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 :


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

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.

Also see

 * Definition:Symmetric Mapping (Linear Algebra), this concept applied to subfields of the field of real numbers.
 * Definition:Semi-Inner Product, where this property is used in the definition of the concept.