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LinearAlgebra

  

FrobeniusForm

  

reduce a square Matrix to Frobenius form (rational canonical form)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

FrobeniusForm(A, out, options, outopts)

RationalCanonicalForm(A, out, options, outopts)

Parameters

A

-

Matrix

out

-

(optional) equation of the form output = obj where obj is one of 'F' or 'Q', or a list containing one or more of these names; selects result objects to compute

options

-

(optional); constructor options for the result object(s)

outopts

-

(optional) equation(s) of the form outputoptions[o] = list where o is one of 'F' or 'Q'; constructor options for the specified result object

Description

• 

The FrobeniusForm(A) command returns the Frobenius form F of square Matrix A.

• 

This function can also be invoked using the RationalCanonicalForm command.

• 

The Frobenius form Matrix F has the following structure:

F = DiagonalMatrix([C[1], C[2],..., C[k]])

  

where the Ci are companion Matrices associated with polynomials p1,p2,..,pk where the pi are a factorization of the characteristic polynomial of A with the property that pi divides pi1, for i = 2..k.

  

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

  

The columns of Q form a rational canonical basis for A.

• 

The output option (out) determines the content of the returned expression sequence.

  

Depending on what is included in the output option, an expression sequence containing one or more of the factors F (the Frobenius form), or Q (the transformation Matrix) can be returned. If output is a list, the objects are returned in the same order as specified in the list.

  

The returned Matrix objects have the property that MatrixInverseQ.A.Q=F.

• 

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result(s). These options may also be provided in the form outputoptions[o]=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions[o] option, the latter takes precedence (regardless of the order).

  

The following list indicates permissible values for index [o] of outputoptions with their corresponding meaning.

  

 

F

Frobenius form

Q

transformation Matrix

Examples

withLinearAlgebra:

A0,1,1,1,1|2,2,0,2,4|0,0,1,1,3|6,0,3,1,3|2,2,2,2,4

A:=0206212002101321211214334

(1)

FrobeniusFormA

0040010200012000000200010

(2)

factorCharacteristicPolynomialA,x

x2x2+22

(3)

MBandMatrix2,2,2,2,1,1,0

M:=2100021000200002

(4)

F,QFrobeniusFormM,output='F','Q'

F,Q:=00801012001600002,0010014012401241

(5)

Q1.M.Q

00801012001600002

(6)

See Also

LinearAlgebra

LinearAlgebra[CharacteristicPolynomial]

LinearAlgebra[CompanionMatrix]

LinearAlgebra[DiagonalMatrix]

LinearAlgebra[MatrixInverse]

Matrix

 


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