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Frobenius

inert Frobenius function

 

Calling Sequence

Parameters

Description

Examples

References

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 Ci are companion matrices associated with polynomials p1,p2,..,pk with the property that pi divides pi1, 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

AMatrix1+x,1+x2,1+x2,1+x4

A:=1+xx2+1x2+1x4+1

(1)

FFrobeniusA,'P'mod2

F:=0x5+x1x4+x

(2)

P

11+x0x2+1

(3)

Test the result

mapNormal,InversePmod2.A.PFmod2

0000

(4)

A1Matrix34RootOf_Z2+1x2+12RootOf_Z2+1x54RootOf_Z2+1,4+4RootOf_Z2+1x2+6+3RootOf_Z2+1x6+2RootOf_Z2+1,2+6RootOf_Z2+1x2+53RootOf_Z2+1x+2+2RootOf_Z2+1,35RootOf_Z2+1x2+4+4RootOf_Z2+1x+6+2RootOf_Z2+1:

F1evalaFrobeniusA1,'P1'

F1:=01114543RootOf_Z2+1+211168RootOf_Z2+1x3+1145x4+442x2RootOf_Z2+12119x31482xRootOf_Z2+1+796x2144RootOf_Z2+11726x62211133RootOf_Z2+1+239x216x+4+7RootOf_Z2+1+11xRootOf_Z2+1

(5)

P1

11253+4RootOf_Z2+125x2+5x+318RootOf_Z2+1+10xRootOf_Z2+10151+3RootOf_Z2+19xRootOf_Z2+1+10x22RootOf_Z2+12x+4

(6)

Test the result

mapevala@Normal,P11.A1.P1F1

0000

(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.

See Also

LinearAlgebra[FrobeniusForm]

LinearAlgebra[Modular]

RootOf

 


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