Dense matrices over univariate polynomials over fields#

The implementation inherits from Matrix_generic_dense but some algorithms are optimized for polynomial matrices.

AUTHORS:

Kwankyu Lee (2016-12-15): initial version with code moved from other files.
Johan Rosenkilde (2017-02-07): added weak_popov_form()
Vincent Neiger (2018-06-13): added basic functions (row/column degrees, leading positions, leading matrix, testing reduced and canonical forms)
Vincent Neiger (2018-09-29): added functions for computing and for verifying minimal approximant bases
Vincent Neiger (2020-04-01): added functions for computing and for verifying minimal kernel bases
Vincent Neiger (2021-03-11): added matrix-wise basic functions for univariate polynomials (shifts, reverse, truncate, get coefficient of specified degree)
Vincent Neiger (2021-07-29): added popov_form(). Added more options to weak_popov_form() (column-wise, ordered, zero rows).
Vincent Neiger (2021-08-07): added inverse_series_trunc(), solve_{left/right}_series_trunc(), {left/right}_quo_rem(), reduce().
Vincent Neiger (2024-02-13): added basis_completion(), _is_basis_completion(), _basis_completion_via_reversed_approx().

class sage.matrix.matrix_polynomial_dense.Matrix_polynomial_dense#

Bases: Matrix_generic_dense

Dense matrix over a univariate polynomial ring over a field.

For a field \(\Bold{K}\), we consider matrices over the univariate polynomial ring \(\Bold{K}[x]\).

They are often used to represent bases of some \(\Bold{K}[x]\)-modules. In this context, there are two possible representations which are both commonly used in the literature.

Working column-wise: each column of the matrix is a vector in the basis; then, a \(\Bold{K}[x]\)-submodule of \(\Bold{K}[x]^{m}\) of rank \(n\) is represented by an \(m \times n\) matrix, whose columns span the module (via \(\Bold{K}[x]\)-linear combinations). This matrix has full rank, and \(n \leq m\).
Working row-wise: each row of the matrix is a vector in the basis; then, a \(\Bold{K}[x]\)-submodule of \(\Bold{K}[x]^{n}\) of rank \(m\) is represented by an \(m \times n\) matrix, whose rows span the module (via \(\Bold{K}[x]\)-linear combinations). This matrix has full rank, and \(m \leq n\).

For the rest of this class description, we assume that one is working row-wise. For a given such module, all its bases are equivalent under left-multiplication by a unimodular matrix, that is, a square matrix which has determinant in \(\Bold{K}\setminus\{0\}\).

There are bases which are called reduced or minimal: their rows have the minimal degree possible among all bases of this module; here the degree of a row is the maximum of the degrees of the entries of the row. An equivalent condition is that the leading matrix of this basis has full rank (see leading_matrix(), reduced_form(), is_reduced()). There is a unique minimal basis, called the Popov basis of the module, which satisfies some additional normalization condition (see popov_form(), is_popov()).

These notions can be extended via a more general degree measure, involving a tuple of integers which is called shift and acts as column degree shifts in the definition of row degree. Precisely, for given \(s_1,\ldots,s_n \in \ZZ\) and a row vector \([p_1 \; \cdots \; p_n] \in \Bold{K}[x]^{1 \times n}\), its shifted row degree is the maximum of \(\deg(p_j) + s_j\) for \(1 \leq j \leq n\) (see row_degrees()). Then, reduced bases and Popov bases are defined similarly, with respect to this notion of degree.

Another important canonical basis is the Hermite basis, which is an upper triangular matrix satisfying a normalization condition similar to that for the Popov basis. In fact, if \(d\) is the largest degree appearing in the Hermite basis, then the Hermite basis coincide with the shifted Popov basis with the shifts \(((n-1)d,\ldots,2d,d,0)\).

basis_completion(row_wise=True, algorithm='approximant')#

Return a Smith form-preserving nonsingular completion of a basis of this matrix: row-wise completing a row basis if row_wise is True; column-wise completing a column basis if it is False.

For a more detailed description, consider the row-wise case (the column-wise case is the same up to matrix transposition). Let \(A\) be the input matrix, \(m \times n\) over univariate polynomials \(\Bold{K}[x]\), for some field \(\Bold{K}\), and let \(r\) be the rank of \(A\), which is unknown a priori. This computes a matrix \(C\) of dimensions \((n-r) \times n\) such that stacking both matrices one above the other, say \([[A],[C]]\), gives a matrix of maximal rank \(n\) and with the same nontrivial Smith factors as \(A\). In particular, \(C\) has full row rank, and the rank of the input matrix may be recovered from the number of rows of \(C\).

As a consequence, if \(B\) is a basis of the module generated by the rows of \(A\) (for example \(B = A\) if \(A\) has full row rank), then \([[B],[C]]\) is nonsingular, and its determinant is the product of the nonzero Smith factors of \(A\) up to multiplication by a nonzero element of \(\Bold{K}\).

In particular, for \(A\) with full row rank: if the rows \(A\) can be completed into a basis of \(\Bold{K}[x]^{n}\) (or equivalently, \(A\) has unimodular column bases, or also, if the rows of \(A\) generate all polynomial vectors in the rational row space of \(A\)), then \(C\) provides such a completion. In this case, \([[A],[C]]\) is unimodular: it is invertible over \(\Bold{K}[x]\), and \(det([[A],[C]])\) is a nonzero element of the base field \(\Bold{K}\).

INPUT:

row_wise – (optional, default: True) boolean, if True then compute a row-wise completion, else compute a column-wise completion.
algorithm – (optional, default: "approximant") selects the approach for computing the completion; currently supported: "approximant" and "smith".

OUTPUT: a matrix over the same base ring as the input matrix, which forms a completion as defined above.

ALGORITHM:

approximant: the approximant-based algorithm follows the ideas in [ZL2014] , based on polynomial reversals combined with the computation of a minimal kernel basis and a minimal approximant basis.
smith: the Smith form-based algorithm computes the Smith form of this matrix along with corresponding unimodular transformations, from which a completion is readily obtained.

EXAMPLES:

Three polynomials whose GCD is \(1\) can be completed into a unimodular matrix:

sage: ring.<x> = GF(7)[]
sage: mat = matrix([[x*(x-1)*(x-2), (x-2)*(x-3)*(x-4), (x-4)*(x-5)*(x-6)]])
sage: mat
[      x^3 + 4*x^2 + 2*x   x^3 + 5*x^2 + 5*x + 4   x^3 + 6*x^2 + 4*x + 6]
sage: rcomp = mat.basis_completion(); rcomp
[        5*x^2 + 4*x + 1             5*x^2 + 2*x                   5*x^2]
[          2*x^3 + 4*x^2 2*x^3 + 6*x^2 + 2*x + 1       2*x^3 + x^2 + 3*x]
sage: basis = mat.stack(rcomp); basis
[      x^3 + 4*x^2 + 2*x   x^3 + 5*x^2 + 5*x + 4   x^3 + 6*x^2 + 4*x + 6]
[        5*x^2 + 4*x + 1             5*x^2 + 2*x                   5*x^2]
[          2*x^3 + 4*x^2 2*x^3 + 6*x^2 + 2*x + 1       2*x^3 + x^2 + 3*x]
sage: basis.determinant()
6

The following matrix has rank \(2\) and trivial Smith form. It can be completed row-wise into a \(3 \times 3\) unimodular matrix (column-wise, there is nothing to complete):

sage: mat = matrix(ring, 2, 3, \
        [[x^2 + 5*x + 5,   3*x^2 + x + 3, 4*x^2 + 5*x + 4], \
         [5*x^2 + 4*x,   3*x^2 + 4*x + 5, 5*x^2 + 5*x + 3]])
sage: rcomp = mat.basis_completion(); rcomp
[  2*x^2 + 1 4*x^2 + 3*x 2*x^2 + 3*x]
sage: mat.stack(rcomp).determinant()
3
sage: mat.basis_completion(row_wise=False)
[]

The following matrix has rank 1 and its nonzero Smith factor is \(x+3\). A row-wise completion has a single nonzero row, whereas a column-wise completion has two columns; in both cases, the Smith form is preserved:

sage: mat = matrix(ring, 3, 2, \
        [[    x^3 + x^2 + 5*x + 5,         2*x^3 + 2*x + 4], \
         [  3*x^3 + 2*x^2 + x + 3,   6*x^3 + 5*x^2 + x + 1], \
         [2*x^3 + 5*x^2 + 3*x + 4, 4*x^3 + 6*x^2 + 5*x + 6]])
sage: mat.smith_form(transformation=False)
[x + 3     0]
[    0     0]
[    0     0]
sage: rcomp = mat.basis_completion(); rcomp
[x + 1   2*x]
sage: ccomp = mat.basis_completion(row_wise=False); ccomp
[3*x + 1 4*x + 4]
[    2*x 5*x + 1]
[    6*x       x]
sage: rcomp.stack(mat).smith_form(transformation=False)
[    1     0]
[    0 x + 3]
[    0     0]
[    0     0]
sage: ccomp.augment(mat).smith_form(transformation=False)
[    1     0     0     0]
[    0     1     0     0]
[    0     0 x + 3     0]

