Linear and multilinear algebra; matrix theory Definition

Sesquilinear form

Definition

Let $V$ be a vector space over $\C$ . A sesquilinear form on $V$ is a map $\varphi \colon V \times V \to \C$ such that, for all $u, v, w \in V$ and $a \in \C$ :

$\varphi(u + v, w) = \varphi(u, w) + \varphi(v, w)$ and $\varphi(au, w) = a\, \varphi(u, w)$ .
$\varphi(u, v + w) = \varphi(u, v) + \varphi(u, w)$ and $\varphi(u, aw) = \bar a\, \varphi(u, w)$ .

Remarks

Here the form is linear in the first argument; many authors, especially in physics, use linearity in the second argument instead. A sesquilinear form is Hermitian when $\varphi(v, u) = \overline{\varphi(u, v)}$ for all $u, v$ . On $\C^n$ , each sesquilinear form is $\varphi(x, y) = y^* M x$ for one $n \times n$ complex matrix $M$ . A complex inner product is a positive definite Hermitian sesquilinear form.

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Vector space definition
Sesquilinear form definition

Tag	`0W44K2VNBD`
URI	`m2://0W44K2VNBD`
Kind	definition
Writing	W1 stated
MSC	15A63
Defines	Sesquilinear form
Uses	Vector space

Sesquilinear form

Definition

Remarks

Reading path

Further reading

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