Dense matrices over multivariate polynomials over fields¶

This implementation inherits from Matrix_generic_dense, i.e. it is not optimized for speed only some methods were added.

AUTHOR:

Martin Albrecht <malb@informatik.uni-bremen.de>

class sage.matrix.matrix_mpolynomial_dense.Matrix_mpolynomial_dense[source]¶

Bases: Matrix_generic_dense

Dense matrix over a multivariate polynomial ring over a field.

determinant(algorithm=None)[source]¶

Return the determinant of this matrix.

INPUT:

algorithm – ignored

EXAMPLES:

We compute the determinant of the arbitrary \(3x3\) matrix:

Sage

sage: R = PolynomialRing(QQ, 9, 'x')
sage: A = matrix(R, 3, R.gens())
sage: A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: A.determinant()
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(9), 'x')
>>> A = matrix(R, Integer(3), R.gens())
>>> A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
>>> A.determinant()
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

Sage Live

R = PolynomialRing(QQ, 9, 'x')
A = matrix(R, 3, R.gens())
A
A.determinant()

We check if two implementations agree on the result:

Sage

sage: R.<x,y> = QQ[]
sage: C = matrix(R, [[-6/5*x*y - y^2, -6*y^2 - 1/4*y],
....:                [  -1/3*x*y - 3, x*y - x]])
sage: C.determinant()
-6/5*x^2*y^2 - 3*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 - 18*y^2 - 3/4*y

sage: C.change_ring(R.change_ring(QQbar)).det()
(-6/5)*x^2*y^2 + (-3)*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 + (-18)*y^2 + (-3/4)*y

Python

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> C = matrix(R, [[-Integer(6)/Integer(5)*x*y - y**Integer(2), -Integer(6)*y**Integer(2) - Integer(1)/Integer(4)*y],
...                [  -Integer(1)/Integer(3)*x*y - Integer(3), x*y - x]])
>>> C.determinant()
-6/5*x^2*y^2 - 3*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 - 18*y^2 - 3/4*y

>>> C.change_ring(R.change_ring(QQbar)).det()
(-6/5)*x^2*y^2 + (-3)*x*y^3 + 6/5*x^2*y + 11/12*x*y^2 + (-18)*y^2 + (-3/4)*y

Sage Live

R.<x,y> = QQ[]
C = matrix(R, [[-6/5*x*y - y^2, -6*y^2 - 1/4*y],
               [  -1/3*x*y - 3, x*y - x]])
C.determinant()
C.change_ring(R.change_ring(QQbar)).det()

Finally, we check whether the Singular interface is working:

Sage

sage: R.<x,y> = RR[]
sage: C = matrix(R, [[0.368965517352886*y^2 + 0.425700773972636*x, -0.800362171389760*y^2 - 0.807635502485287],
....:                [0.706173539423122*y^2 - 0.915986060298440, 0.897165181570476*y + 0.107903328188376]])
sage: C.determinant()
0.565194587390682*y^4 + 0.33102301536914...*y^3 + 0.381923912175852*x*y - 0.122977163520282*y^2 + 0.0459345303240150*x - 0.739782862078649

Python

>>> from sage.all import *
>>> R = RR['x, y']; (x, y,) = R._first_ngens(2)
>>> C = matrix(R, [[RealNumber('0.368965517352886')*y**Integer(2) + RealNumber('0.425700773972636')*x, -RealNumber('0.800362171389760')*y**Integer(2) - RealNumber('0.807635502485287')],
...                [RealNumber('0.706173539423122')*y**Integer(2) - RealNumber('0.915986060298440'), RealNumber('0.897165181570476')*y + RealNumber('0.107903328188376')]])
>>> C.determinant()
0.565194587390682*y^4 + 0.33102301536914...*y^3 + 0.381923912175852*x*y - 0.122977163520282*y^2 + 0.0459345303240150*x - 0.739782862078649

Sage Live

R.<x,y> = RR[]
C = matrix(R, [[0.368965517352886*y^2 + 0.425700773972636*x, -0.800362171389760*y^2 - 0.807635502485287],
               [0.706173539423122*y^2 - 0.915986060298440, 0.897165181570476*y + 0.107903328188376]])
C.determinant()

ALGORITHM: Calls Singular, libSingular or native implementation.

echelon_form(algorithm='row_reduction', **kwds)[source]¶

Return an echelon form of self using chosen algorithm.

