Frobenius - Maple Programming Help

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Frobenius

inert Frobenius function

 Calling Sequence Frobenius(A) Frobenius(A, 'P')

Parameters

 A - square Matrix 'P' - (optional) assigned the transformation matrix

Description

 • 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 ${p}_{1},{p}_{2},\left(\right)..\left(\right),{p}_{k}$ 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.

Examples

 > $A≔\mathrm{Matrix}\left(\left[\left[1+x,1+{x}^{2}\right],\left[1+{x}^{2},1+{x}^{4}\right]\right]\right)$
 ${A}{:=}\left[\begin{array}{cc}{1}{+}{x}& {{x}}^{{2}}{+}{1}\\ {{x}}^{{2}}{+}{1}& {{x}}^{{4}}{+}{1}\end{array}\right]$ (1)
 > $F≔\mathrm{Frobenius}\left(A,'P'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2$
 ${F}{:=}\left[\begin{array}{cc}{0}& {{x}}^{{5}}{+}{x}\\ {1}& {{x}}^{{4}}{+}{x}\end{array}\right]$ (2)
 > $P$
 $\left[\begin{array}{cc}{1}& {1}{+}{x}\\ {0}& {{x}}^{{2}}{+}{1}\end{array}\right]$ (3)

Test the result

 > $\mathrm{map}\left(\mathrm{Normal},\left(\mathrm{Inverse}\left(P\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}P-F\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (4)
 > $\mathrm{A1}≔\mathrm{Matrix}\left(\left[\left[\left(-3-4\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right){x}^{2}+\left(1-2\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right)x-5-4\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right),\left(-4+4\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right){x}^{2}+\left(6+3\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right)x-6+2\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right],\left[\left(2+6\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right){x}^{2}+\left(5-3\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right)x+2+2\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right),\left(-3-5\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right){x}^{2}+\left(4+4\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right)x+6+2\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+1\right)\right]\right]\right):$
 > $\mathrm{F1}≔\mathrm{evala}\left(\mathrm{Frobenius}\left(\mathrm{A1},'\mathrm{P1}'\right)\right)$
 ${\mathrm{F1}}{:=}\left[\begin{array}{cc}{0}& {-}\frac{{1}}{{1145}}{}\left({43}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{21}\right){}\left({-}{1168}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){}{{x}}^{{3}}{+}{1145}{}{{x}}^{{4}}{+}{442}{}{{x}}^{{2}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){-}{2119}{}{{x}}^{{3}}{-}{1482}{}{x}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{796}{}{{x}}^{{2}}{-}{144}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){-}{1726}{}{x}{-}{622}\right)\\ {1}& {-}\frac{{1}}{{13}}{}\left({3}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{2}\right){}\left({39}{}{{x}}^{{2}}{-}{16}{}{x}{+}{4}{+}{7}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{11}{}{x}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right)\end{array}\right]$ (5)
 > $\mathrm{P1}$
 $\left[\begin{array}{cc}{1}& {-}\frac{{1}}{{25}}{}\left({3}{+}{4}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right){}\left({25}{}{{x}}^{{2}}{+}{5}{}{x}{+}{31}{-}{8}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{10}{}{x}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right)\\ {0}& \frac{{1}}{{5}}{}\left({1}{+}{3}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right){}\left({-}{9}{}{x}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){+}{10}{}{{x}}^{{2}}{-}{2}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){-}{2}{}{x}{+}{4}\right)\end{array}\right]$ (6)

Test the result

 > $\mathrm{map}\left(\mathrm{evala}@\mathrm{Normal},{\mathrm{P1}}^{-1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{A1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{P1}-\mathrm{F1}\right)$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (7)

References

 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.