# Definition:Conjugate Symmetric Mapping

## Definition

Let $\C$ be the field of complex numbers.

Let $\F$ be a subfield of $\C$.

Let $V$ be a vector space over $\F$

Let $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping.

Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is conjugate symmetric if and only if:

$\forall x, y \in V: \quad \innerprod x y = \overline {\innerprod y x}$

where $\overline {\innerprod y x}$ denotes the complex conjugate of $\innerprod x y$.

## Also known as

• Hermitian symmetric mapping

This property as a noun is referred to as conjugate symmetry.