NormalMatrixFormm

NormalMatrixFormm is a proposed canonical representation in linear algebra for square matrices over a field. It aims to unify and extend classical normal forms such as the Jordan form, Schur form, and the rational canonical form by selecting a single, uniquely determined representative from each similarity class through a fixed set of normalization rules.

Construction of NormalMatrixFormm begins with the usual similarity classification. For a matrix A, one considers its Jordan form J = P^{-1} A P. A normalization procedure N is then applied to J to produce a canonical block structure J'. The NormalMatrixFormm of A is defined as the similarity transform A -> P J' P^{-1}. The exact normalization rules are domain-specific and can vary, but common guidelines include standardizing leading entries within blocks, fixing certain off-diagonal patterns, and arranging blocks in a prescribed eigenvalue order.

Key properties include invariance under similarity: if A and B are similar, they share the same NormalMatrixFormm. Consequently, NormalMatrixFormm encodes the eigenvalues, algebraic and geometric multiplicities, and the nilpotent structure of A, while abstracting away the particular basis.

Examples illustrate its behavior in simple cases. For a diagonal matrix diag(λ1, …, λn) with distinct eigenvalues, NormalMatrixFormm coincides with a diagonally ordered diag(λ1, …, λn). For a single Jordan block J_k(λ), NormalMatrixFormm produces a standardized Jordan block with a fixed canonical pattern. The identity matrix maps to itself.

Applications and limitations: NormalMatrixFormm can aid symbolic computation, matrix classification, and instructional exposition by providing a uniform representative. However, as with other canonical forms, different communities may adopt varying normalization rules, and numerical implementations must address stability concerns.

See also: Jordan form, Schur form, rational canonical form.