Formingsimilar

Formingsimilar is a term used in algebraic geometry and invariant theory to describe an equivalence relation on homogeneous forms. Two forms f and g of the same degree d in n variables are forming-similar if there exists an invertible linear transformation T in GL(n) and a nonzero scalar α such that f(Tx) = α g(x) for all x. Equivalently, after a linear change of coordinates and up to a nonzero scalar, the two forms represent the same projective form.

This relation partitions the space of degree-d forms into orbits under the action of GL(n) × C*, and defines projective equivalence classes of forms. Invariant properties under forming-similarity are those that are preserved up to the combined action of coordinate changes and scalar rescaling.

Special case: quadratic forms. If f and g are quadratic, with Q_f and Q_g the corresponding symmetric matrices, then f and g are forming-similar if there exists T with Q_f = α T^T Q_g T for some α ≠ 0. Invariants such as rank are preserved; determinant transforms according to det(Q_f) = α^n det(T)^2 det(Q_g), and discriminants or signatures map in a predictable way under scaling and basis change.

Applications include classification of plane and space forms up to projective similarity, invariant theory, and shape recognition in computer vision, where objects are identified up to linear changes of coordinates and uniform scaling.

See also: similarity (linear algebra), quadratic form, projective geometry, invariant theory.