Dot product over complex vectors: Conjugate first or second?

Solution 1:

If we consider column vectors in $\mathbb{C}^n$, it's more natural to define the standard inner product by $$ \langle\mathbf{x}\mid\mathbf{y}\rangle = \mathbf{x}^H\mathbf{y} $$ rather than $\mathbf{y}^H\mathbf{x}$ ($H$ denotes the conjugate transpose, notation for the inner product varies among authors), so it's “naturally” antilinear in the first variable and linear in the second.

If one identifies coordinate vectors with rows (writing maps on the right), then the “natural” way becomes the opposite.

It's just a convention; just learn how to translate from one to the other.

Solution 2:

In fact I've got a book in which the dot product is considered antilinear in the first argument despite other books having antilinearity in the second one.

However this ambiguity is in no way dangerous, because you may make a composition of operators to have linearity in the argument that you want.

The book in question is Richtmyer, Morton. Difference Methods for Initial-Value Problems

Solution 3:

Wikipedia reads: conjugating the second argument "is how the inner product was originally defined and is still used in some old-school math communities. However, all of engineering and computer science, and most of physics and modern mathematics now define the inner product to be linear in the second argument and conjugate-linear in the first argument because this is more compatible with several other conventions in mathematics." https://en.wikipedia.org/wiki/Inner_product_space#Notes

That is why I have given up $x\cdot y = y^Hx$ for the more logical $x\cdot y = x^H y$, where $x^H$ is the Hermitian (conjugated transpose).

Nevertheless, this is how inner product is defined. For mathematicians, usually dot product = inner product, whereas for physicists, dot product and cross product do not involve any conjugation! Beware!

Matlab conjugates, Mathematica does not, Maple conjugates in the dot product but not in the cross product. In Matlab, $i'=-i$. More on this and on the conjugation of the cross product: https://math.stackexchange.com/a/3567679/