By default only a usual row reduction with no divisions or column swaps is returned.

If Gauss-Bareiss algorithm is chosen, column swaps are recorded and can be retrieved via swapped_columns().

INPUT:

algorithm – string, which algorithm to use (default: ‘row_reduction’). Valid options are:
- 'row_reduction' (default) – reduce as far as possible, only divide by constant entries
- 'frac' – reduced echelon form over fraction field
- 'bareiss' – fraction free Gauss-Bareiss algorithm with column swaps

OUTPUT:

The row echelon form of A depending on the chosen algorithm, as an immutable matrix. Note that self is not changed by this command. Use A.echelonize() to change \(A\) in place.

EXAMPLES:

Sage

sage: P.<x,y> = PolynomialRing(GF(127), 2)
sage: A = matrix(P, 2, 2, [1, x, 1, y])
sage: A
[1 x]
[1 y]
sage: A.echelon_form()
[     1      x]
[     0 -x + y]

Python

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(127)), Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> A = matrix(P, Integer(2), Integer(2), [Integer(1), x, Integer(1), y])
>>> A
[1 x]
[1 y]
>>> A.echelon_form()
[     1      x]
[     0 -x + y]

Sage Live

P.<x,y> = PolynomialRing(GF(127), 2)
A = matrix(P, 2, 2, [1, x, 1, y])
A
A.echelon_form()

The reduced row echelon form over the fraction field is as follows:

Sage

sage: A.echelon_form('frac') # over fraction field
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> A.echelon_form('frac') # over fraction field
[1 0]
[0 1]

Sage Live

A.echelon_form('frac') # over fraction field

Alternatively, the Gauss-Bareiss algorithm may be chosen:

Sage

sage: E = A.echelon_form('bareiss'); E
[    1     y]
[    0 x - y]

Python

>>> from sage.all import *
>>> E = A.echelon_form('bareiss'); E
[    1     y]
[    0 x - y]

Sage Live

E = A.echelon_form('bareiss'); E

After the application of the Gauss-Bareiss algorithm the swapped columns may inspected:

Sage

sage: E.swapped_columns(), E.pivots()
((0, 1), (0, 1))
sage: A.swapped_columns(), A.pivots()
(None, (0, 1))

Python

>>> from sage.all import *
>>> E.swapped_columns(), E.pivots()
((0, 1), (0, 1))
>>> A.swapped_columns(), A.pivots()
(None, (0, 1))

Sage Live

E.swapped_columns(), E.pivots()
A.swapped_columns(), A.pivots()

Another approach is to row reduce as far as possible:

Sage

sage: A.echelon_form('row_reduction')
[     1      x]
[     0 -x + y]

Python

>>> from sage.all import *
>>> A.echelon_form('row_reduction')
[     1      x]
[     0 -x + y]

Sage Live

A.echelon_form('row_reduction')

echelonize(algorithm='row_reduction', **kwds)[source]¶

Transform self into a matrix in echelon form over the same base ring as self.

If Gauss-Bareiss algorithm is chosen, column swaps are recorded and can be retrieved via swapped_columns().

INPUT:

algorithm – string, which algorithm to use. Valid options are:
- 'row_reduction' – reduce as far as possible, only divide by constant entries
- 'bareiss' – fraction free Gauss-Bareiss algorithm with column swaps

EXAMPLES:

Sage

sage: P.<x,y> = PolynomialRing(QQ, 2)
sage: A = matrix(P, 2, 2, [1/2, x, 1, 3/4*y+1])
sage: A
[      1/2         x]
[        1 3/4*y + 1]

sage: B = copy(A)
sage: B.echelonize('bareiss'); B
[              1       3/4*y + 1]
[              0 x - 3/8*y - 1/2]

sage: B = copy(A)
sage: B.echelonize('row_reduction'); B
[               1              2*x]
[               0 -2*x + 3/4*y + 1]

sage: P.<x,y> = PolynomialRing(QQ, 2)
sage: A = matrix(P,2,3,[2,x,0,3,y,1]); A
[2 x 0]
[3 y 1]

sage: E = A.echelon_form('bareiss'); E
[1 3 y]
[0 2 x]
sage: E.swapped_columns()
(2, 0, 1)
sage: A.pivots()
(0, 1, 2)

Python

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> A = matrix(P, Integer(2), Integer(2), [Integer(1)/Integer(2), x, Integer(1), Integer(3)/Integer(4)*y+Integer(1)])
>>> A
[      1/2         x]
[        1 3/4*y + 1]

