[Math] Is the dot product not applicable for complex vectors

Question

Is the concept of dot products not applicable for vectors involving imaginary #s?

Are dot products a subset of inner products?

Answer 1

You get a slightly different dot product in complex vectors.

If $\mathbf x= (x_1,x_2,\dots,x_n)$ and $\mathbf y=(y_1,y_2,\dots,y_n)$ then the dot product is $$\mathbf x\cdot \mathbf y=x_1\overline{y_1}+x_2\overline{y_2}+\cdots+x_n\overline{y_n}$$

This has the feature that $\mathbf x\cdot\mathbf x$ is always real and $\mathbf x\cdot \mathbf y=\overline{\mathbf y\cdot \mathbf x}$.

It's interesting to note that the "real part" of $\mathbf x\cdot \mathbf y$ is just the standard dot product if you considered $x$ and $y$ to be real vectors of dimension $2n$ in the obvious way.

And yes, the dot product is just a special case of an inner product. (Sometimes the dot product is called "the inner product," which can be confusing.)