Frobenius Form

Frobenius form, also called Frobenius normal form, has two distinct uses in matrix theory.

For a square matrix over a field, it is another name for rational canonical form, a block diagonal matrix of companion matrices whose monic polynomial invariant factors divide their successors (Geck 2020). It is the same for similar matrices and is unique when the divisibility ordering is fixed.

For a nonnegative matrix , Frobenius normal form instead means a block triangular form obtained by simultaneously permuting rows and columns (Butkovič et al. 2012). A permutation matrix can be chosen so that

where each diagonal block is an irreducible matrix, including a possible zero block. The blocks correspond to the strongly connected components of the directed graph with an edge from to when . The ordering of these blocks need not be unique. This decomposition is different from rational canonical form, which permits an arbitrary nonsingular matrix as the change of basis rather than just a permutation matrix.