Semilinear transformation

As a semi- linear map is called in linear algebra a picture of a vector space over a field to another vector space over the same body, which up to a Körperautomorphismus in this sense "almost" a linear mapping is linear. In geometry, more generally semi- linear maps between vector spaces links are defined by possibly different skew fields as pictures in the same sense, which are linear up to a Schiefkörpermonomorphismus.

Each linear map is semi -linear. Just then any semi- linear map on a vector space (or links vector space ) is linear even if the body ( or skew field ) permits a single automorphism of the identity. Have this property, for example, all prime field, the field of real numbers and all Euclidean, in particular the real closed fields. A semi- linear function (also Semlinearform ) is a semilinear map a - ( left ) vector space in the (skew - ) body itself as a one-dimensional vector space.

If you choose solid bases of vector spaces each semilinear map can be uniquely represented as a consecutive execution of a linear map, that is a matrix, and the application of the respective (skew - ) Körperautomorphismus on each coordinate.

The most important for applications outside of geometry in the narrower sense, as for Sesquilinearformen, cases are the semi- linear maps between complex spaces, that is, between vector spaces, with respect to the complex conjugation. For these cases, the term described in this article is also referred to as "anti- linear map " or " conjugate linear mapping ," in the projective case is called a bijective semi -linear self-map then Antiprojektivität, in these terms, the figure must each semi- linear, but may not linearly be, in other words, the associated Körperautomorphismus may not be the identity map.

Each semi- linear mapping provides the synthetic geometry a representation of the homogeneous portion of a straight true image of an at least two-dimensional Desargues affine geometry with more than two points on each line in a different affine geometry or a matrix representation of a at least two dimensional, Desargues projective geometry to another projective geometry with respect to each a fixed predetermined in values ​​and target space coordinate system. Here the morphism from the definition and the representation is also a Schiefkörpermonomorphismus, ie an injective homomorphism between skew fields can be. The image space can also be a left - vector space over a "larger " skew field and the value space over a field which is isomorphic to a part of the body.

Bijective semi -linear self-maps of at least two-dimensional, Desargues affine or projective space in this sense are exactly the matrix representations for the collineations of this space, possibly together with a Schiefkörperautomorphismus.

Definition

An illustration of a - ( left ) vector space over the field (or skew field ) to a left - vector space is called semilinear map, if a (skew - ) Körperautomorphismus exists with which it satisfies the following two conditions. For any and all of the following applies:

Representation

It should be a skew field and, let - or -dimensional vector spaces over links. Be a semilinear map. Then there exist, for any vector space basis and any vector space basis of unique matrices and a Schiefkörperautomorphismus, so that, with respect to an arbitrary coordinate vector in the coordinate representation of the base

When the image vector is represented as a coordinate vector relative to the base. The matrices are uniquely determined by the bases and the said relationship to each, but different from each other in general. As automorphism, the same can be used independently of the chosen bases in both representations. Uniquely determined he is to the relationship, if true for the image of the semi -linear mapping. Compare also collineation.

Examples and counter-examples

• There are vector spaces over the complex numbers. A picture

• It should be. The non- identical involutory automorphism

• A collineation of a projective translation plane of Lenz class IV can not be represented by a semi- linear map, because the plane is not coordinatizable by a skew field.
• A antiunitärer operator is a semilinear map on a complex Hilbert space with respect to the complex conjugation, which is obtained by successive application of a unitary operator and the coordinate- wise complex conjugation. Alternatively, antiunitary operators characterized as semi -linear, surjective isometries. They play in the mathematical description of quantum mechanics as symmetries - albeit against unitary operators less important - role (see also theorem of Wigner ). The time conversion is an example of such a symmetry.

Generalization

More generally, a ring and a linear operator, it means an additive mapping - semilinear if

For all and true.