Hermitian form

A sesquilinear form over a pair of complex vector spaces $(V,W)$ is a function $B\colon V\times W\to\mathbb{C}$ satisfying the following properties:

1.

$B(\mathbf{v}_{1}+\mathbf{v}_{2},\mathbf{w})=B(\mathbf{v}_{1},\mathbf{w})+B(% \mathbf{v}_{2},\mathbf{w})$
2.

$B(\mathbf{v},\mathbf{w}_{1}+\mathbf{w}_{2})=B(\mathbf{v},\mathbf{w}_{1})+B(% \mathbf{v},\mathbf{w}_{2})$
3.

$B(c\mathbf{v},d\mathbf{w})=cB(\mathbf{v},\mathbf{w})\overline{d}$

for all $\mathbf{v},\mathbf{v}_{1},\mathbf{v}_{2}\in V$ , $\mathbf{w},\mathbf{w}_{1},\mathbf{w}_{2}\in W$ , and $c,d\in\mathbb{C}$ . The vector spaces $V$ and $W$ are often identical, although the definition does not require them to be the same vector space.

A sesquilinear form $B\colon V\times V\to\mathbb{C}$ over a single vector space $V$ is called a Hermitian form if it is complex conjugate symmetric: namely, if $B(\mathbf{v}_{1},\mathbf{v}_{2})=\overline{B(\mathbf{v}_{2},\mathbf{v}_{1})}$ .

An inner product over a complex vector space is a positive definite Hermitian form.

Title	Hermitian form
Canonical name	HermitianForm
Date of creation	2013-03-22 12:25:47
Last modified on	2013-03-22 12:25:47
Owner	djao (24)
Last modified by	djao (24)
Numerical id	8
Author	djao (24)
Entry type	Definition
Classification	msc 47A07
Classification	msc 15A63
Classification	msc 11E39
Synonym	sesquilinear form
Synonym	sesqui-linear form
Related topic	InnerProduct