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, non-negative, 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.)

Answer 2

"The dot product" is commonly taken to be the standard inner product in $R^n$. It doesn't work on $C^n$ since conjugate symmetry ($\overline{\langle x,y\rangle}=\langle y,x\rangle$) does not apply. Here's a counter example from $C^1$ $$\overline{\langle 1+2i,1+2i\rangle}=4-3i\neq -3+4i=\langle 1+2i,1+2i\rangle$$ where the arguments have been switched in the second inner product.

You can extend it to complex spaces though, by taking the complex conjugate of the second vector.