[devmotion]

At a Hermitian A, the eigen/eigvals frules and rrules for StridedMatrix delegated to the rules for Hermitian(A). Their (co)tangents are those of x -> eigen(Hermitian(x)), which reads only the upper triangle of x. That is wrong for unstructured matrices: the rrules returned upper-triangular cotangents and the frules ignored the lower triangle of ΔA (FluxML/Zygote.jl#1369).

This PR:

• Eigenvalues: tangents uᵢ' ΔA uᵢ and cotangents U Diagonal(Δλ) U', with the type of the primal eigenvalues (so they are only projected onto the reals when the primal eigenvalues are real-typed).
• Eigenvectors: at Hermitian A, eigen uses the Hermitian solver, whose eigenvectors follow a different phase convention than those of non-Hermitian matrices. They are therefore only differentiable along Hermitian directions, so the rules use the Hermitian part of the eigenvector (co)tangents (and the Hermitian phase convention).
• Both cases now share the code path of the generic rules (V' instead of V \ since V is unitary); _sorteig!_fwd became unused and is removed.
• Fixes the failures on Julia 1.14 (pre): LinearAlgebra.jl#1717 made eigenvalues of complex matrices complex-typed also on the Hermitian path, so the old rules returned real tangents for complex primals. It also exposed a Core.Box in the eigen pullback (self-recursive thunk method), which made the eigvals rrule non-inferable; the pullback is now a single method.
• Tests compare against finite differences of the unwrapped eigvals (whenever the eigenvalue type admits all directions) and of eigen ∘ hermitianpart for eigenvectors, with non-Hermitian ΔA, instead of x -> eigen(Matrix(Hermitian(x))). Includes a regression test for Zygote#1369 (∇ sum(eigvals(A)) == I).

Supersedes #841 (thanks @CarloLucibello for the analysis there), which fixed the values-only rrule for real matrices.

Verified locally: rulesets/LinearAlgebra/factorization passes on Julia 1.10, 1.13 and 1.14.0-alpha1 (the new tests fail against the released version).

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.64%. Comparing base (e6e8fae) to head (ef12a50).

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #853      +/-   ##
==========================================
+ Coverage   96.57%   96.64%   +0.06%     
==========================================
  Files          28       28              
  Lines        3618     3604      -14     
==========================================
- Hits         3494     3483      -11     
+ Misses        124      121       -3     

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