##  [Rational Canonical Form](/rational-canonical-form-1) 

 Definition

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

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

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

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

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

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

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

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

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.