>>> B = copy(A)
>>> B.echelonize('bareiss'); B
[              1       3/4*y + 1]
[              0 x - 3/8*y - 1/2]

>>> B = copy(A)
>>> B.echelonize('row_reduction'); B
[               1              2*x]
[               0 -2*x + 3/4*y + 1]

>>> P = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> A = matrix(P,Integer(2),Integer(3),[Integer(2),x,Integer(0),Integer(3),y,Integer(1)]); A
[2 x 0]
[3 y 1]

>>> E = A.echelon_form('bareiss'); E
[1 3 y]
[0 2 x]
>>> E.swapped_columns()
(2, 0, 1)
>>> A.pivots()
(0, 1, 2)

Sage Live

P.<x,y> = PolynomialRing(QQ, 2)
A = matrix(P, 2, 2, [1/2, x, 1, 3/4*y+1])
A
B = copy(A)
B.echelonize('bareiss'); B
B = copy(A)
B.echelonize('row_reduction'); B
P.<x,y> = PolynomialRing(QQ, 2)
A = matrix(P,2,3,[2,x,0,3,y,1]); A
E = A.echelon_form('bareiss'); E
E.swapped_columns()
A.pivots()

pivots()[source]¶

Return the pivot column positions of this matrix as a list of integers.

This returns a list, of the position of the first nonzero entry in each row of the echelon form.

OUTPUT: list of Python ints

EXAMPLES:

Sage

sage: matrix([PolynomialRing(GF(2), 2, 'x').gen()]).pivots()
(0,)
sage: K = QQ['x,y']
sage: x, y = K.gens()
sage: m = matrix(K, [(-x, 1, y, x - y), (-x*y, y, y^2 - 1, x*y - y^2 + x), (-x*y + x, y - 1, y^2 - y - 2, x*y - y^2 + x + y)])
sage: m.pivots()
(0, 2)
sage: m.rank()
2

Python

>>> from sage.all import *
>>> matrix([PolynomialRing(GF(Integer(2)), Integer(2), 'x').gen()]).pivots()
(0,)
>>> K = QQ['x,y']
>>> x, y = K.gens()
>>> m = matrix(K, [(-x, Integer(1), y, x - y), (-x*y, y, y**Integer(2) - Integer(1), x*y - y**Integer(2) + x), (-x*y + x, y - Integer(1), y**Integer(2) - y - Integer(2), x*y - y**Integer(2) + x + y)])
>>> m.pivots()
(0, 2)
>>> m.rank()
2

Sage Live

matrix([PolynomialRing(GF(2), 2, 'x').gen()]).pivots()
K = QQ['x,y']
x, y = K.gens()
m = matrix(K, [(-x, 1, y, x - y), (-x*y, y, y^2 - 1, x*y - y^2 + x), (-x*y + x, y - 1, y^2 - y - 2, x*y - y^2 + x + y)])
m.pivots()
m.rank()

swapped_columns()[source]¶

Return which columns were swapped during the Gauss-Bareiss reduction.

OUTPUT:

Return a tuple representing the column swaps during the last application of the Gauss-Bareiss algorithm (see echelon_form() for details).

The tuple as length equal to the rank of self and the value at the \(i\)-th position indicates the source column which was put as the \(i\)-th column.

If no Gauss-Bareiss reduction was performed yet, None is returned.

EXAMPLES:

Sage

sage: R.<x,y> = QQ[]
sage: C = random_matrix(R, 2, 2, terms=2)
sage: while C.rank() != 2:
....:     C = random_matrix(R, 2, 2, terms=2)
sage: C.swapped_columns()
sage: E = C.echelon_form('bareiss')
sage: sorted(E.swapped_columns())
[0, 1]

Python

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> C = random_matrix(R, Integer(2), Integer(2), terms=Integer(2))
>>> while C.rank() != Integer(2):
...     C = random_matrix(R, Integer(2), Integer(2), terms=Integer(2))
>>> C.swapped_columns()
>>> E = C.echelon_form('bareiss')
>>> sorted(E.swapped_columns())
[0, 1]

Sage Live

R.<x,y> = QQ[]
C = random_matrix(R, 2, 2, terms=2)
while C.rank() != 2:
    C = random_matrix(R, 2, 2, terms=2)
C.swapped_columns()
E = C.echelon_form('bareiss')
sorted(E.swapped_columns())