Definition:Symmetric Mapping (Linear Algebra)

Definition
Let $\R$ be the field of real numbers.

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

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 symmetric :


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

Also see

 * Definition:Conjugate Symmetric Mapping, this concept generalised to subfields of the field of complex 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 symmetry.