A canonical representative for the similarity class of a linear operator over a field, constructed from the invariant factors (monic divisors of the characteristic polynomial) as a direct sum of companion blocks; it classifies matrices up to similarity without assuming polynomial splitting.

Principle

Decompose the vector space into cyclic modules for the operator corresponding to invariant factors; each invariant factor determines a companion matrix block and their direct sum (block-diagonal) is unique up to ordering and provides a canonical form under similarity over the base field.

Demonstration

Given a linear map with invariant factors f1 | f2 | ... | fk, assemble the companion matrices of those polynomials on the diagonal. This block-diagonal matrix has characteristic polynomial equal to the product of the fi and is similar to the original operator over the same field.

Misapplication

Using rational canonical form to infer explicit eigenvectors or to claim diagonalizability: companion blocks need not be diagonal and the form expresses similarity class but not an eigenbasis; one may incorrectly treat companion blocks as Jordan blocks.

Consequence

Provides a field-independent classification of similarity classes and is the appropriate normal form when eigenvalues do not lie in the base field; it allows algorithmic computation of invariants like the minimal polynomial and elementary divisors.

Reversal

The Jordan canonical form refines the rational canonical form when the characteristic polynomial splits: companion blocks then further decompose into Jordan blocks reflecting geometric multiplicities and nilpotent sizes.

Boundary

Applicable over any field and does not require algebraic closure; it does not, however, give orthogonality or unitary reductions and is less directly tied to spectral projections than diagonal or Schur decompositions.

Semantic Tension

Tension with Jordan form: both give canonical representatives of similarity, but Jordan form requires splitting and yields a more granular nilpotent block structure, while rational canonical form emphasizes invariant factors and works over the original field.

Synthesis

Rational canonical form encodes the module-theoretic structure of a linear operator via invariant factors and companion blocks, offering a canonical similarity representative valid over the base field and serving as the robust classification when eigenvalue splitting is not available.