Definition:Conjugate Symmetric Mapping
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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.
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.
Sources
- 1989: R.M. Dudley: Real Analysis and Probability: $\S 5.3$: Hilbert Spaces