Here are a few more examples, similar to the above but over fields other than GF(7):

sage: ring.<x> = QQ[]
sage: mat = matrix([[x*(x-1)*(x-2), (x-2)*(x-3)*(x-4), (x-4)*(x-5)*(x-6)]])
sage: mat
[        x^3 - 3*x^2 + 2*x   x^3 - 9*x^2 + 26*x - 24 x^3 - 15*x^2 + 74*x - 120]
sage: rcomp = mat.basis_completion(algorithm="smith"); rcomp
[        -1/12*x - 1/12         -1/12*x + 5/12                      0]
[                  1/12                   1/12 1/24*x^2 - 13/24*x + 2]
sage: mat.stack(rcomp).determinant()
1

sage: mat = matrix([[x*(x-1), x*(x-2)], \
                    [x*(x-2), x*(x-3)], \
                    [(x-1)*(x-2), (x-1)*(x-3)]])
sage: mat.smith_form(transformation=False)
[1 0]
[0 x]
[0 0]
sage: ccomp = mat.basis_completion(row_wise=False, algorithm="smith")
sage: ccomp
[1/2*x - 1/2]
[  1/2*x - 1]
[1/2*x - 3/2]
sage: ccomp.augment(mat).smith_form(transformation=False)
[    1     0     0]
[    0     1     0]
[    0     0 1/2*x]

sage: field.<a> = NumberField(x**2 - 2)
sage: ring.<y> = field[]
sage: mat = matrix([[3*a*y - 1, (-8*a - 1)*y - 2*a + 1]])
sage: rcomp = mat.basis_completion(algorithm="smith"); rcomp
[ 39/119*a - 30/119 -99/119*a + 67/119]
sage: mat.stack(rcomp).determinant()
1

coefficient_matrix(d, row_wise=True)#

Return the constant matrix which is obtained from this matrix by taking the coefficient of its entries with degree specified by \(d\).

if \(d\) is an integer, this selects the coefficient of \(d\) for all entries;
if \(d\) is a list \((d_1,\ldots,d_m)\) and row_wise is True, this selects the coefficient of degree \(d_i\) for all entries of the \(i`th row for each `i\);
if \(d\) is a list \((d_1,\ldots,d_n)\) and row_wise is False, this selects the coefficient of degree \(d_i\) for all entries of the \(j`th column for each `j\).

INPUT:

d – a list of integers, or an integer,
row_wise – (optional, default: True) boolean, if True (resp. False) then \(d\) should be a list of length equal to the row (resp. column) dimension of this matrix.

OUTPUT: a matrix over the base field.

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.coefficient_matrix(2)
[5 0 0 0]
[6 0 0 0]
[4 0 2 1]
sage: M.coefficient_matrix(0) == M.constant_matrix()
True

Row-wise and column-wise coefficient extraction are available:

sage: M.coefficient_matrix([3,2,1])
[1 0 0 0]
[6 0 0 0]
[6 5 5 5]

sage: M.coefficient_matrix([2,0,1,3], row_wise=False)
[5 5 6 0]
[6 1 0 0]
[4 1 5 0]

Negative degrees give zero coefficients:

sage: M.coefficient_matrix([-1,0,1,3], row_wise=False)
[0 5 6 0]
[0 1 0 0]
[0 1 5 0]

Length of list of degrees is checked:

sage: M.coefficient_matrix([2,1,1,2])
Traceback (most recent call last):
...
ValueError: length of input degree list should be the row
dimension of the input matrix

sage: M.coefficient_matrix([3,2,1], row_wise=False)
Traceback (most recent call last):
...
ValueError: length of input degree list should be the column
dimension of the input matrix

column_degrees(shifts=None)#

Return the (shifted) column degrees of this matrix.

For a given polynomial matrix \(M = (M_{i,j})_{i,j}\) with \(m\) rows and \(n\) columns, its column degrees is the tuple \((d_1,\ldots,d_n)\) where \(d_j = \max_i(\deg(M_{i,j}))\) for \(1\leq j \leq n\). Thus, \(d_j=-1\) if the \(j\)-th column of \(M\) is zero, and \(d_j \geq 0\) otherwise.

For given shifts \(s_1,\ldots,s_m \in \ZZ\), the shifted column degrees of \(M\) is \((d_1,\ldots,d_n)\) where \(d_j = \max_i(\deg(M_{i,j})+s_i)\). Here, if the \(j\)-th column of \(M\) is zero then \(d_j = \min(s_1,\ldots,s_m)-1\); otherwise \(d_j\) is larger than this value.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].

OUTPUT: a list of integers.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.column_degrees()
[3, -1, 0]

sage: M.column_degrees(shifts=[0,2])
[5, -1, 0]

A zero column in a polynomial matrix can be identified in the (shifted) column degrees as the entries equal to min(shifts)-1:

sage: M.column_degrees(shifts=[-2,1])
[4, -3, -2]

The column degrees of an empty matrix (\(0\times n\) or \(m\times 0\)) is not defined:

sage: M = Matrix(pR, 0, 3)
sage: M.column_degrees()
Traceback (most recent call last):
...
ValueError: empty matrix does not have column degrees

sage: M = Matrix(pR, 3, 0)
sage: M.column_degrees()
Traceback (most recent call last):
...
ValueError: empty matrix does not have column degrees

See also

The documentation of row_degrees().

constant_matrix()#

Return the constant coefficient of this matrix seen as a polynomial with matrix coefficients; this is also this matrix evaluated at zero.

OUTPUT: a matrix over the base field.

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.constant_matrix()
[1 5 4 0]
[1 1 2 0]
[4 1 5 6]

degree()#

Return the degree of this matrix.

For a given polynomial matrix, its degree is the maximum of the degrees of all its entries. If the matrix is nonzero, this is a nonnegative integer; here, the degree of the zero matrix is -1.

OUTPUT: an integer.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.degree()
3

The zero matrix has degree -1:

sage: M = Matrix(pR, 2, 3)
sage: M.degree()
-1

For an empty matrix, the degree is not defined:

sage: M = Matrix(pR, 3, 0)
sage: M.degree()
Traceback (most recent call last):
...
ValueError: empty matrix does not have a degree

degree_matrix(shifts=None, row_wise=True)#

Return the matrix of the (shifted) degrees in this matrix.

For a given polynomial matrix \(M = (M_{i,j})_{i,j}\), its degree matrix is the matrix \((\deg(M_{i,j}))_{i,j}\) formed by the degrees of its entries. Here, the degree of the zero polynomial is \(-1\).

For given shifts \(s_1,\ldots,s_m \in \ZZ\), the shifted degree matrix of \(M\) is either \((\deg(M_{i,j})+s_j)_{i,j}\) if working row-wise, or \((\deg(M_{i,j})+s_i)_{i,j}\) if working column-wise. In the former case, \(m\) has to be the number of columns of \(M\); in the latter case, the number of its rows. Here, if \(M_{i,j}=0\) then the corresponding entry in the shifted degree matrix is \(\min(s_1,\ldots,s_m)-1\). For more on shifts and working row-wise versus column-wise, see the class documentation.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, if True then shifts apply to the columns of the matrix and otherwise to its rows (see the class description for more details).

OUTPUT: an integer matrix.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.degree_matrix()
[ 1 -1  0]
[ 3 -1 -1]

sage: M.degree_matrix(shifts=[0,1,2])
[ 1 -1  2]
[ 3 -1 -1]

The zero entries in the polynomial matrix can be identified in the (shifted) degree matrix as the entries equal to min(shifts)-1:

sage: M.degree_matrix(shifts=[-2,1,2])
[-1 -3  2]
[ 1 -3 -3]

Using row_wise=False, the function supports shifts applied to the rows of the matrix (which, in terms of modules, means that we are working column-wise, see the class documentation):

sage: M.degree_matrix(shifts=[-1,2], row_wise=False)
[ 0 -2 -1]
[ 5 -2 -2]

hermite_form(include_zero_rows=True, transformation=False)#

Return the Hermite form of this matrix.

