The Frobenius function is a placeholder for representing the Frobenius form (or Rational Canonical form) of a square matrix. It is used in conjunction with either mod or evala.

The Frobenius function returns the square matrix  which has the following structure: F = diag(C[1], C[2],.., C[k]) where the  are companion matrices associated with polynomials  with the property that  divides , for  = 2..k.

If called in the form Frobenius(A, 'P'), then P will be assigned the transformation matrix corresponding to the Frobenius form, that is, the matrix P such that inverse(P) * A * P = F.

The call Frobenius(A) mod p computes the Frobenius form of A modulo p which is a prime integer. The entries of A must have rational coefficients or coefficients from an algebraic extension of the integers modulo p.

The call evala(Frobenius(A)) computes the Frobenius form of the square matrix A where the entries of A are algebraic numbers (or functions) defined by RootOfs.

Martin, K., and Olazabal, J.M. "An Algorithm to Compute the Change Basis for the Rational Form of K-endomorphisms." Extracta Mathematicae, (August 1991): 142-144.

Ozello, Patrick. "Calcul Exact des Formes de Jordan et de Frobenius d'une Matrice." PhD Thesis, Joseph Fourier University, Grenoble, France, 1987.