Improve performance for dot(AdjTrans,AdjTrans) (#39004)

* Improve performance for dot(AdjTrans,AdjTrans)

* generalize to Adjoint/Transpose

Author: dkarrasch

stdlib/LinearAlgebra/src/generic.jl

@@ -918,6 +918,9 @@ function dot(x::AbstractArray, y::AbstractArray)
     s
 end
 
+dot(x::Adjoint, y::Adjoint) = conj(dot(parent(x), parent(y)))
+dot(x::Transpose, y::Transpose) = dot(parent(x), parent(y))
+
 """
     dot(x, A, y)
 

stdlib/LinearAlgebra/test/generic.jl

@@ -453,6 +453,24 @@ end
     @test all(!isnan, lmul!(false, Any[NaN]))
 end
 
+@testset "adjtrans dot" begin
+    for t in (transpose, adjoint)
+        x, y = t(rand(ComplexF64, 10)), t(rand(ComplexF64, 10))
+        X, Y = copy(x), copy(y)
+        @test dot(x, y) ≈ dot(X, Y)
+        x, y = t([rand(ComplexF64, 2, 2) for _ in 1:5]), t([rand(ComplexF64, 2, 2) for _ in 1:5])
+        X, Y = copy(x), copy(y)
+        @test dot(x, y) ≈ dot(X, Y)
+        x, y = t(rand(ComplexF64, 10, 5)), t(rand(ComplexF64, 10, 5))
+        X, Y = copy(x), copy(y)
+        @test dot(x, y) ≈ dot(X, Y)
+        x = t([rand(ComplexF64, 2, 2) for _ in 1:5, _ in 1:5])
+        y = t([rand(ComplexF64, 2, 2) for _ in 1:5, _ in 1:5])
+        X, Y = copy(x), copy(y)
+        @test dot(x, y) ≈ dot(X, Y)
+    end
+end
+
 @testset "generalized dot #32739" begin
     for elty in (Int, Float32, Float64, BigFloat, ComplexF32, ComplexF64, Complex{BigFloat})
         n = 10