Endomorphism

In linear algebra, an endomorphism is a linear mapping &phi; of a linear space V into itself, where V is assumed to be over the field  of numbers F. (Outside of pure mathematics F is usually either the field of real or complex numbers). A mapping &phi; is linear if
 * &phi;(c1&thinsp;v1+c2&thinsp;v2)= c1&thinsp;&phi;(v1)+ c2&thinsp;&phi;(v2)

for any c1, c2 in F and v1, v2 in V.

Since the product of two endomorphisms &phi; and &psi; is again an endomorphism of V, the multiplication &phi;&deg;&psi; associates with any two endomorphisms of V a third endomorphism. This multiplication has the following properties:
 * 1) Associative law: &chi;&deg;(&psi;&deg;&phi;) = (&chi;&deg;&psi;)&deg;&phi;.
 * 2) Distributive laws: (c1&thinsp;&psi;1+c2&thinsp;&psi;2)&deg;&phi; =  c1&thinsp;&psi;1&deg;&phi; + c2&thinsp;&psi;2&deg;&phi;   and   &psi;&deg;(c1&thinsp;&phi;1 +c2&thinsp;&phi;2) =   c1&thinsp;&psi;&deg;&phi;1 + c2&thinsp;&thinsp;&psi;&deg;&phi;2, where  c1 and c2 belong to F.
 * 3) There exists an endomorphism &iota; (the identity map)  such that &phi;&deg;&iota; = &iota;&deg;&phi; = &phi; for every endomorphism &phi;.

Note that the product is not commutative, i.e., in general &psi;&deg;&phi; &ne;&phi;&deg;&psi;. The set of all endomorphisms forms an associative algebra. That is, the set is a linear space with multiplication. This algebra is often denoted by EndF(V) or by L(V,V).

An endomorphism &phi; is called an automorphism of V if it is invertible, that is, there exists an endomorphism &phi;&minus;1 such that &phi;&minus;1&deg;&phi; = &phi;&deg;&phi;&minus;1 = &iota;. The set of automorphisms is not a linear space (a linear combination of automorphisms is not generally invertible). The set is, however, a group, denoted by GL(V, F) (the general linear group of automorphisms on V).

If V is finite dimensional, say of dimension n, the algebra EndF(V) is algebra-isomorphic with the algebra of n&times;n matrices with elements in F. The element &iota; is in 1-1 correspondence with the n&times;n identity matrix I. The elements of GL(V, F) are in 1-1 correspondence with non-singular (invertible) matrices.