# Definition for Hermitian inner product?

I have found two different papers (both are well cited) Classification of Self-Orthogonal Codes over $$\mathbb{F}_3$$ and $$\mathbb{F}_4$$ (pg.3) and Convolutional and Tail-Biting Quantum Error-Correcting Codes (end of pg.4) that give different definitions for Hermitian inner product in $$\mathbb{F}_4$$.

The first paper (which has 380+ cites) gives the following:

$$(\boldsymbol u, \boldsymbol v)_H = u_1v^2_1 + u_2v^2_2 + \ldots + u_nv^2_n,$$ where $$\boldsymbol u = \begin{pmatrix}u_1 & u_2 & \ldots & u_n\end{pmatrix}$$ and $$\boldsymbol v = \begin{pmatrix}v_1 & v_2 & \ldots & v_n\end{pmatrix}$$.

The takeaway is that the second parameter (i.e., $$\boldsymbol v$$) is conjugated and $$\boldsymbol u$$ is left the same.

The second paper (which has 90 cites) gives the following definition:

The Hermitian inner product of two elements $$a, b \in \mathbb{F}_4$$ is defined as $$\langle a, b\rangle = a^\dagger b \in \mathbb{F}_4$$ where where $${}^{\dagger}$$ denotes conjugation.

In other words, $$a^\dagger = a^2$$. The takeaway is that the first argument is conjugated, not the second.

If we try $$\langle \bar{\omega}, \omega\rangle$$ using the first definition (assuming $$u = \begin{pmatrix}\bar{\omega}\end{pmatrix}$$ and $$\boldsymbol v = \begin{pmatrix}\omega\end{pmatrix}$$): $$\langle \bar{\omega}, \omega\rangle = \bar{\omega}\omega^2 = \bar{\omega}\bar{\omega}=\omega.$$ Using the second definition, we get:

$$\langle \bar{\omega}, \omega\rangle = \bar{\omega}^2\omega = \omega\omega=\bar{\omega}$$

Why are the two definitions different? Are they even describing the same inner product? Which definition should I follow and why not the other?

That said, both definitions are "correct" and they describe the same inner product up to notational differences, but they follow different conventions. In a general mathematical context, it is more common to define Hermitian forms $$\langle \cdot, \cdot \rangle$$ to be linear in the first argument. In a physics context (especially where quantum mechanics is involved), it is more common to follow the opposite convention whereby $$\langle \cdot, \cdot \rangle$$ is linear in the second argument (in analogy to "bra-ket" notation).