Sesquilinearform

As a sesquilinear (Latin sesqui = one and a half ) is called in linear algebra is a function that associates two vectors a scalar value, and the semi -linear is linear in one of the other of its two arguments.

The two arguments can come from different vector spaces, which, however, a common Skalarkörper must be based; a sesquilinear form is a mapping; it is a linear form with respect to the one and a semi linear form with respect to the other argument. For the sequence of linear and semilinear argument, there are different conventions; in physics it is customary to call the semilinear argument first.

Over the real numbers is consistent with the concept of sesquilinear which the bilinear form.

Definition

There are vector spaces over the complex numbers.

A picture

Called sesquilinear if semilinear in the first and linear in the second argument, ie

And

Sometimes instead linearity also in the first and semi -linearity is required in the second argument; However, this difference is purely formal.

This definition can be generalized also to vector spaces over other bodies or modules over a ring once on the base or ring an excellent automorphism or at least endomorphism

Is given. A candidate for such endomorphisms is the Frobeniushomomorphismus in positive characteristic.

The constant zero mapping is a sesquilinear form, we write. Pointwise sums and scalar multiples of Sesquilinearformen are Sesquilinearformen again. The amount of Sesquilinearformen thus forms a vector space.

Hermitian sesquilinear

A sesquilinear form is called Hermitian if

Applies here denotes the complex conjugation.

This definition is analogous to the definition of the symmetric bilinear form. The adjective Hermitian derives from the mathematician Charles Hermite.

Examples

The inner product on a complex vector space is a Hermitian sesquilinear form with symmetry, so even a Hermitian form, see also Krein space.

Unique determination by the diagonal

Statement

The definition of the sesquilinear may seem more complicated than that of the bilinear form. One feature that distinguishes Sesquilinearformen of unsymmetrical bilinear forms, is the following:

It follows, for all, then, so.

Counterexample

For bilinear forms, the claim is false. This can be seen in the following example. Be and set

This is obviously bilinear and it applies, as well.

Conclusion

Be a Hilbert space and a bounded linear operator. Then a sesquilinear form is ( every sesquilinear can thus write ). Now, if and only if ( " ⇐ " is trivial opposite direction: ).

It follows that an operator is zero if and only if for all. However, this statement applies only to the main body of the complex numbers over the real numbers, the condition is also necessary that T is self-adjoint.

Sesquilinearformen on moduli

The concept of the sesquilinear form can be generalized to arbitrary modules, wherein any Antiautomorphismus on the underlying does not necessarily commutative ring takes the place of the complex conjugation. Be modules over the same ring and a Antiautomorphismus. A picture is said to be - sesquilinear if any apply to, and the following conditions: