Why the usage of $H$ in the dot product $x^H y$ instead of $T$ or $'$ for transpose?

Question

Given two vectors $v, u \in \mathbb{R}^n$ (i.e. column vectors), then the dot product of them is defined like this

$$\sum_{i=1}^n v_i * u_i$$

Or usually it can be expressed in matrix-product notation as follows $$v^Tu$$

I've now encountered a place where instead of $T$, $H$ is used, that is

$$v^Hu$$

I noticed that $H$ is also used in my slides in the formula of the outer product, so this is not intrinsic to the inner product.

Why is $H$ being used? Does it have a particular meaning?

Answer 1

$H$ stands for "Hermitian tranposition", which is the same operation as transposition for real matrices. But when dealing with complex ones, you have to conjugate each coefficient as well.

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