In linear algebra , the sesquilinear form (Latin sesqui = one and a half) is a function that assigns a scalar value to two vectors and which is linear in one of its two arguments and semilinear in the other. A classic example is the through
defined mapping , the complex standard scalar product . Here, the dash denotes the complex conjugation .
The two arguments can come from different vector spaces , but they must be based on a common scalar field ; a sesquilinear form is an illustration ; it is a linear form with respect to one argument and a semi- linear form with respect to the other argument. There are different conventions for the order of linear and semi-linear arguments; in physics it is customary to put the semilinear argument first.
Above the real numbers , the concept of the sesquilinear form agrees with that of the bilinear form .
Let there be vector spaces over the complex numbers .
is called sesquilinear form if is semilinear in the first and linear in the second argument, that is
There are , and .
Sometimes linearity in the first and semi-linearity in the second argument is required instead; however, this difference is of a purely formal nature.
This definition can also be generalized to vector spaces over other bodies or modules over a ring as soon as there is an excellent automorphism or at least endomorphism on the basic body or ring
given is. A candidate for such endomorphisms is the Frobenius homomorphism with positive characteristics .
The constant zero mapping is a sesquilinear form, we write . Pointwise sums and scalar multiples of sesquilinear forms are again sesquilinear forms. The set of sesquilinear forms thus forms a vector space.
Hermitian sesquilinear form
A sesquilinear form is called Hermitian, if
applies. This definition is analogous to the definition of the symmetrical bilinear form . The adjective "Hermitesch" is derived from the mathematician Charles Hermite .
An inner product over a complex vector space is a sesquilinear form with Hermitian symmetry, i.e. even a Hermitian form , see also Krein space .
The so-called polarization formula plays an important role
which shows that the shape is already uniquely determined by its values on the diagonal, ie on pairs of the shape .
The polarization formula only applies to sesquilinear forms, but not to general bilinear forms.
A direct consequence of the polarization formula is the fact that the form already disappears when for all .
Or to put it another way: if for everyone , then , well .
This statement does not apply to general bilinear forms, consequently there cannot be a polarization formula. This can be seen in the following example. Be and sit
is apparently bilinear and it applies
to everyone . On the other hand is
Let be a Hilbert space and a bounded linear operator . Then is a bounded sesquilinear form. The narrowness means that (here ). Conversely, it follows from
Fréchet-Riesz's representation theorem that every bounded sesquilinear form determines a bounded operator
such that for all .
In particular, disappears exactly when disappears. This can also be easily seen directly as follows: if so follows for all , so
. The converse follows immediately from the definition of .
With the polarization identity it follows that an operator is zero if and only if for all . However, this statement only applies to the basic field of complex numbers ; the condition that T is self-adjoint is also necessary for the real numbers .
Sesquilinear forms on modules
The concept of the sesquilinear form can be generalized to any modules , with the complex conjugation being replaced by any anti-automorphism on the underlying, not necessarily commutative ring . Have modules over the same ring and an anti-automorphism . A picture is said to be -Sesquilinearform if for any , and the following conditions apply:
↑ D. Werner: functional analysis 5th, extended edition. Springer, 2004, ISBN 3-540-21381-3 , Korollar V.5.8, p. 236.
^ Nicolas Bourbaki : Algèbre (= Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 10 .