Frobenius Normal Form - A Rational Normal Form Generalizing The Jordan Normal Form

A Rational Normal Form Generalizing The Jordan Normal Form

The Frobenius normal form does not reflect any form of factorization of the characteristic polynomial, even if it does exist over the ground field F. This implies that it is invariant when F is replaced by a different field (as long as it contains the entries of the original matrix A). On the other hand this makes the Frobenius normal form rather different than other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form (if A is diagonalizable) or more generally the Jordan normal form (if the characteristic polynomial splits into linear factors). For instance, the Frobenius normal form of a diagonal matrix with distinct diagonal entries is just the companion matrix of its characteristic polynomial.

There is another way to define a normal form, that like the Frobenius normal form is always defined over the same field F as A, but that does reflect a possible factorization of the characteristic polynomial (or equivalently the minimal polynomial) into irreducible factors over F, and which reduces to the Jordan normal form in case this factorization only contain linear factors (corresponding to eigenvalues). This form is sometimes called the generalized Jordan normal form, or primary rational canonical form. It is based on the fact that the vector space can be canonically decomposed into a direct sum of stable subspaces corresponding to the distinct irreducible factors P of the characteristic polynomial (as stated by the lemme des noyaux), where the characteristic polynomial of each summand is a power of the corresponding P. These summands can be further decomposed, non-canonically, as a direct sum of cyclic F-modules (like is done for the Frobenius normal form above), where the characteristic polynomial of each summand is still a (generally smaller) power of P. The primary rational canonical form is a block diagonal matrix corresponding to such a decomposition into cyclic modules, with a particular form called generalized Jordan block in the diagonal blocks, corresponding to a particular choice of a basis for the cyclic modules. This generalized Jordan block is itself a block matrix of the form

where C is the companion matrix of the irreducible polynomial P, and U is a matrix whose sole nonzero entry is a 1 in the upper right hand corner. For the case of a linear irreducible factor P = x − λ, these blocks are reduced to single entries C = λ and U = 1 and, one finds a (transposed) Jordan block. In any generalized Jordan block, all enrties immediately below the main diagonal are 1. A basis of the cyclic module giving rise to this form is obtained by choosing a generating vector v (one that is not annihilated by Pk−1(A) where the minimal polynomial of the cyclic module is Pk), and taking as basis

where d = deg(P).