To respond to the issue of complex conjugation, Dot
is doing what it was designed to do. It operates on general tensors and, more importantly, is vararg (in other words, not restricted to two arguments). The scalar product under discussion above, in contrast, has arity 2, that is, must have exactly two arguments.
To get the Hermitian inner product one can use Inner
, as below.
aa = {1 + I, 3 - I, -5 + 7*I};
bb = {-2, -3 - I, 6*I};
Inner[#1*Conjugate[#2] &, aa, bb, Plus]
There is an open suggestion that this be documented better. I'll add a note referencing this Community thread.
(* Out[1705]= 32 + 34 I *)