See is_hermite() for a definition of Hermite forms. If the input is a matrix \(A\), then its Hermite form is the unique matrix \(H\) in Hermite form such that \(UA = H\) for some unimodular matrix \(U\).

INPUT:

include_zero_rows – boolean (default: True); if False, the zero rows in the output matrix are deleted.
transformation – boolean (default: False); if True, return the transformation matrix.

OUTPUT:

the Hermite normal form \(H\) of this matrix \(A\) .
(optional) transformation matrix \(U\) such that \(UA = H\) .

EXAMPLES:

sage: M.<x> = GF(7)[]
sage: A = matrix(M, 2, 3, [x, 1, 2*x, x, 1+x, 2])
sage: A.hermite_form()
[      x       1     2*x]
[      0       x 5*x + 2]
sage: A.hermite_form(transformation=True)
(
[      x       1     2*x]  [1 0]
[      0       x 5*x + 2], [6 1]
)
sage: A = matrix(M, 2, 3, [x, 1, 2*x, 2*x, 2, 4*x])
sage: A.hermite_form(transformation=True, include_zero_rows=False)
([  x   1 2*x], [0 4])
sage: H, U = A.hermite_form(transformation=True,
....:                       include_zero_rows=True); H, U
(
[  x   1 2*x]  [0 4]
[  0   0   0], [5 1]
)
sage: U * A == H
True
sage: H, U = A.hermite_form(transformation=True,
....:                       include_zero_rows=False)
sage: U * A
[  x   1 2*x]
sage: U * A == H
True

See also

is_hermite() , popov_form() .

inverse_series_trunc(d)#

Return a matrix polynomial approximation of precision d of the inverse series of this matrix polynomial.

Here matrix polynomial means that self is seen as a univariate polynomial with matrix coefficients, meaning that this method has the same output as if one: 1) converts this matrix to a univariate polynomial with matrix coefficients, 2) calls

sage.rings.polynomial.polynomial_element.Polynomial.inverse_series_trunc()

on that univariate polynomial, and 3) converts back to a matrix of polynomials.

Raises a ZeroDivisionError if the constant matrix of self is not invertible (i.e. has zero determinant); raises an ArithmeticError if self is nonsquare; and raises a ValueError if the precision d is not positive.

INPUT: a positive integer \(d\) .

OUTPUT: the unique polynomial matrix \(B\) of degree less than \(d\) such that \(AB\) and \(BA\) are the identity matrix modulo \(x^d\), where \(A\) is self.

ALGORITHM: This uses Newton iteration, performing about \(\log(d)\) polynomial matrix multiplications in size \(m \times m\) and in degree less than \(2d\), where \(m\) is the row dimension of self.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: A = Matrix(pR, 3, 3,
....:            [[4*x+5,           5*x^2 + x + 1, 4*x^2 + 4],
....:             [6*x^2 + 6*x + 6, 4*x^2 + 5*x,   4*x^2 + x + 3],
....:             [3*x^2 + 2,       4*x + 1,       x^2 + 3*x]])
sage: B = A.inverse_series_trunc(4); B
[    x^3 + 5*x^2 + x + 4   x^3 + 5*x^2 + 6*x + 4         6*x^2 + 5*x + 3]
[        4*x^2 + 5*x + 6     6*x^3 + x^2 + x + 6       3*x^3 + 2*x^2 + 2]
[5*x^3 + 5*x^2 + 6*x + 6 4*x^3 + 2*x^2 + 6*x + 4   6*x^3 + x^2 + 6*x + 1]
sage: (B*A).truncate(4) == 1
True

sage: A.inverse_series_trunc(0)
Traceback (most recent call last):
...
ValueError: the precision must be positive

sage: A[:2,:].inverse_series_trunc(4)
Traceback (most recent call last):
...
ArithmeticError: the input matrix must be square

sage: A[0,:] = A[0,:] - A[0,:](0) + A[1,:](0) + A[2,:](0)
sage: A.inverse_series_trunc(4)
Traceback (most recent call last):
...
ZeroDivisionError: the constant matrix term self(0) must be invertible

Todo

in the current state of polynomial matrix multiplication (July 2021), it would be highly beneficial to use conversions and rely on polynomials with matrix coefficients when the matrix size is “large” and the degree “small”, see github issue #31472#comment:5.

is_constant()#

Return whether this polynomial matrix is constant, that is, all its entries are constant.

OUTPUT: a boolean.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.is_constant()
False
sage: M = Matrix(pR, [[1,5,2], [3,1,5]]); M.is_constant()
True
sage: M = Matrix.zero(pR, 3, 5); M.is_constant()
True

is_hermite(row_wise=True, lower_echelon=False, include_zero_vectors=True)#

Return a boolean indicating whether this matrix is in Hermite form.

If working row-wise, a polynomial matrix is said to be in Hermite form if it is in row echelon form with all pivot entries monic and such that all entries above a pivot have degree less than this pivot. Being in row echelon form means that all zero rows are gathered at the bottom of the matrix, and in each nonzero row the pivot (leftmost nonzero entry) is strictly to the right of the pivot of the row just above this row.

Note that, for any integer \(d\) strictly greater than all degrees appearing in the Hermite form, then the Hermite form coincides with the shifted Popov form with the shifts \(((n-1)d,\ldots,2d,d,0)\), where \(n\) is the column dimension.

If working column-wise, a polynomial matrix is said to be in Hermite form if it is in column echelon form with all pivot entries monic and such that all entries to the left of a pivot have degree less than this pivot. Being in column echelon form means that all zero columns are gathered at the right-hand side of the matrix, and in each nonzero column the pivot (topmost nonzero entry) is strictly below the pivot of the column just to the left of this row.

Optional arguments provide support of alternative definitions, concerning the choice of upper or lower echelon forms and concerning whether zero rows (resp. columns) are allowed.

INPUT:

row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
lower_echelon – (optional, default: False) boolean, False if working with upper triangular Hermite forms, True if working with lower triangular Hermite forms.
include_zero_vectors – (optional, default: True) boolean, False if one does not allow zero rows (resp. zero columns) in Hermite forms.

OUTPUT: a boolean.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[x^4+6*x^3+4*x+4, 3*x+6,     3  ],
....:                 [0,               x^2+5*x+5, 2  ],
....:                 [0,               0,         x+5]])

sage: M.is_hermite()
True
sage: M.is_hermite(row_wise=False)
True
sage: M.is_hermite(row_wise=False, lower_echelon=True)
False

sage: N = Matrix(pR, [[x+5, 0,               0        ],
....:                 [2,   x^4+6*x^3+4*x+4, 0        ],
....:                 [3,   3*x^3+6,         x^2+5*x+5]])
sage: N.is_hermite()
False
sage: N.is_hermite(lower_echelon=True)
True
sage: N.is_hermite(row_wise=False)
False
sage: N.is_hermite(row_wise=False, lower_echelon=True)
False

Rectangular matrices with zero rows are supported. Zero rows (resp. columns) can be forbidden, and otherwise they should be at the bottom (resp. the right-hand side) of the matrix:

sage: N[:,1:].is_hermite(lower_echelon=True)
False
sage: N[[1,2,0],1:].is_hermite(lower_echelon=True)
True
sage: N[:2,:].is_hermite(row_wise=False, lower_echelon=True)
True
sage: N[:2,:].is_hermite(row_wise=False,
....:                    lower_echelon=True,
....:                    include_zero_vectors=False)
False

See also

hermite_form() .

is_minimal_approximant_basis(pmat, order, shifts=None, row_wise=True, normal_form=False)#

Return whether this matrix is an approximant basis in shifts-ordered weak Popov form for the polynomial matrix pmat at order order.

If normal_form is True, then the polynomial matrix must furthermore be in shifts-Popov form. An error is raised if the input dimensions are not sound. If a single integer is provided for order, then it is interpreted as a list of repeated integers with this value. (See minimal_approximant_basis() for definitions and more details.)

INPUT:

pmat – a polynomial matrix.
order – a list of positive integers, or a positive integer.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, if True then the basis considered row-wise and operates on the left of pmat; otherwise it is column-wise and operates on the right of pmat.
normal_form – (optional, default: False) boolean, if True then checks for a basis in shifts-Popov form.

OUTPUT: a boolean.

ALGORITHM:

Verification that the matrix is formed by approximants is done via a truncated matrix product; verification that the matrix is square, nonsingular and in shifted weak Popov form is done via is_weak_popov(); verification that the matrix generates the module of approximants is done via the characterization in Theorem 2.1 of [GN2018] .

