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 $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping.

Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is conjugate symmetric iff:


 * $\forall x, y \in V: \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle}$

where $\overline{\left \langle {y, x} \right \rangle}$ denotes the complex conjugate of $\left \langle {x, y} \right \rangle$.

Also known as

 * Hermitian symmetric mapping

Also see

 * Definition:Symmetric Mapping, 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.

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