Involution (math)

In mathematics, involution means a self- inverse mapping . The name is derived from the Latin word involvere "to wrap up".

definition

A picture with a matching definition and target quantity is said to be an involution if all the following applies: . ${\displaystyle f\colon A\rightarrow A}$ ${\displaystyle A}$ ${\displaystyle x\in A}$ ${\displaystyle f(f(x))=x}$

This requirement can also be formulated more compactly than or . Here referred to the identity on . ${\displaystyle f\circ f=\operatorname {id} _{A}}$ ${\displaystyle f^{2}=\operatorname {id} _{A}}$ ${\displaystyle \operatorname {id} _{A}}$ ${\displaystyle A}$

properties

• Every involution is a bijection and it applies . ${\displaystyle f^{-1}=f}$
• If and are involutions, then their composition is itself an involution if and only if holds. ${\displaystyle f\colon A\to A}$ ${\displaystyle g\colon A\to A}$ ${\displaystyle f\circ g}$ ${\displaystyle f\circ g=g\circ f}$
• If there is a bijection of the finite set (i.e. an element of the symmetrical group ), then it is involutor if and only if it can be written as the product of disjunct swaps . In this case one speaks of a self-inverse permutation . ${\displaystyle \pi }$ ${\displaystyle \mathbb {N} _{n}=\{1,\dotsc ,n\}}$ ${\displaystyle S_{n}}$ ${\displaystyle \pi }$

Involutions on vector spaces

Let be a finite-dimensional vector space over the body . ${\displaystyle V}$ ${\displaystyle K}$

• A (linear) self-mapping is involutive if and only if the minimal polynomial of has the form , or . This means in particular: ${\displaystyle f\in \operatorname {End} (V)}$ ${\displaystyle f}$ ${\displaystyle x^{2}-1}$ ${\displaystyle x-1}$ ${\displaystyle x+1}$
  • If the characteristic of the basic body is different from 2, then every involutive endomorphism is diagonalizable and all its eigenvalues are in . ${\displaystyle K}$ ${\displaystyle \{-1;+1\}}$
  • Each involution is a representation of the group Z / 2Z in the general linear group GL (V). ${\displaystyle f\in \operatorname {End} (V)}$
  • Over bodies with characteristic 2 there are involutionary endomorphisms that cannot be diagonalized. Thus, in the two-dimensional vector space , the matrix gives an involution that cannot be diagonalized. ${\displaystyle K}$ ${\displaystyle \mathbb {F} _{2}^{2}}$ ${\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$

Examples

Negative and reciprocal

The illustrations

${\displaystyle \mathbb {R} \to \mathbb {R} ,\quad x\mapsto -x}$

and

${\displaystyle \mathbb {R} ^{\times }\to \mathbb {R} ^{\times },\quad x\mapsto {\frac {1}{x}}}$

are involutions because it applies

${\displaystyle -(-x)=x}$ for all ${\displaystyle x\in \mathbb {R} }$

and

${\displaystyle {\frac {1}{1/x}}=x}$for everyone . ${\displaystyle x\neq 0}$

If in general an Abelian group is , then the mapping (with additive notation) or (with multiplicative notation) is a group automorphism and an involution. For a non-Abelian group, this mapping is also an involution, but not a group homomorphism (nonetheless a group anti- homomorphism ). ${\displaystyle G}$ ${\displaystyle g\mapsto -g}$ ${\displaystyle g\mapsto g^{-1}}$

The negation in classical logic is also an involution, because the following applies:

${\displaystyle \forall x(\lnot \lnot x\rightarrow x)}$

The complex conjugation

When calculating with complex numbers , the formation of the conjugate complex number is an involution: For a complex number with is the conjugate complex number ${\displaystyle z=a+b\mathrm {i} }$ ${\displaystyle a,b\in \mathbb {R} }$

${\displaystyle {\bar {z}}=z^{*}=a-b\mathrm {i} .}$

Performing the conjugation again delivers . ${\displaystyle {\overline {\overline {z}}}=z^{**}=a+b\mathrm {i} =z}$

The quaternion conjugation

To the quaternion

${\displaystyle x=x_{0}+x_{1}\mathrm {i} +x_{2}\mathrm {j} +x_{3}\mathrm {k} }$

with becomes the conjugated quaternion through ${\displaystyle x_{0},x_{1},x_{2},x_{3}\in \mathbb {R} }$

${\displaystyle {\bar {x}}=x_{0}-x_{1}\mathrm {i} -x_{2}\mathrm {j} -x_{3}\mathrm {k} }$

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

${\displaystyle {\overline {x\cdot y}}={\bar {y}}\cdot {\bar {x}}}$

this conjugation is called antiautomorphism .

Performing the conjugation again delivers

${\displaystyle {\overline {\overline {x}}}=x.}$

So it is an involution.

Both properties together result in an involutive anti-automorphism .

Transposing matrices

In the set of square matrices above a ring is the transpose ${\displaystyle R^{n\times n}}$ ${\displaystyle R}$

${\displaystyle \cdot ^{T}\colon R^{n\times n}\rightarrow R^{n\times n}}$, ${\displaystyle A\mapsto A^{T}}$

an involution. Since there is a ring, even an involutive anti-automorphism. ${\displaystyle R^{n\times n}}$

Calculate in F ₂

In the additive group of the remainder class field , the mapping is an involution: ${\displaystyle \mathbb {F} _{2}}$ ${\displaystyle x\mapsto x+1}$

${\displaystyle (x+1)+1=x.}$

geometry

In geometry , point and line reflections are involutions.

Involuntary Ciphers

Involuntary ciphers have the peculiarity that the algorithms for encryption and decryption are identical. They are therefore particularly easy to use. A simple example from cryptology is the shift code ROT13 , in which each letter is replaced by the letter shifted by 13 places in the alphabet for encryption . Applying this method twice results in a shift of 26 letters and thus again the original plain text . In history, however, there were also much more complex involutive encryption methods. The best-known example is the German ENIGMA encryption machine , which was used in communications by the German military during World War II .

The logical function Exclusive Or is also self-inverse and is therefore used in encryption algorithms such as One Time Pad .

Body involution

A body involution is usually understood to be an involution that is also a body automorphism .

So one demands a body involution over a body ${\displaystyle \sigma }$ ${\displaystyle K}$

${\displaystyle \sigma ^{2}=\operatorname {id} _{K}}$

as well as for everyone ${\displaystyle a,b\in K}$

${\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b)}$

and

${\displaystyle \sigma (ab)=\sigma (a)\sigma (b).}$

The best known nontrivial body involution is the conjugation over the complex numbers . For this reason the same notation is used for a body involution as for the complex conjugation: instead of is often written. ${\displaystyle \sigma (a)}$ ${\displaystyle {\overline {a}}}$

Another example is the automorphism of the body

${\displaystyle \mathbb {Q} \left({\sqrt {2}}\right)=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},}$

the through

${\displaystyle a+b{\sqrt {2}}\mapsto a-b{\sqrt {2}}}$

is defined. Note that in contrast to the complex conjugation, it does not receive the amount:

${\displaystyle |7-5{\sqrt {2}}|\approx 0{,}1,}$ but ${\displaystyle |7+5{\sqrt {2}}|\approx 14{,}1.}$

literature

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