EXAMPLES:

sage: pR.<x> = GF(97)[]

We consider the following example from [Arne Storjohann, Notes on computing minimal approximant bases, 2006]:

sage: order = 8; shifts = [1,1,0,0,0]
sage: pmat = Matrix(pR, 5, 1, [
....:     pR([35,  0, 41, 87,  3, 42, 22, 90]),
....:     pR([80, 15, 62, 87, 14, 93, 24,  0]),
....:     pR([42, 57, 90, 87, 22, 80, 71, 53]),
....:     pR([37, 72, 74,  6,  5, 75, 23, 47]),
....:     pR([36, 10, 74,  1, 29, 44, 87, 74])])
sage: appbas = Matrix(pR, [
....:     [x+47,   57, 58*x+44,     9*x+23,      93*x+76],
....:     [  15, x+18, 52*x+23,     15*x+58,     93*x+88],
....:     [  17,   86, x^2+77*x+16, 76*x+29,     90*x+78],
....:     [  44,   36, 3*x+42,      x^2+50*x+26, 85*x+44],
....:     [   2,   22, 54*x+94,     73*x+24,     x^2+2*x+25]])
sage: appbas.is_minimal_approximant_basis(                                  # needs sage.libs.pari
....:     pmat, order, shifts, row_wise=True, normal_form=True)
True

The matrix \(x^8 \mathrm{Id}_5\) is square, nonsingular, in Popov form, and its rows are approximants for pmat at order 8. However, it is not an approximant basis since its rows generate a module strictly contained in the set of approximants for pmat at order 8:

sage: M = x^8 * Matrix.identity(pR, 5)
sage: M.is_minimal_approximant_basis(pmat, 8)                               # needs sage.libs.pari
False

Since pmat is a single column, with nonzero constant coefficient, its column-wise approximant bases at order 8 are all \(1\times 1\) matrices \([c x^8]\) for some nonzero field element \(c\):

sage: M = Matrix(pR, [x^8])
sage: M.is_minimal_approximant_basis(
....:     pmat, 8, row_wise=False, normal_form=True)
True

Exceptions are raised if input dimensions are not sound:

sage: appbas.is_minimal_approximant_basis(pmat, [8,8], shifts)
Traceback (most recent call last):
...
ValueError: order length should be the column dimension
            of the input matrix

sage: appbas.is_minimal_approximant_basis(
....:     pmat, order, shifts, row_wise=False)
Traceback (most recent call last):
...
ValueError: shifts length should be the column dimension
            of the input matrix

sage: Matrix(pR, [x^8]).is_minimal_approximant_basis(pmat, 8)
Traceback (most recent call last):
...
ValueError: column dimension should be the row dimension of the
input matrix

See also

minimal_approximant_basis() .

is_minimal_kernel_basis(pmat, shifts=None, row_wise=True, normal_form=False)#

Return whether this matrix is a left kernel basis in shifts-ordered weak Popov form for the polynomial matrix pmat.

If normal_form is True, then the kernel basis must furthermore be in shifts-Popov form. An error is raised if the input dimensions are not sound.

INPUT:

pmat – a polynomial matrix.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, if True then the basis is considered row-wise and operates on the left of pmat; otherwise it is column-wise and operates on the right of pmat.
normal_form – (optional, default: False) boolean, if True then checks for a basis in shifts-Popov form.

OUTPUT: a boolean.

ALGORITHM:

Verification that the matrix has full rank and is in shifted weak Popov form is done via is_weak_popov(); verification that the matrix is a left kernel basis is done by checking that the rank is correct, that the product is indeed zero, and that the matrix is saturated, i.e. it has unimodular column bases (see Lemma 6.10 of https://arxiv.org/pdf/1807.01272.pdf for details).

EXAMPLES:

sage: pR.<x> = GF(97)[]
sage: pmat = Matrix(pR, [[1], [x], [x**2]])

sage: kerbas = Matrix(pR, [[x,-1,0], [0,x,-1]])
sage: kerbas.is_minimal_kernel_basis(pmat)
True

A matrix in Popov form which has the right rank, all rows in the kernel, but does not generate the kernel:

sage: kerbas = Matrix(pR, [[x**2,0,-1], [0,x,-1]])
sage: kerbas.is_minimal_kernel_basis(pmat)
False

Shifts and right kernel bases are supported (with row_wise), and one can test whether the kernel basis is normalized in shifted-Popov form (with normal_form):

sage: kerbas = Matrix(pR, [[-x,-x**2], [1,0], [0,1]])
sage: kerbas.is_minimal_kernel_basis(
....:     pmat.transpose(), row_wise=False,
....:     normal_form=True, shifts=[0,1,2])
True

is_popov(shifts=None, row_wise=True, up_to_permutation=False, include_zero_vectors=True)#

Return a boolean indicating whether this matrix is in (shifted) Popov form.

If working row-wise (resp. column-wise), a polynomial matrix is said to be in Popov form if it has no zero row above a nonzero row (resp. no zero column to the left of a nonzero column), the leading positions of its nonzero rows (resp. columns) are strictly increasing, and for each row (resp. column) the pivot entry is monic and has degree strictly larger than the other entries in its column (resp. row).

Since other conventions concerning the ordering of the rows (resp. columns) are sometimes useful, an optional argument allows one to test whether the matrix is in Popov form up to row (resp. column) permutation. For example, there is an alternative definition which replaces “leading positions strictly increasing” by “row (resp. column) degree nondecreasing, and for rows (resp. columns) of same degree, leading positions increasing”.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
up_to_permutation – (option, default: False) boolean, True if testing Popov form up to row permutation (if working row-wise).
include_zero_vectors – (optional, default: True) boolean, False if one does not allow zero rows (resp. zero columns) in Popov forms.

OUTPUT: a boolean.

REFERENCES:

For the square case, without shifts: [Pop1972] and [Kai1980] (Section 6.7.2). For the general case: [BLV2006] .

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[x^4+6*x^3+4*x+4, 3*x+6,     3  ],
....:                 [x^2+6*x+6,       x^2+5*x+5, 2  ],
....:                 [3*x,             6*x+5,     x+5]])
sage: M.is_popov()
True

sage: M.is_popov(shifts=[0,1,2])
True

sage: M[:,:2].is_popov()
False

sage: M[:2,:].is_popov(shifts=[0,1,2])
True

sage: M = Matrix(pR, [[x^4+3*x^3+x^2+2*x+6, x^3+5*x^2+5*x+1],
....:                 [6*x+1,               x^2+4*x+1      ],
....:                 [6,                   6              ]])
sage: M.is_popov(row_wise=False)
False

sage: M.is_popov(shifts=[0,2,3], row_wise=False)
True

One can forbid zero rows (or columns if not working row-wise):

sage: N = Matrix(pR, [[x^4+3*x^3+x^2+2*x+6, 6*x+1     ],
....:                 [5*x^2+5*x+1,         x^2+4*x+1 ],
....:                 [0,                   0         ]])

sage: N.is_popov()
True

sage: N.is_popov(include_zero_vectors=False)
False

One can verify Popov form up to row permutation (or column permutation if not working row-wise):

sage: M.swap_columns(0, 1)
sage: M.is_popov(shifts=[0,2,3], row_wise=False)
False

sage: M.is_popov(shifts=[0,2,3], row_wise=False,
....:            up_to_permutation=True)
True

sage: N.swap_rows(0, 2)

sage: N.is_popov()
False

sage: N.is_popov(up_to_permutation=True)
True

is_reduced(shifts=None, row_wise=True, include_zero_vectors=True)#

Return a boolean indicating whether this matrix is in (shifted) reduced form.

An \(m \times n\) univariate polynomial matrix \(M\) is said to be in shifted row reduced form if it has \(k\) nonzero rows with \(k \leq n\) and its shifted leading matrix has rank \(k\). Equivalently, when considering all the matrices obtained by left-multiplying \(M\) by a unimodular matrix, then the shifted row degrees of \(M\) – once sorted in nondecreasing order – is lexicographically minimal.

Similarly, \(M\) is said to be in shifted column reduced form if it has \(k\) nonzero columns with \(k \leq m\) and its shifted leading matrix has rank \(k\).

Sometimes, one forbids \(M\) to have zero rows (resp. columns) in the above definitions; an optional parameter allows one to adopt this more restrictive setting.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
include_zero_vectors – (optional, default: True) boolean, False if one does not allow zero rows in row reduced forms (resp. zero columns in column reduced forms).

OUTPUT: a boolean value.

REFERENCES:

[Wol1974] (Section 2.5, without shifts) and [VBB1992] (Section 3).

