# 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. 1.

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

2. 2.

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

3. 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 HermitianForm 2013-03-22 12:25:47 2013-03-22 12:25:47 djao (24) djao (24) 8 djao (24) Definition msc 47A07 msc 15A63 msc 11E39 sesquilinear form sesqui-linear form InnerProduct