with ForwardDiff/finite differences and is discontinuous at exact symmetry:

julia> using LinearAlgebra, Zygote

julia> Zygote.jacobian(eigvals, [1 2; 2+1e-13 3])[1]   # ~symmetric → correct (split)
2×4 Matrix{Float64}:
 0.723607  -0.447214  -0.447214  0.276393
 0.276393   0.447214   0.447214  0.723607

julia> Zygote.jacobian(eigvals, [1 2; 2 3])[1]         # exactly symmetric → WRONG (zero column)
2×4 Matrix{Float64}:
 0.723607  0.0  -0.894427  0.276393
 0.276393  0.0   0.894427  0.723607

Fix

eigen_rev! already produces (via _hermitrizelike!) a cotangent whose off-diagonal eigenvalue sensitivity is split evenly across both triangles. We simply materialise that full matrix instead of collapsing it onto one triangle.

Scope: real matrices, eigenvalues only (T <: Real && ΔV isa AbstractZero), i.e. the eigvals path. The eigenvector phase convention differs between the symmetric (syev) and general (geev) algorithms, and the complex-Hermitian case can't be validated against FiniteDifferences (the eigenvalues leave the reals, so to_vec mismatches), so those paths are deliberately left unchanged. The Symmetric/Hermitian wrapper rules in symmetric.jl are untouched and still return structured tangents.

Verification

• test_rrule(eigvals, Matrix(Hermitian(randn(n, n)))) now passes (previously failed on the symmetric input).
• End-to-end: jacobian(eigvals, [1 2; 2 3]) now matches finite differences.
• Updated the dense "hermitian" eigvals/eigen tests so the real eigenvalue gradient is checked against the unwrapped eigvals/eigen (general-matrix convention); the eigenvector and complex paths keep the Matrix(Hermitian(·)) reference.
• Full factorization.jl (2977) and symmetric.jl (5499) test suites pass.

🤖 Generated with Claude Code

[devmotion]

Why not just limiting this whole branch (on the outer level) to Hermitian and Symmetric{<:Real} instead of ishermitian?