Endomorphism

In universal algebra , an endomorphism (from the Greek ἔνδον éndon 'inside' and μορφή morphē 'shape', 'form') is a homomorphism of a mathematical structure in itself. If it is also an isomorphism , it is also called automorphism . ${\ displaystyle f \ colon A \ to A}$ ${\ displaystyle A}$ ${\ displaystyle f}$

In category theory , any morphism whose source and destination coincide is called an endomorphism of the object in question.

The totality of the endomorphisms of an object is denoted by and always forms a monoid (the endomorphism monoid or the endomorphism half-group ), in additive categories even a (unitary) ring . ${\ displaystyle A}$ ${\ displaystyle \ operatorname {End} (A)}$

definition

Algebraic structures

Let be an algebraic structure , i.e. a non-empty set together with a finite number of links with corresponding arithmetic . Such an algebraic structure could for example be a vector space , a group or a ring . Then, in algebra, an endomorphism is understood to be a mapping of the set onto itself, which is a homomorphism , that is, it applies ${\ displaystyle (A, (f_ {i}))}$ ${\ displaystyle A}$ ${\ displaystyle f_ {i}}$ ${\ displaystyle \ sigma _ {i}}$ ${\ displaystyle (A, (+, \ cdot))}$ ${\ displaystyle (A, *)}$ ${\ displaystyle (A, (+, *))}$ ${\ displaystyle \ phi \ colon A \ to A}$ ${\ displaystyle A}$

{\ displaystyle \ phi \ left (f_ {i} (a_ {1}, \ dotsc, a _ {\ sigma _ {i}}) \ right) = f_ {i} (\ phi (a_ {1}), \ dotsc, \ phi (a _ {\ sigma _ {i}}))}

for everyone and everyone . ${\ displaystyle i}$ ${\ displaystyle a_ {1}, \ dotsc, a _ {\ sigma _ {i}} \ in A}$

Category theory

Be an object of a category . A morphism that operates on an object is called an endomorphism. ${\ displaystyle X}$ ${\ displaystyle f \ colon X \ to X}$ ${\ displaystyle X}$

For categories of homomorphisms between algebraic structures, the definition is equivalent to that in the previous section.

Special structures

Vector spaces

General

In linear algebra , an endomorphism of a - vector space is a - linear mapping . Here -linear (or simply linear, if it is clear which body is meant) means that the mapping ${\ displaystyle K}$ ${\ displaystyle V}$ ${\ displaystyle K}$ ${\ displaystyle f \ colon V \ to V}$ ${\ displaystyle K}$

{\ displaystyle f \ left (ax + y \ right) = af \ left (x \ right) + f \ left (y \ right)}

fulfilled for all and all . Together with the addition of the images and the composition as a multiplication, the set of all endomorphisms forms a ring, which is called the endomorphism ring. If the linear mappings are described by matrices, then with the matrix addition and the matrix multiplication the matrix ring is obtained, which is isomorphic to the endomorphism ring. ${\ displaystyle a \ in K}$ ${\ displaystyle x, y \ in V}$

If the underlying vector space is a topological vector space and one considers the vector space of the continuous endomorphisms, which in the case of infinite-dimensional vector spaces is generally a real subspace of the endomorphism space, then one can induce a topology on this vector space of all continuous endomorphisms so that the addition and the multiplication of the ring are continuous . Thus the endomorphism ring is a topological ring .

example

The derivation is an endomorphism on the vector space of the polynomials of a maximum of third degree with real coefficients. As a basis for choosing the monomial basis . This can be mapped isomorphically to the canonical base of . The 1 is in the i-th position of the 4- tuple . So each polynomial can be represented as a 4-tuple, for example . Now you can concatenate with and get a matrix notation for the differential: ${\ displaystyle \ textstyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}}}$ ${\ displaystyle V = \ mathbb {R} [x] _ {3}}$ ${\ displaystyle V}$ ${\ displaystyle \ textstyle \ left \ {1, x, x ^ {2}, x ^ {3} \ right \}}$ ${\ displaystyle \ mathbb {R} ^ {4}}$ ${\ displaystyle \ Phi \ left (x ^ {i} \ right) = (0, \ dotsc, 1, \ dotsc, 0) ^ {t} \ in \ mathbb {R} ^ {4}}$ ${\ displaystyle \ mathbb {R} [x] _ {3}}$ ${\ displaystyle \ Phi \ left (4x ^ {3} + 2x + 5 \ right) = (4,0,2,5) ^ {t}}$ ${\ displaystyle \ Phi}$ ${\ displaystyle \ textstyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}}}$

{\ displaystyle \ Phi \ circ {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ circ \ Phi ^ {- 1} = {\ begin {pmatrix} 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \ \ 0 & 0 & 1 & 0 \ end {pmatrix}}}

.

If you apply this matrix to the above example , you get what corresponds to the polynomial ; this could have been obtained by applying the derivative directly. ${\ displaystyle (4,0,2,5) ^ {t}}$ ${\ displaystyle (0,12,0,2) ^ {t}}$ ${\ displaystyle 12x ^ {2} +2}$

groups

An endomorphism on a group is a group homomorphism from to , that is, for applies to all . ${\ displaystyle G}$ ${\ displaystyle \ phi}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle \ phi \ colon G \ to G}$ ${\ displaystyle \ phi (gh) = \ phi (g) \ phi (h)}$ ${\ displaystyle g, h \ in G}$

literature

Gerd Fischer: Linear Algebra. ISBN 3-528-03217-0 .
M. Sh. Tsalenko: Endomorphism . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).