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.
|
•
|
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
|
|
>
|
|
| (1) |
>
|
|
| (2) |
>
|
|
| (3) |
Test the result
>
|
|
| (4) |
>
|
![A1 := Matrix([[(-3-4*RootOf(_Z^2+1))*x^2+(1-2*RootOf(_Z^2+1))*x-5-4*RootOf(_Z^2+1), (-4+4*RootOf(_Z^2+1))*x^2+(6+3*RootOf(_Z^2+1))*x-6+2*RootOf(_Z^2+1)], [(2+6*RootOf(_Z^2+1))*x^2+(5-3*RootOf(_Z^2+1))*x+2+2*RootOf(_Z^2+1), (-3-5*RootOf(_Z^2+1))*x^2+(4+4*RootOf(_Z^2+1))*x+6+2*RootOf(_Z^2+1)]])](/support/helpjp/helpview.aspx?si=5044/file00049/math141.png)
|
>
|
|
![F1 := Matrix([[0, -(1/1145)*(43*RootOf(_Z^2+1)+21)*(-1726*x+442*x^2*RootOf(_Z^2+1)-1482*x*RootOf(_Z^2+1)-1168*x^3*RootOf(_Z^2+1)-2119*x^3+796*x^2-622-144*RootOf(_Z^2+1)+1145*x^4)], [1, -(1/13)*(2+3*RootOf(_Z^2+1))*(39*x^2-16*x+11*x*RootOf(_Z^2+1)+4+7*RootOf(_Z^2+1))]])](/support/helpjp/helpview.aspx?si=5044/file00049/math148.png)
| (5) |
>
|
|
| (6) |
Test the result
>
|
|
| (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.
|
|
|
Download Help Document
Was this information helpful?