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.is_reduced()
False

sage: M.is_reduced(shifts=[0,1,2])
True

sage: M.is_reduced(shifts=[2,0], row_wise=False)
True

sage: M.is_reduced(shifts=[2,0], row_wise=False,
....:              include_zero_vectors=False)
False

sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0], [0, 1, 0]])
sage: M.is_reduced(shifts=[2,0,0], row_wise=False)
True

See also

leading_matrix() , reduced_form() .

is_weak_popov(shifts=None, row_wise=True, ordered=False, include_zero_vectors=True)#

Return a boolean indicating whether this matrix is in (shifted) (ordered) weak Popov form.

If working row-wise (resp. column-wise), a polynomial matrix is said to be in weak Popov form if the leading positions of its nonzero rows (resp. columns) are pairwise distinct. For the ordered weak Popov form, these positions must be strictly increasing, except for the possibly repeated -1 entries which are at the end. For the shifted variants, see the class description for an introduction to shifts.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
ordered – (optional, default: False) boolean, True if checking for an ordered weak Popov form.
include_zero_vectors – (optional, default: True) boolean, False if one does not allow zero rows (resp. zero columns) in (ordered) weak Popov forms.

OUTPUT: a boolean.

REFERENCES:

[Kai1980] (Section 6.7.2, square case without shifts), [MS2003] (without shifts), [BLV1999] .

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix([ [x^3+3*x^2+6*x+6, 3*x^2+3*x+6, 4*x^2+x+3],
....:              [5,               1,           0        ],
....:              [2*x^2+2,         2*x+5,       x^2+4*x+6] ])
sage: M.is_weak_popov()
True

One can check whether the leading positions, in addition to being pairwise distinct, are actually in increasing order:

sage: M.is_weak_popov(ordered=True)
True

sage: N = M.with_swapped_rows(1, 2)
sage: N.is_weak_popov()
True
sage: N.is_weak_popov(ordered=True)
False

Shifts and orientation (row-wise or column-wise) are supported:

sage: M.is_weak_popov(shifts=[2,3,1])
False

sage: M.is_weak_popov(shifts=[0,2,0], row_wise=False,
....:                 ordered=True)
True

Rectangular matrices are supported:

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.is_weak_popov(shifts=[0,2,1,3])
True

sage: M.is_weak_popov(shifts=[0,2,1,3], ordered=True)
True

Zero rows (resp. columns) can be forbidden:

sage: M = Matrix([
....:   [      6*x+4,       0,             5*x+1, 0],
....:   [          2, 5*x + 1,       6*x^2+3*x+1, 0],
....:   [2*x^2+5*x+5,       1, 2*x^3+4*x^2+6*x+4, 0]
....:   ])
sage: M.is_weak_popov(shifts=[2,1,0], row_wise=False,
....:                 ordered=True)
True

sage: M.is_weak_popov(shifts=[2,1,0], row_wise=False,
....:                 include_zero_vectors=False)
False

See also

weak_popov_form() .

leading_matrix(shifts=None, row_wise=True)#

Return the (shifted) leading matrix of this matrix.

Let \(M\) be a univariate polynomial matrix in \(\Bold{K}[x]^{m \times n}\). Working row-wise and without shifts, its leading matrix is the matrix in \(\Bold{K}^{m \times n}\) formed by the leading coefficients of the entries of \(M\) which reach the degree of the corresponding row.

More precisely, if working row-wise, let \(s_1,\ldots,s_n \in \ZZ\) be a shift, and let \((d_1,\ldots,d_m)\) denote the shifted row degrees of \(M\). Then, the shifted leading matrix of \(M\) is the matrix in \(\Bold{K}^{m \times n}\) whose entry \(i,j\) is the coefficient of degree \(d_i-s_j\) of the entry \(i,j\) of \(M\).

If working column-wise, let \(s_1,\ldots,s_m \in \ZZ\) be a shift, and let \((d_1,\ldots,d_n)\) denote the shifted column degrees of \(M\). Then, the shifted leading matrix of \(M\) is the matrix in \(\Bold{K}^{m \times n}\) whose entry \(i,j\) is the coefficient of degree \(d_j-s_i\) of the entry \(i,j\) of \(M\).

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).

OUTPUT: a matrix over the base field.

REFERENCES:

[Wol1974] (Section 2.5, without shifts) and [VBB1992] (Section 3).

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.leading_matrix()
[3 0 0]
[1 0 0]

sage: M.leading_matrix().base_ring()
Finite Field of size 7

sage: M.leading_matrix(shifts=[0,1,2])
[0 0 1]
[1 0 0]

sage: M.leading_matrix(row_wise=False)
[0 0 1]
[1 0 0]

sage: M.leading_matrix(shifts=[-2,1], row_wise=False)
[0 0 1]
[1 0 0]

sage: M.leading_matrix(shifts=[2,0], row_wise=False)
[3 0 1]
[1 0 0]

leading_positions(shifts=None, row_wise=True, return_degree=False)#

Return the (shifted) leading positions (also known as the pivot indices), and optionally the (shifted) pivot degrees of this matrix.

If working row-wise, for a given shift \(s_1,\ldots,s_n \in \ZZ\), taken as \((0,\ldots,0)\) by default, and a row vector of univariate polynomials \([p_1,\ldots,p_n]\), the leading position of this vector is the index \(j\) of the rightmost nonzero entry \(p_j\) such that \(\deg(p_j) + s_j\) is equal to the shifted row degree of the vector. Then the pivot degree of the vector is the degree \(\deg(p_j)\).

For the zero row, both the leading positions and degree are \(-1\). For a \(m \times n\) polynomial matrix, the leading positions and pivot degrees are the two lists containing the leading positions and the pivot degrees of its rows.

The definition is similar if working column-wise (instead of rightmost nonzero entry, we choose the bottommost nonzero entry).

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
return_degree – (optional, default: False) boolean, True implies that the pivot degrees are returned.

OUTPUT: a list of integers if return_degree=False; a pair of lists of integers otherwise.

REFERENCES:

[Kai1980] (Section 6.7.2, without shifts).

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.leading_positions()
[0, 0]

sage: M.leading_positions(return_degree=True)
([0, 0], [1, 3])

sage: M.leading_positions(shifts=[0,5,2], return_degree=True)
([2, 0], [0, 3])

sage: M.leading_positions(row_wise=False, return_degree=True)
([1, -1, 0], [3, -1, 0])

sage: M.leading_positions(shifts=[1,2], row_wise=False,
....:                     return_degree=True)
([1, -1, 0], [3, -1, 0])

In case several entries in the row (resp. column) reach the shifted row (resp. column) degree, the leading position is chosen as the rightmost (resp. bottommost) such entry:

sage: M.leading_positions(shifts=[0,5,1], return_degree=True)
([2, 0], [0, 3])

sage: M.leading_positions(shifts=[2,0], row_wise=False,
....:                     return_degree=True)
([1, -1, 0], [3, -1, 0])

The leading positions and pivot degrees of an empty matrix (\(0\times n\) or \(m\times 0\)) is not defined:

sage: M = Matrix(pR, 0, 3)
sage: M.leading_positions()
Traceback (most recent call last):
...
ValueError: empty matrix does not have leading positions

sage: M.leading_positions(row_wise=False)
Traceback (most recent call last):
...
ValueError: empty matrix does not have leading positions

sage: M = Matrix(pR, 3, 0)
sage: M.leading_positions(row_wise=False)
Traceback (most recent call last):
...
ValueError: empty matrix does not have leading positions

left_quo_rem(B)#

Return, if it exists, the quotient and remainder \((Q,R)\) such that self is \(BQ+R\), where \(R\) has row degrees less than those of \(B\) entrywise.

