Definition:Symmetric Mapping (Linear Algebra)

This page is about Symmetric Mapping in the context of Linear Algebra. For other uses, see Symmetry.

Definition

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

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 symmetric if and only if:

$\forall x, y \in V: \innerprod x y = \innerprod y x$

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.

This property as a noun is referred to as symmetry.