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MatrixPolynomialAlgebra

  

ColumnReducedForm

  

compute a column-reduced form of a Matrix

  

RowReducedForm

  

compute a row-reduced form of a Matrix

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ColumnReducedForm(A, x, U)

RowReducedForm(A, x, U)

Parameters

A

-

Matrix

x

-

variable name of the polynomial domain

U

-

(optional) name to return unimodular multiplier

Description

• 

The ColumnReducedForm(A,x) command computes a column-reduced form of an m x n rectangular matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q (that is, univariate polynomials in x with coefficients in Q(a1,...,an)).

• 

The RowReducedForm(A,x) command computes a row-reduced form over such domains.

• 

A column-reduced form is one in which the column leading coefficient matrix has the same column rank as the rank of the matrix of polynomials. A row reduced form has the same properties but with respect to the leading row.

• 

The column-reduced form is obtained by elementary column operations, which include interchanging columns, multiplying a column by a unit, or subtracting a polynomial multiple of one column from another. The row-reduced form uses similar row operations. The method used is a fraction-free algorithm by Beckermann and Labahn.

• 

The optional third argument returns a unimodular matrix of elementary operations having the property that P=A.U in the column-reduced case and P=U.A in the row-reduced case.

Examples

withMatrixPolynomialAlgebra:

Az3z2,z32z2+2z2|z32z21,z33z2+3z4

A:=z3z2z32z21z32z2+2z2z33z2+3z4

(1)

PColumnReducedFormA,z

P:=z1+3z14z

(2)

dDegree[column]P,z

d:=1,1

(3)

CCoeff[column]P,z,d

C:=1301

(4)

PColumnReducedFormA,z,'U'

P:=z1+3z14z

(5)

mapexpand,PA.U

0000

(6)

LinearAlgebra[Determinant]C

1

(7)

LinearAlgebra[Determinant]U

12

(8)

PRowReducedFormA,z

P:=z23z2z+112

(9)

dDegree[row]P,z

d:=2,0

(10)

CCoeff[row]P,z,d

C:=1312

(11)

PRowReducedFormA,z,'U'

P:=z23z2z+112

(12)

mapexpand,PU.A

0000

(13)

LinearAlgebra[Determinant]C

1

(14)

LinearAlgebra[Determinant]U

12

(15)

References

  

Beckermann, B. and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications. Vol. 22 No. 1, (2000): 114-144.

See Also

expand

indets

LinearAlgebra[Determinant]

map

Matrix

MatrixPolynomialAlgebra

MatrixPolynomialAlgebra[Coeff]

MatrixPolynomialAlgebra[Degree]

MatrixPolynomialAlgebra[HermiteForm]

MatrixPolynomialAlgebra[PopovForm]

 


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