Fix matrix-function primal type for Hermitian{<:Real} on Julia 1.12

[CarloLucibello]

Summary

Julia 1.12 changed real-valued matrix functions of Hermitian{<:Real} (e.g. exp, log, sqrt, cos, sin, ...) to return Hermitian instead of Symmetric:

julia> exp(Hermitian(ones(2,2))) |> typeof
Hermitian{Float64, Matrix{Float64}}   # was Symmetric on Julia ≤ 1.11

_matfun unconditionally wrapped the real-eltype result as Symmetric, so the rrule/frule primal no longer matched f(A). This surfaced as a type mismatch downstream — see FluxML/Zygote.jl#1592:

julia> Zygote.pullback(exp, Hermitian(ones(2,2)))[1] |> typeof
Symmetric{Float64, Matrix{Float64}}   # should be Hermitian

Fix

In _matfun, when the input is real, wrap the result in the same Symmetric/Hermitian type as the input when the output is real. The complex-valued case (e.g. acosh of eigenvalues < 1) is still wrapped as Symmetric, which is what Julia returns for both real Symmetric and Hermitian inputs. Behavior on Julia < 1.12 is unchanged (always Symmetric).

The inference assertions in the matrix-function tests are updated to expect the input wrapper type on 1.12+.

Testing

Pkg.test(test_args=["symmetric"]) passes on Julia 1.12 (5499 tests, 0 errors).

🤖 Generated with Claude Code

[devmotion]

Ref #833

As mentioned in #833 (comment), AD doesn't guarantee that primal values are exactly identical to the values obtained without AD. So it's unclear why Hermitian{<:Real} would be better than Symmetric{<:Real}.