bugfix for dot of Hermitian{noncommutative} (#52333)

Co-authored-by: Daniel Karrasch <daniel.karrasch@posteo.de>

Author: stevengj

src/symmetric.jl

@@ -453,7 +453,7 @@ function triu(A::Symmetric, k::Integer=0)
     end
 end
 
-for (T, trans, real) in [(:Symmetric, :transpose, :identity), (:Hermitian, :adjoint, :real)]
+for (T, trans, real) in [(:Symmetric, :transpose, :identity), (:(Hermitian{<:Union{Real,Complex}}), :adjoint, :real)]
     @eval begin
         function dot(A::$T, B::$T)
             n = size(A, 2)

test/symmetric.jl

@@ -4,6 +4,11 @@ module TestSymmetric
 
 using Test, LinearAlgebra, Random
 
+const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
+
+isdefined(Main, :Quaternions) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "Quaternions.jl"))
+using .Main.Quaternions
+
 Random.seed!(1010)
 
 @testset "Pauli σ-matrices: $σ" for σ in map(Hermitian,
@@ -462,6 +467,17 @@ end
     end
 end
 
+# bug identified in PR #52318: dot products of quaternionic Hermitian matrices,
+# or any number type where conj(a)*conj(b) ≠ conj(a*b):
+@testset "dot Hermitian quaternion #52318" begin
+    A, B = [Quaternion.(randn(3,3), randn(3, 3), randn(3, 3), randn(3,3)) |> t -> t + t' for i in 1:2]
+    @test A == Hermitian(A) && B == Hermitian(B)
+    @test dot(A, B) ≈ dot(Hermitian(A), Hermitian(B))
+    A, B = [Quaternion.(randn(3,3), randn(3, 3), randn(3, 3), randn(3,3)) |> t -> t + transpose(t) for i in 1:2]
+    @test A == Symmetric(A) && B == Symmetric(B)
+    @test dot(A, B) ≈ dot(Symmetric(A), Symmetric(B))
+end
+
 #Issue #7647: test xsyevr, xheevr, xstevr drivers.
 @testset "Eigenvalues in interval for $(typeof(Mi7647))" for Mi7647 in
         (Symmetric(diagm(0 => 1.0:3.0)),