linalg(deprecated)/frobenius - Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : linalg(deprecated)/frobenius

linalg(deprecated)

  

frobenius

  

compute the Frobenius form of a matrix

  

ratform

  

a synonym for frobenius

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

frobenius(A)

frobenius(A, 'P')

Parameters

A

-

square matrix

'P'

-

(optional) assigned the transformation matrix

Description

• 

Important: The linalg package has been deprecated. Use the superseding command, LinearAlgebra[FrobeniusForm], instead.

  

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

• 

The function frobenius(A) or ratform(A) computes and returns the Frobenius form F of a matrix A. This is also known as the Rational Canonical form of a matrix.

• 

F 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 inversePAP=F.

• 

The Frobenius form defined in this way is unique (if we require that pi divides pi1).

• 

If the sizes of the blocks C1,C2,...,Ck are n1,n2,...,nk respectively, then the columns of the matrix P are the vectors [f1,Af1,..,An11f1,f2,...,fk,Afk,...,Ank1fk] 

• 

These functions are part of the linalg package, and so can be used in the form frobenius(..) or ratform(..) only after performing the command with(linalg), or with(linalg,frobenius) or with(linalg,ratform). These functions can always be accessed in the long form linalg[frobenius](..) or linalg[ratform](..).

Examples

Important: The linalg package has been deprecated. Use the superseding command, LinearAlgebra[FrobeniusForm], instead.

withlinalg,frobenius:

Aarray9,21,15,4,2,0,10,21,14,4,2,0,8,16,11,4,2,0,6,12,9,3,3,0,4,8,6,0,5,0,2,4,3,0,1,3

A−921−15420−1021−14420−816−11420−612−9330−48−6050−24−3013

(1)

frobeniusA,P

00001501000−47001005600010−320000190000003

(2)

printP

2175533732926512005126530840926564433126502511537533532259953557015311199953011211265302732659014726522407326545650726507347265220412656813926517216126535721926505163265154892654781126512192926525961126502714265814226525098265645422651405382651

(3)

References

  

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

  

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.

See Also

Frobenius

LinearAlgebra

LinearAlgebra[FrobeniusForm]