This method directly calls right_quo_rem() on transposed matrices, and transposes the result. See right_quo_rem() for a complete documentation and more examples.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: A = Matrix(pR, 3, 2,
....:            [[      3*x^3 + 3*x,         2*x^3 + 4],
....:             [  3*x^3 + 6*x + 5, 6*x^3 + 5*x^2 + 1],
....:             [  2*x^3 + 2*x + 6,   3*x^2 + 2*x + 2]])
sage: B = Matrix(pR, 3, 3,
....:            [[              3,       x + 3,               6],
....:             [3*x^3 + 3*x + 1, 4*x^2 + 3*x,   6*x^3 + x + 4],
....:             [  4*x^2 + x + 4, 3*x^2 + 4*x, 3*x^2 + 3*x + 2]])
sage: Q, R = A.left_quo_rem(B); Q, R
(
[2*x^2 + 4*x + 6 6*x^2 + 4*x + 1]  [              3               1]
[    3*x^2 + 5*x   2*x^2 + x + 5]  [              6 5*x^2 + 2*x + 3]
[    6*x^2 + 3*x 4*x^2 + 6*x + 1], [        2*x + 3         6*x + 3]
)
sage: rdegR = R.row_degrees(); rdegB = B.row_degrees()
sage: A == B*Q+R and all(rdegR[i] < rdegB[i] for i in range(3))
True

sage: A[:2,:].left_quo_rem(B)
Traceback (most recent call last):
...
ValueError: row dimension of self should be the row dimension of
the input matrix

Rectangular or rank-deficient matrices are supported but there may be no quotient and remainder (unless the matrix has full row rank, see right_quo_rem()):

sage: Q, R = A[:2,:].left_quo_rem(B[:2,:]); Q, R
(
[      3*x + 3       2*x + 1]
[  3*x^2 + 5*x 2*x^2 + x + 5]  [            5             0]
[            0             0], [4*x^2 + x + 2     4*x^2 + x]
)
sage: rdegR = R.row_degrees(); rdegB = B[:2,:].row_degrees()
sage: A[:2,:] == B[:2,:]*Q+R
True
sage: all([rdegR[i] < rdegB[i] for i in range(len(rdegR))])
True

sage: A.left_quo_rem(B[:,:2])
Traceback (most recent call last):
...
ValueError: division of these matrices does not admit a remainder
with the required degree property

See also

right_quo_rem() , reduce() .

minimal_approximant_basis(order, shifts=None, row_wise=True, normal_form=False)#

Return an approximant basis in shifts-ordered weak Popov form for this polynomial matrix at order order.

Assuming we work row-wise, if \(F\) is an \(m \times n\) polynomial matrix and \((d_0,\ldots,d_{n-1})\) are positive integers, then an approximant basis for \(F\) at order \((d_0,\ldots,d_{n-1})\) is a polynomial matrix whose rows form a basis of the module of approximants for \(F\) at order \((d_0,\ldots,d_{n-1})\). The latter approximants are the polynomial vectors \(p\) of size \(m\) such that the column \(j\) of \(p F\) has valuation at least \(d_j\), for all \(0 \le j \le n-1\).

If normal_form is True, then the output basis \(P\) is furthermore in shifts-Popov form. By default, \(P\) is considered row-wise, that is, its rows are left-approximants for self; if row_wise is False then its columns are right-approximants for self. It is guaranteed that the degree of the output basis is at most the maximum of the entries of order, independently of shifts.

An error is raised if the input dimensions are not sound: if working row-wise (resp. column-wise), the length of order must be the number of columns (resp. rows) of self, while the length of shifts must be the number of rows (resp. columns) of self.

If a single integer is provided for order, then it is converted into a list of repeated integers with this value.

INPUT:

order – a list of positive integers, or a positive integer.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, if True then the output basis is considered row-wise and operates on the left of self; otherwise it is column-wise and operates on the right of self.
normal_form – (optional, default: False) boolean, if True then the output basis is in shifts-Popov form.

OUTPUT: a polynomial matrix.

ALGORITHM:

The implementation is inspired from the iterative algorithms described in [VBB1992] and [BL1994] ; for obtaining the normal form, it relies directly on Lemmas 3.3 and 4.1 in [JNSV2016] .

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: order = [4, 3]; shifts = [-1, 2, 0]
sage: F = Matrix(pR, [[5*x^3 + 4*x^2 + 4*x + 6, 5*x^2 + 4*x + 1],
....:                 [        2*x^2 + 2*x + 3, 6*x^2 + 6*x + 3],
....:                 [4*x^3         +   x + 1, 4*x^2 + 2*x + 3]])
sage: P = F.minimal_approximant_basis(order, shifts)
sage: P.is_minimal_approximant_basis(F, order, shifts)
True

By default, the computed basis is not required to be in normal form (and will not be except in rare special cases):

sage: P.is_minimal_approximant_basis(F, order, shifts,
....:                                normal_form=True)
False
sage: P = F.minimal_approximant_basis(order, shifts,
....:                                 normal_form=True)
sage: P.is_minimal_approximant_basis(F, order, shifts,
....:                                normal_form=True)
True

If shifts are not specified, they are chosen as uniform \([0,\ldots,0]\) by default. Besides, if the orders are all the same, one can rather give a single integer:

sage: (F.minimal_approximant_basis(3) ==
....:  F.minimal_approximant_basis([3,3], shifts=None))
True

One can work column-wise by specifying row_wise=False:

sage: P = F.minimal_approximant_basis([5,2,2], [0,1],
....:                                 row_wise=False)
sage: P.is_minimal_approximant_basis(F, [5,2,2], shifts=[0,1],
....:                                row_wise=False)
True
sage: (F.minimal_approximant_basis(3, row_wise=True) ==
....:  F.transpose().minimal_approximant_basis(
....:      3, row_wise=False).transpose())
True

Errors are raised if the input dimensions are not sound:

sage: P = F.minimal_approximant_basis([4], shifts)
Traceback (most recent call last):
...
ValueError: order length should be the column dimension

sage: P = F.minimal_approximant_basis(order, [0,0,0,0])
Traceback (most recent call last):
...
ValueError: shifts length should be the row dimension

An error is raised if order does not contain only positive integers:

sage: P = F.minimal_approximant_basis([1,0], shifts)
Traceback (most recent call last):
...
ValueError: order should consist of positive integers

minimal_kernel_basis(shifts=None, row_wise=True, normal_form=False)#

Return a left kernel basis in shifts-ordered weak Popov form for this polynomial matrix.

Assuming we work row-wise, if \(F\) is an \(m \times n\) polynomial matrix, then a left kernel basis for \(F\) is a polynomial matrix whose rows form a basis of the left kernel of \(F\), which is the module of polynomial vectors \(p\) of size \(m\) such that \(p F\) is zero.

If normal_form is True, then the output basis \(P\) is furthermore in shifts-Popov form. By default, \(P\) is considered row-wise, that is, its rows are left kernel vectors for self; if row_wise is False then its columns are right kernel vectors for self.

An error is raised if the input dimensions are not sound: if working row-wise (resp. column-wise), the length of shifts must be the number of rows (resp. columns) of self.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, if True then the output basis considered row-wise and operates on the left of self; otherwise it is column-wise and operates on the right of self.
normal_form – (optional, default: False) boolean, if True then the output basis is in shifts-Popov form.

OUTPUT: a polynomial matrix.

ALGORITHM: uses minimal approximant basis computation at a sufficiently large order so that the approximant basis contains a kernel basis as a submatrix.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: pmat = Matrix([[(x+1)*(x+3)], [(x+1)*(x+3)+1]])
sage: pmat.minimal_kernel_basis()
[6*x^2 + 3*x + 3   x^2 + 4*x + 3]

sage: pmat = Matrix([[(x+1)*(x+3)], [(x+1)*(x+4)]])
sage: pmat.minimal_kernel_basis()
[6*x + 3   x + 3]

sage: pmat.minimal_kernel_basis(row_wise=False)
[]

sage: pmat = Matrix(pR, [[1, x, x**2]])
sage: pmat.minimal_kernel_basis(row_wise=False, normal_form=True)
[x 0]
[6 x]
[0 6]

sage: pmat.minimal_kernel_basis(row_wise=False, normal_form=True,
....:                           shifts=[0,1,2])
[  6*x 6*x^2]
[    1     0]
[    0     1]

Some particular cases (matrix is zero, dimension is zero, column is zero):

sage: Matrix(pR, 2, 1).minimal_kernel_basis()
[1 0]
[0 1]

sage: Matrix(pR, 2, 0).minimal_kernel_basis()
[1 0]
[0 1]

sage: Matrix(pR, 0, 2).minimal_kernel_basis()
[]

sage: Matrix(pR, 3, 2, [[1,0],[1,0],[1,0]]).minimal_kernel_basis()
[6 1 0]
[6 0 1]

sage: Matrix(pR, 3, 2, [[x,0],[1,0],[x+1,0]]).minimal_kernel_basis()
[6 x 0]
[6 6 1]

popov_form(transformation=False, shifts=None, row_wise=True, include_zero_vectors=True)#

Return the (shifted) Popov form of this matrix.

See is_popov() for a definition of Popov forms. If the input matrix is \(A\), the (shifted) Popov form of \(A\) is the unique matrix \(P\) in (shifted) Popov form and such that \(UA = P\) for some unimodular matrix \(U\). The latter matrix is called the transformation, and the first optional argument allows one to specify whether to return this transformation. We refer to the description of weak_popov_form() for an explanation of the option include_zero_vectors .

