Hermitian form
A sesquilinear form^{} over a pair of complex vector spaces $(V,W)$ is a function $B:V\times W\to \u2102$ 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 $c,d\in \u2102$. 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 $B:V\times V\to \u2102$ 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  20130322 12:25:47 
Last modified on  20130322 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  sesquilinear form 
Related topic  InnerProduct 