bugfix for dot of Hermitian{noncommutative}

stdlib/LinearAlgebra/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)

stdlib/LinearAlgebra/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)),

test/testhelpers/Quaternions.jl

@@ -20,6 +20,7 @@ Base.abs2(q::Quaternion) = q.s*q.s + q.v1*q.v1 + q.v2*q.v2 + q.v3*q.v3
 Base.float(z::Quaternion{T}) where T = Quaternion(float(z.s), float(z.v1), float(z.v2), float(z.v3))
 Base.abs(q::Quaternion) = sqrt(abs2(q))
 Base.real(::Type{Quaternion{T}}) where {T} = T
+Base.real(q::Quaternion) = q.s
 Base.conj(q::Quaternion) = Quaternion(q.s, -q.v1, -q.v2, -q.v3)
 Base.isfinite(q::Quaternion) = isfinite(q.s) & isfinite(q.v1) & isfinite(q.v2) & isfinite(q.v3)
 Base.zero(::Type{Quaternion{T}}) where T = Quaternion{T}(zero(T), zero(T), zero(T), zero(T))
@@ -33,7 +34,9 @@ Base.:(*)(q::Quaternion, w::Quaternion) = Quaternion(q.s*w.s - q.v1*w.v1 - q.v2*
                                             q.s*w.v2 - q.v1*w.v3 + q.v2*w.s + q.v3*w.v1,
                                             q.s*w.v3 + q.v1*w.v2 - q.v2*w.v1 + q.v3*w.s)
 Base.:(*)(q::Quaternion, r::Real) = Quaternion(q.s*r, q.v1*r, q.v2*r, q.v3*r)
-Base.:(*)(q::Quaternion, b::Bool) = b * q # remove method ambiguity
+Base.:(*)(q::Quaternion, r::Bool) = Quaternion(q.s*r, q.v1*r, q.v2*r, q.v3*r) # remove method ambiguity
+Base.:(*)(r::Real, q::Quaternion) = q * r
+Base.:(*)(r::Bool, q::Quaternion) = q * r # remove method ambiguity
 Base.:(/)(q::Quaternion, w::Quaternion) = q * conj(w) * (1.0 / abs2(w))
 Base.:(\)(q::Quaternion, w::Quaternion) = conj(q) * w * (1.0 / abs2(q))