spezialized conj of Transpose/Adjoint (#33609)

* specialized conj for Adjoint/Transpose

* add tests

* implement @dkarrasch's suggestions

* add test case where M' != Adjoint(M)

stdlib/LinearAlgebra/src/adjtrans.jl

@@ -281,3 +281,7 @@ pinv(v::TransposeAbsVec, tol::Real = 0) = pinv(conj(v.parent)).parent
 /(u::TransposeAbsVec, A::AbstractMatrix) = transpose(transpose(A) \ u.parent)
 /(u::AdjointAbsVec, A::Transpose{<:Any,<:AbstractMatrix}) = adjoint(conj(A.parent) \ u.parent) # technically should be adjoint(copy(adjoint(copy(A))) \ u.parent)
 /(u::TransposeAbsVec, A::Adjoint{<:Any,<:AbstractMatrix}) = transpose(conj(A.parent) \ u.parent) # technically should be transpose(copy(transpose(copy(A))) \ u.parent)
+
+## complex conjugate
+conj(A::Transpose) = adjoint(A.parent)
+conj(A::Adjoint) = transpose(A.parent)

stdlib/LinearAlgebra/test/adjtrans.jl

@@ -504,4 +504,25 @@ using .Main.OffsetArrays
     @test_throws BoundsError s[1, 4]
 end
 
+@testset "specialized conj of Adjoint/Transpose" begin
+    realmat = [1 2; 3 4]
+    complexmat = ComplexF64[1+im 2; 3 4-im]
+    nested = [[complexmat] [-complexmat]; [0complexmat] [3complexmat]]
+    @testset "AdjOrTrans{...,$(typeof(i))}" for i in (
+                                                      realmat, vec(realmat),
+                                                      complexmat, vec(complexmat),
+                                                      nested, vec(nested),
+                                                     )
+        for (t,type) in ((transpose, Adjoint), (adjoint, Transpose))
+            M = t(i)
+            @test conj(M) isa type
+            @test conj(M) == conj(collect(M))
+            @test conj(conj(M)) === M
+        end
+    end
+    # test if `conj(transpose(::Hermitian))` is a no-op
+    hermitian = Hermitian([1 2+im; 2-im 3])
+    @test conj(transpose(hermitian)) === hermitian
+end
+
 end # module TestAdjointTranspose