# Definition A Hermitian form $\braket{.|.}$ is a [[Quadratic Forms on Vector Spaces|quadratic form]] on a [[vector space]] $V$ $ \begin{align} \braket{.|.}: &V\times V \rightarrow \mathbb{F} , \\ &(\pmb{x}, \pmb{y}) \mapsto \braket{\pmb{x}|\pmb{y}} \end{align} $ where $\mathbb{F}$ is the field over which $V$ is defined. It has the following properties (in physicists' convention) 1. *Linearity in its second argument:* $\braket{\pmb{z}|c\pmb{x}+\pmb{y}} = c\braket{\pmb{z}|\pmb{x}} + \braket{\pmb{z}|\pmb{y}}$ 2. *Antilinearity in its first argument:* $\braket{c\pmb{x}+\pmb{y}|\pmb{z}} = c^* \braket{\pmb{x}|\pmb{z}} + \braket{\pmb{y}|\pmb{z}}$ 3. *Hermitian:* $\braket{\pmb{x}|\pmb{y}} = (\braket{\pmb{y}|\pmb{x}})^*$ Note that 1 and 3 imply 2. # Non-Degenerate Hermitian Form If the Hermitian form is additionally [[Non-Degeneracy (Quadratic Forms)|non-degenerate]], then it's called (obviously) a non-degenerate Hermitian form. If it is acting on a real vector space, we have $\braket{\pmb{x}|\pmb{y}} = \braket{\pmb{y}|\pmb{x}}$, and the Hermitian form is symmetric (i.e. a [[Bilinear Form]]). Physicists call it a metric in this case (e.g., the [[Minkowski Metric|Minkowski metric]]). If additionally we add with positive definiteness, we end up with an [[inner product]]. ## Examples The most common example is the Hermitian scalar product on $\mathbb{C}^n$ ([[Complex Coordinate Space|C^n]]) $ \braket{\pmb{x}|\pmb{y}} \equiv \sum_{i=1}^n (x^i)^* w^i $ The Hermitian scalar product used in quantum mechanics is the most common example in physics.