INPUT:

transformation – (optional, default: False). If this is True, the transformation matrix \(U\) will be returned as well.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
include_zero_vectors – (optional, default: True) boolean, False if zero rows (resp. zero columns) should be discarded from the Popov forms.

OUTPUT:

A polynomial matrix which is the Popov form of self if transformation is False; otherwise two polynomial matrices which are the Popov form of self and the corresponding unimodular transformation.

ALGORITHM:

This method implements the Mulders-Storjohann algorithm of [MS2003] for transforming a weak Popov form into Popov form, straightforwardly extended to the case of shifted forms.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [
....:     [      6*x+4,       5*x^3+5*x,       6*x^2+2*x+2],
....:     [4*x^2+5*x+2, x^4+5*x^2+2*x+4, 4*x^3+6*x^2+6*x+5]])

sage: # needs sage.combinat
sage: P, U = M.popov_form(transformation=True)
sage: P
[            4 x^2 + 4*x + 1             3]
[            0       4*x + 1 x^2 + 6*x + 1]
sage: U
[            x             2]
[5*x^2 + x + 6       3*x + 2]
sage: P.is_popov() and U.is_invertible() and U*M == P
True

Demonstrating shifts and specific case of Hermite form:

sage: # needs sage.combinat
sage: P = M.popov_form(shifts=[0,2,4]); P
[              4*x^2 + 3*x + 4 x^4 + 3*x^3 + 5*x^2 + 5*x + 5                             0]
[                            6               5*x^2 + 6*x + 5                             1]
sage: P.is_popov(shifts=[0,2,4])
True
sage: P == M.popov_form(shifts=[-6,-4,-2])
True
sage: dd = sum(M.row_degrees()) + 1
sage: M.popov_form(shifts=[2*dd,dd,0]) == M.hermite_form()
True

Column-wise form is the row-wise form of the transpose:

sage: M.popov_form() == M.T.popov_form(row_wise=False).T
True

Zero vectors can be discarded:

sage: M.popov_form(row_wise=False)
[x + 2     6     0]
[    0     1     0]

sage: # needs sage.combinat
sage: P, U = M.popov_form(transformation=True,
....:                     row_wise=False,
....:                     include_zero_vectors=False)
sage: P
[x + 2     6]
[    0     1]
sage: U
[        3*x^2 + 6*x + 3         5*x^2 + 4*x + 4 3*x^3 + 3*x^2 + 2*x + 4]
[                      3                       1                 2*x + 1]
[                5*x + 2                       2                       6]
sage: M*U[:,:2] == P and (M*U[:,2]).is_zero()
True

reduce(B, shifts=None, row_wise=True, return_quotient=False)#

Reduce self, i.e. compute its normal form, modulo the row space of \(B\) with respect to shifts.

If self is a \(k \times n\) polynomial matrix (written \(A\) below), and the input \(B\) is an \(m \times n\) polynomial matrix, this computes the normal form \(R\) of \(A\) with respect the row space of \(B\) and the monomial order defined by shifts (written \(s\) below). This means that the \(i\) th row of \(R\) is equal to the \(i\) th row of \(A\) up to addition of an element in the row space of \(B\), and if \(J = (j_1,\ldots,j_r)\) are the \(s\)-leading positions of the \(s\)-Popov form \(P\) of \(A\), then the submatrix \(R_{*,J}\) (submatrix of \(R\) formed by its columns in \(J\)) has column degrees smaller entrywise than the column degrees of \(P_{*,J}\).

If the option row_wise is set to False, the same operation is performed, but with everything considered column-wise: column space of \(B\), \(i\) th column of \(R\) and \(A\), column-wise \(s\)-leading positions and \(s\)-Popov form, and submatrices \(R_{J,*}\) and \(P_{J,*}\).

The operation above can be seen as a matrix generalization of division with remainder for univariate polynomials. If the option return_quotient is set to True, this method returns both the normal form \(R\) and a quotient matrix \(Q\) such that \(A = QB + R\) (or \(A = BQ + R\) if row_wise is False). Whereas the remainder is unique, this quotient is not unique unless \(B\) has a trivial left kernel i.e. has full row rank (or right kernel, full column rank if row_wise is False).

This method checks whether \(B\) is in \(s\)-Popov form, and if not, computes the \(s\)-Popov form \(P\) of \(B\), which can take some time. Therefore, if \(P\) is already known or is to be re-used, this method should be called directly with \(P\), yielding the same normal form \(R\) since \(P\) and \(B\) have the same row space (or column space, if row_wise is False).

A ValueError is raised if the dimensions of the shifts and/or of the matrices are not conformal.

INPUT:

B – polynomial matrix.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
return_quotient – (optional, default: False). If this is True, the quotient will be returned as well.

OUTPUT: a polynomial matrix if return_quotient=False, two polynomial matrices otherwise.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: B = Matrix(pR, [
....:     [      6*x+4,       5*x^3+5*x,       6*x^2+2*x+2],
....:     [4*x^2+5*x+2, x^4+5*x^2+2*x+4, 4*x^3+6*x^2+6*x+5]])
sage: A = Matrix(pR, 1, 3, [
....:     [3*x^4+3*x^3+4*x^2+5*x+1, x^4+x^3+5*x^2+4*x+4, 4*x^4+2*x^3+x]])

sage: Q, R = A.reduce(B,return_quotient=True); R
[3*x^4 + 3*x^3 + 4*x + 3                 2*x + 2                 2*x + 6]
sage: A == Q*B + R
True
sage: P = B.popov_form(); P.leading_positions(return_degree=True)
([1, 2], [2, 2])
sage: R.degree_matrix()
[4 1 1]
sage: A.reduce(P) == R
True
sage: A.reduce(P[:,:2])
Traceback (most recent call last):
...
ValueError: column dimension of self should be the column
dimension of the input matrix

Demonstrating shifts:

sage: Qs, Rs = A.reduce(B, shifts=[0,2,4], return_quotient=True); Rs
[3*x^4 + 3*x^3 + 6*x + 2             4*x^3 + 5*x                       0]
sage: A == Qs*B + Rs
True
sage: Ps = B.popov_form(shifts=[0,2,4])
sage: Ps.leading_positions(shifts=[0,2,4], return_degree=True)
([1, 2], [4, 0])
sage: Rs.degree_matrix()
[ 4  3 -1]
sage: A.reduce(Ps, shifts=[0,2,4]) == Rs
True

If return_quotient is False, only the normal form is returned:

sage: R == A.reduce(B) and Rs == A.reduce(B, shifts=[0,2,4])
True

Demonstrating column-wise normal forms, with a matrix \(A\) which has several columns, and a matrix \(B\) which does not have full column rank (its column-wise Popov form has a zero column):

sage: A = Matrix(pR, 2, 2,
....:     [[5*x^3 + 2*x^2 + 4*x + 1,           x^3 + 4*x + 4],
....:      [2*x^3 + 5*x^2 + 2*x + 4,         2*x^3 + 3*x + 2]])
sage: (Q,R) = A.reduce(B,row_wise=False, return_quotient=True); R
[0 3]
[0 0]
sage: A == B*Q + R
True
sage: P = B.popov_form(row_wise=False); P
[x + 2     6     0]
[    0     1     0]
sage: P.leading_positions(row_wise=False, return_degree=True)
([0, 1, -1], [1, 0, -1])
sage: R.degree_matrix()
[-1  0]
[-1 -1]

See also

left_quo_rem() , right_quo_rem() .

reduced_form(transformation=None, shifts=None, row_wise=True, include_zero_vectors=True)#

Return a row reduced form of this matrix (resp. a column reduced form if the optional parameter row_wise is set to False).

An \(m \times n\) univariate polynomial matrix \(M\) is said to be in (shifted) row reduced form if it has \(k\) nonzero rows with \(k \leq n\) and its (shifted) leading matrix has rank \(k\). See is_reduced() for more information.

Currently, the implementation of this method is a direct call to weak_popov_form().

INPUT:

transformation – (optional, default: False). If this is True, the transformation matrix \(U\) will be returned as well: this is a unimodular matrix over \(\Bold{K}[x]\) such that self equals \(UR\), where \(R\) is the output matrix.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
include_zero_vectors – (optional, default: True) boolean, False if one does not allow zero rows in row reduced forms (resp. zero columns in column reduced forms).

OUTPUT:

A polynomial matrix \(R\) which is a reduced form of self if transformation=False; otherwise two polynomial matrices \(R, U\) such that \(UA = R\) and \(R\) is reduced and \(U\) is unimodular where \(A\) is self.

