normalformen

Normalformen, or normal forms, are canonical representations of mathematical objects within a chosen equivalence relation, obtained by applying a specific class of transformations. The aim is to simplify the object while preserving essential structure or properties, so that objects in the same class share the same normal form.

In linear algebra, common normal forms include the diagonal and Jordan normal forms of square matrices, reached by similarity transformations. A matrix is diagonalizable if it is similar to a diagonal matrix; otherwise it has a Jordan form with Jordan blocks. Other normal forms for integer matrices include the Smith normal form and the Hermite normal form, which provide canonical representatives under specified equivalences and are useful in studying modules and abelian groups.

In differential equations and dynamical systems, normal form theory simplifies systems near equilibrium points by removing nonessential nonlinear terms through near-identity changes of variables. This helps reveal the local dynamics. Techniques such as the Poincaré-Dulac method yield resonant normal forms, while linearization and, when needed, center manifolds provide successive approximations to the behavior near equilibria.

In algebraic geometry and singularity theory, normal forms classify singularities up to coordinate changes, producing canonical models. Simple singularities are often organized into types such as A-D-E, and the study includes versal or miniversal deformations that describe nearby structural changes.

In logic and theoretical computer science, normal forms organize formulas and grammars for analysis and computation. Examples include conjunctive and disjunctive normal forms, prenex normal form in logic, and Chomsky normal form or Greibach normal form in formal grammars, which facilitate automated reasoning and parsing.