Hermitian sesquilinear form

The Hermitean product , Hermitean sesquilinear form or simply Hermitean form (after Charles Hermite ) is a special kind of sesquilinear form in linear algebra similar to the symmetrical bilinear form .

definition

Let be a vector space over the body . A Hermitian sesquilinear form is an illustration ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$

${\displaystyle \langle \,,\,\rangle \colon V\times V\to \mathbb {C} }$,

which fulfills the following conditions for all out and for all out : ${\displaystyle x,y,z}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle \mathbb {C} }$

1. ${\displaystyle \;\langle x,\,a\cdot y+z\rangle =a\langle x,y\rangle +\langle x,z\rangle }$( linear in one argument);
2. ${\displaystyle \;\langle a\cdot x+y,z\rangle ={\overline {a}}\langle x,z\rangle +\langle y,z\rangle }$( semilinear in the other argument);
3. ${\displaystyle \;\langle x,y\rangle ={\overline {\langle y,x\rangle }}}$ (Hermitian symmetry).

It denotes complex conjugation . ${\displaystyle {\overline {x}}}$

There are different conventions for the order of linear and semilinear arguments.

With property (3), (1) already follows from (2) and (2) from (1). For the sake of clarity, however, both (1) and (2) are mentioned here as conditions.

A Hermitian sesquilinear form is a sesquilinear form for which the third property also applies.

The term Hermitean sesquilinear form is only relevant over the field of complex numbers ; over the field of real numbers , every Hermitian sesquilinear form is a symmetric bilinear form . The inner product over a complex vector space is a Hermitian sesquilinear form. Analogously to this, a sesquilinear form on any module is also called Hermitian if there is any involutive anti-automorphism on the ring on which the module is based. If it lies in the center of the ring, then the sesquilinear form is called -hermitian if and only if applies. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \langle x,y\rangle =\sigma (\langle y,x\rangle )}$ ${\displaystyle \sigma }$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$ ${\displaystyle \langle x,y\rangle =\varepsilon \sigma (\langle y,x\rangle )}$

polarization

A polarization formula applies to Hermitian sesquilinear forms . The consequence of this is, in particular, that such a shape is already determined by its values ​​on the diagonal.

Hermitean standard shape

By

${\displaystyle \langle {\vec {x}},{\vec {y}}\rangle =\langle (x_{1},x_{2},\dotsc ,x_{n}),(y_{1},y_{2},\dotsc ,y_{n})\rangle ={\bar {x}}_{1}y_{1}+{\bar {x}}_{2}y_{2}+\dotsb +{\bar {x}}_{n}y_{n}=\sum _{k=1}^{n}{\bar {x}}_{k}y_{k}}$

defined figure is called Hermitean standard form .

See also

• Hermitian matrix

literature

• VL Popov: Hermitian form . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 49 .