Endform

Endform is a term that appears in some mathematical writings as a generic label for the canonical form of a linear endomorphism. An endomorphism is a linear map from a vector space to itself, and choosing a basis turns such a map into a square matrix. The endform, in this sense, is a standard matrix representative chosen from the similarity class of that endomorphism.

In linear algebra, two endomorphisms are considered equivalent if they are similar, meaning they represent the same linear transformation under different bases. A canonical form provides a unique or nearly unique representative within each equivalence class. Classic examples of endforms include the Jordan canonical form, which exists when the field is algebraically closed, and the rational canonical form, which exists over any field. These forms reveal structural features such as eigenvalues, nilpotency, and invariant factors.

Computing an endform typically involves determining the minimal polynomial, eigenvalues, and invariant factors of the endomorphism. The Jordan form groups blocks by eigenvalues and sizes, emphasizing geometric and algebraic multiplicities, while the rational canonical form uses companion matrices corresponding to invariant factors, offering a basis-independent description even when the field lacks certain roots.

Notes and usage: the term endform is not standard in all texts; many writers refer directly to Jordan form, rational canonical form, or invariant factors. Endform is often understood as an umbrella notion noting that various canonical representations classify endomorphisms up to similarity.

See also: Endomorphism, Canonical form, Jordan form, Rational canonical form, Invariant factors.