carry out (c)transpose in Ax_mul_Bx methods for Hermitian and Symmetric

- Symmetric and Hermitian{<:Real} are invariant to transpose so it is a no-op
- ctranspose(Hermitian{<:Complex}) is faster to carry out then ending up in generic_matmatmul
- Hermitian and Symmetric{<:Real} are invariant to ctranspose so it is a no-op
- transpose(Hermitian{<:Complex}) is faster to carry out then ending up in generic_matmatmul

Author: fredrikekre

base/linalg/symmetric.jl

@@ -301,6 +301,26 @@ A_mul_B!(C::StridedMatrix{T}, A::StridedMatrix{T}, B::Hermitian{T,<:StridedMatri
 
 *(A::HermOrSym, B::HermOrSym) = A*full(B)
 
+# Fallbacks to avoid generic_matvecmul!/generic_matmatmul!
+## Symmetric and Hermitian{<:Real} are invariant to transpose and for
+## Hermitian{<:Real} it is faster to carry out the transposition
+At_mul_B( A::HermOrSym, B::AbstractVector) = transpose(A)*B
+At_mul_B( A::HermOrSym, B::AbstractMatrix) = transpose(A)*B
+At_mul_Bt(A::HermOrSym, B::AbstractMatrix) = transpose(A)*B
+At_mul_Bc(A::HermOrSym, B::AbstractMatrix) = transpose(A)*B
+A_mul_Bt( A::AbstractMatrix, B::HermOrSym) = A*transpose(B)
+At_mul_Bt(A::AbstractMatrix, B::HermOrSym) = A*transpose(B)
+Ac_mul_Bt(A::AbstractMatrix, B::HermOrSym) = A*transpose(B)
+## Hermitian and Symmetric{<:Real} are invariant to ctranspose and for
+## Symmetric{<:Complex} it is faster to carry out the ctransposition
+Ac_mul_B( A::HermOrSym, B::AbstractVector) = ctranspose(A)*B
+Ac_mul_B( A::HermOrSym, B::AbstractMatrix) = ctranspose(A)*B
+Ac_mul_Bt(A::HermOrSym, B::AbstractMatrix) = ctranspose(A)*B
+Ac_mul_Bc(A::HermOrSym, B::AbstractMatrix) = ctranspose(A)*B
+A_mul_Bc( A::AbstractMatrix, B::HermOrSym) = A*ctranspose(B)
+At_mul_Bc(A::AbstractMatrix, B::HermOrSym) = A*ctranspose(B)
+Ac_mul_Bc(A::AbstractMatrix, B::HermOrSym) = A*ctranspose(B)
+
 for T in (:Symmetric, :Hermitian), op in (:+, :-, :*, :/)
     # Deal with an ambiguous case
     @eval ($op)(A::$T, x::Bool) = ($T)(($op)(A.data, x), Symbol(A.uplo))