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 $F$ which has the following structure: F = diag(C[1], C[2],.., C[k]) where the $C_i$ are companion matrices associated with polynomials $p_1,p_2,..,p_k$ with the property that $p_i$ divides $p_{i-1}$, for $i$ = 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.

$A≔\mathrm{Matrix}\left(\left[\left[1+x\,1+x^2\right]\,\left[1+x^2\,1+x^4\right]\right]\right)$

$A≔\left[\begin{array}{cc}1+x& x^2+1\\ x^2+1& x^4+1\end{array}\right]$

$F≔\mathrm{Frobenius}\left(A\,'P'\right)\mathbf{mod}2$

$F≔\left[\begin{array}{cc}0& x^5+x\\ 1& x^4+x\end{array}\right]$

$P$

$\left[\begin{array}{cc}1& 1+x\\ 0& x^2+1\end{array}\right]$

$\mathrm{map}\left(\mathrm{Normal}\,\left(\mathrm{Inverse}\left(P\right)\mathbf{mod}2\right)·A·P-F\right)\mathbf{mod}2$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\mathrm{A1}≔\mathrm{Matrix}\left(\left[\left[\left(-3-4\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x^2+\left(1-2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x-5-4\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\,\left(-4+4\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x^2+\left(6+3\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x-6+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right]\,\left[\left(2+6\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x^2+\left(5-3\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x+2+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\,\left(-3-5\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x^2+\left(4+4\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)x+6+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right]\right]\right)\:$

$\mathrm{F1}≔\mathrm{evala}\left(\mathrm{Frobenius}\left(\mathrm{A1}\,'\mathrm{P1}'\right)\right)$

$\mathrm{F1}≔\left[\begin{array}{cc}0& -\frac{\left(43\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+21\right)\left(-1168\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)x^3+1145x^4+442x^2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)-2119x^3-1482x\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+796x^2-144\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)-1726x-622\right)}{1145}\\ 1& -\frac{\left(3\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+2\right)\left(39x^2-16x+4+7\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+11x\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)}{13}\end{array}\right]$

$\mathrm{P1}$

$\left[\begin{array}{cc}1& -\frac{\left(3+4\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)\left(25x^2+5x+31-8\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+10x\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)}{25}\\ 0& \frac{\left(1+3\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)\left(-9x\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)+10x^2-2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)-2x+4\right)}{5}\end{array}\right]$

$\mathrm{map}\left(\mathrm{evala}@\mathrm{Normal}\,{\mathrm{P1}}^{-1}·\mathrm{A1}·\mathrm{P1}-\mathrm{F1}\right)$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

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.