Involution (mathematics)

The term involution called in mathematics a self- inverse mapping. The term derives from the Latin word involvere " wrap " from.

  • 4.1 Negative and reciprocal
  • 4.2 The complex conjugation
  • 4.3 The quaternion conjugation
  • 4.4 Transposing matrices
  • 4.5 Calculating in F2
  • 4.6 geometry
  • 4.7 Involutory ciphers

Motivation

A linear mapping with is a reflection; in the case of a reflection at a point, a line or a plane. Is replaced by an arbitrary non-empty set and waives the linearity, we obtain the notion of involution.

Definition

An illustration with a matching definition and target quantity is said to be an involution if and only if for all.

This requirement can also be formulated as a compact or or. The identity referred to.

Properties

  • Each involution is a bijection and it is.
  • Is a bijection of finite set (ie an element of the symmetric group ), then iff is in involution if it can be written as a product of disjoint transpositions louder. One speaks in this case of a self- inverse permutation.

Involutions on vector spaces

  • Be a finite dimensional vector space over the field. A (linear) self-map if and only involutory if the minimal polynomial of the form, or has. This means that a involutory endomorphism is always diagonalizable, if not the characteristic 2 has, and all its eigenvalues ​​are off.
  • Each involution is a reflection on a subspace.
  • Each involution is a representation of the group Z/2Z in the general linear group GL (V).
  • About bodies with the characteristic 2 there is not diagonalizable endomorphisms involution. Thus, in the two-dimensional vector space by mapping an involution given, which is not diagonalizable.

Examples

Negative and reciprocal

The illustrations

And

Are involutions, because it is

And

Is generally an abelian group, so is the figure ( in additive notation ) or ( in multiplicative notation) Gruppenautomorphismus and an involution. For a non-Abelian group, this figure is even an involution, but not a group homomorphism.

The negation in classical logic is also an involution, since it applies:

The complex conjugation

When calculating with complex numbers forming the conjugate complex number is an involution: For a complex number is the conjugate complex number

Re-executing the conjugation supplies.

The quaternion conjugation

For quaternion

With the conjugate quaternion by

Formed. Because of the reversal of the order (important for non-commutative rings! ) Of the factors in the multiplication

This conjugation is called Antiautomorphismus.

Provides Re-executing the conjugation

So it is an involution.

Both of these features add up to an involutive Antiautomorphismus.

Transposing matrices

In the set of square matrices over a ring, the transposing

An involution. As a ring, even involutional Antiautomorphismus.

Arithmetic in F2

In the additive group of the residue class field mapping is an involution:

Geometry

In geometry are point - and Just reflections involutions.

Involutory ciphers

Involutory ciphers have the peculiarity that the algorithm to encrypt and decrypt are identical. They are thus particularly easy to handle. A simple example from the Cryptology is the ROT13 shift cipher in which to encrypt each letter is replaced by the shifted by 13 places in the alphabet letters. The repeated application of this method results in a shift to 26 letters and so back to the original plaintext. In the story, there was also much more complex involution encryption method. The most famous example is the German Enigma encryption machine that was used in World War II in the communications of the German military.

The logical function exclusive or is also self-inverse and is therefore used in, among other encryption algorithms such as one time pad.

Körperinvolution

Under a Körperinvolution is usually understood as an involution, which is also a Körperautomorphismus.

So from a Körperinvolution on a body is demanded

And for all

And

The best-known non-trivial Körperinvolution is the conjugation over the complex numbers. For this reason, it is often used for a Körperinvolution the same notation as for the complex conjugation: Instead of frequently written.

Another example is the automorphism of the body

By

Is defined. Note that he does not receive the amount in contrast to the complex conjugation:

Function (mathematics) Minimal polynomial (linear algebra) Characteristic (algebra) Group representation Group isomorphism Michiel Hazewinkel Encyclopedia of Mathematics
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