EXAMPLES:

sage: pR.<x> = GF(3)[]
sage: A = matrix(pR, 3, [x,   x^2, x^3,
....:                    x^2, x^1, 0,
....:                    x^3, x^3, x^3])
sage: R = A.reduced_form(); R
[        x           x^2       x^3]
[      x^2             x         0]
[  x^3 + 2*x x^3 + 2*x^2         0]
sage: R.is_reduced()
True
sage: R2 = A.reduced_form(shifts=[6,3,0]); R2
[                x               x^2               x^3]
[                0         2*x^2 + x       2*x^4 + x^3]
[                0                 0 2*x^5 + x^4 + x^3]
sage: R2.is_reduced(shifts=[6,3,0])
True
sage: R2.is_reduced()
False

If the matrix is an \(n \times 1\) matrix with at least one non-zero entry, \(R\) has a single non-zero entry and that entry is a scalar multiple of the greatest-common-divisor of the entries of the matrix:

sage: A = matrix([[x*(x-1)*(x+1)], [x*(x-2)*(x+2)], [x]])
sage: R = A.reduced_form()
sage: R
[x]
[0]
[0]

A zero matrix is already reduced:

sage: A = matrix(pR, 2, [0,0,0,0])
sage: A.reduced_form()
[0 0]
[0 0]

In the following example, the original matrix is already reduced, but the output is a different matrix: currently this method is an alias for weak_popov_form(), which is a stronger reduced form:

sage: R.<x> = QQ['x']
sage: A = matrix([[x,x,x],[0,0,x]]); A
[x x x]
[0 0 x]
sage: A.is_reduced()
True
sage: W = A.reduced_form(); W
[ x  x  x]
[-x -x  0]
sage: W.is_weak_popov()
True

The last example shows the usage of the transformation parameter:

sage: # needs sage.rings.finite_rings
sage: Fq.<a> = GF(2^3)
sage: pR.<x> = Fq[]
sage: A = matrix(pR, [[x^2+a,  x^4+a],
....:                 [  x^3,  a*x^4]])
sage: W, U = A.reduced_form(transformation=True)
sage: W, U
(
[          x^2 + a           x^4 + a]  [1 0]
[x^3 + a*x^2 + a^2               a^2], [a 1]
)
sage: W.is_reduced()
True
sage: U*W == A
True
sage: U.is_invertible()
True

reverse(degree=None, row_wise=True, entry_wise=False)#

Return the matrix which is obtained from this matrix after reversing all its entries with respect to the degree as specified by degree.

Reversing a polynomial with respect to an integer \(d\) follows the convention for univariate polynomials, in particular it uses truncation or zero-padding as necessary if \(d\) differs from the degree of this polynomial.

If entry_wise is True: the input degree and row_wise are ignored, and all entries of the matrix are reversed with respect to their respective degrees.

If entry_wise is False (the default):

if degree is an integer, all entries are reversed with respect to it;
if degree is not provided, then all entries are reversed with respect to the degree of the whole matrix;
if degree is a list \((d_1,\ldots,d_m)\) and row_wise is True, all entries of the \(i`th row are reversed with respect to `d_i\) for each \(i\);
if degree is a list \((d_1,\ldots,d_n)\) and row_wise is False, all entries of the \(j`th column are reversed with respect to `d_j\) for each \(j\).

INPUT:

degree – (optional, default: None) a list of nonnegative integers, or a nonnegative integer,
row_wise – (optional, default: True) boolean, if True (resp. False) then degree should be a list of length equal to the row (resp. column) dimension of this matrix.
entry_wise – (optional, default: False) boolean, if True then the input degree and row_wise are ignored.

OUTPUT: a polynomial matrix.

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.reverse()
[  x^3 + 5*x^2 + 5*x + 1                   5*x^3           4*x^3 +
6*x^2                       0]
[      x^3 + 3*x^2 + 6*x                     x^3
2*x^3                       0]
[4*x^3 + 6*x^2 + 4*x + 2             x^3 + 5*x^2     5*x^3 + 5*x^2
+ 2*x       6*x^3 + 5*x^2 + x]

sage: M.reverse(1)
[  x + 5     5*x 4*x + 6       0]
[  x + 3       x     2*x       0]
[4*x + 6   x + 5 5*x + 5 6*x + 5]

sage: M.reverse(0) == M.constant_matrix()
True

Entry-wise reversing with respect to each entry’s degree:

sage: M.reverse(entry_wise=True)
[  x^3 + 5*x^2 + 5*x + 1                       5
4*x + 6                       0]
[          x^2 + 3*x + 6                       1
2                       0]
[4*x^3 + 6*x^2 + 4*x + 2                   x + 5         5*x^2 +
5*x + 2         6*x^2 + 5*x + 1]

Row-wise and column-wise degree reversing are available:

sage: M.reverse([2,3,1])
[    x^2 + 5*x + 5             5*x^2       4*x^2 + 6*x
0]
[x^3 + 3*x^2 + 6*x               x^3             2*x^3
0]
[          4*x + 6             x + 5           5*x + 5
6*x + 5]

sage: M.reverse(M.column_degrees(), row_wise=False)
[  x^3 + 5*x^2 + 5*x + 1                     5*x             4*x^2
+ 6*x                       0]
[      x^3 + 3*x^2 + 6*x                       x
2*x^2                       0]
[4*x^3 + 6*x^2 + 4*x + 2                   x + 5         5*x^2 +
5*x + 2         6*x^2 + 5*x + 1]

Wrong length or negativity of input degree raise errors:

sage: M.reverse([1,3,1,4]) Traceback (most recent call last): … ValueError: length of input degree list should be the row dimension of the input matrix

sage: M.reverse([5,2,1], row_wise=False) Traceback (most recent call last): … ValueError: length of input degree list should be the column dimension of the input matrix

sage: M.reverse([2,3,-1]) Traceback (most recent call last): … ValueError: degree argument must be a non-negative integer, got -1

right_quo_rem(B)#

Return, if it exists, the quotient and remainder \((Q,R)\) such that self is \(QB+R\), where \(R\) has column degrees less than those of \(B\) entrywise.

If self is a \(k \times m\) polynomial matrix (written \(A\) below), and the input \(B\) is an \(m \times m\) polynomial matrix in column reduced form, then \((Q,R)\) is unique. Both \(Q\) and \(R\) have dimensions \(k \times m\). In this case, this method implements Newton iteration of a reversed polynomial matrix \(B\), generalizing to matrices the fast division of univariate polynomials.

If \(A\) is a \(k \times n\) polynomial matrix, and the input \(B\) is an \(m \times n\) polynomial matrix such that \(B\) has full column rank, or more generally such that the matrix equation \(A = XB\) has a rational solution, then there exists such a \((Q,R)\) but it may not be unique; the algorithm returns one such quotient and remainder. Here \(Q\) is \(k \times m\) and \(R\) is \(k \times n\). In this case, this method follows the folklore approach based on solving the matrix equation \(A = XB\) and splitting \(X\) into its polynomial part and proper fraction part.

Finally, if the matrix equation \(A = XB\) has no rational solution, this method computes the normal form \(R\) and quotient \(Q\) of the rows of \(A\) with respect to the row space of \(B\) (see reduce()). Doing this for a well-chosen shift ensures that either \(R\) does not have column degrees less than those of \(B\), and then there is no valid quotient and remainder, or it does satisfy this degree constraint, and then this \(R\) can be returned as a remainder along with the quotient \(Q\).

A ValueError is raised if the dimensions of self and \(B\) are not conformal, or if there exists no quotient and remainder.

EXAMPLES:

Case where \(B\) is a square, column reduced matrix:

sage: pR.<x> = GF(7)[]
sage: A = Matrix(pR, 2, 3,
....:     [[3*x^3 + 3*x, 3*x^3 + 6*x + 5,   2*x^3 + 2*x + 6],
....:      [2*x^3 + 4,   6*x^3 + 5*x^2 + 1, 3*x^2 + 2*x + 2]])
sage: B = Matrix(pR, 3, 3,
....:     [[4*x^2 + 3*x + 3, 3*x^2 + 3*x + 1,   4*x^2 + x + 4],
....:      [6*x^2 + 2*x + 3,     4*x^2 + 3*x,     3*x^2 + 4*x],
....:      [5*x^2 + 3*x + 6,   6*x^2 + x + 4, 3*x^2 + 3*x + 2]])
sage: B.is_reduced(row_wise=False)
True
sage: Q, R = A.right_quo_rem(B); Q, R
(
[    4*x   x + 2 6*x + 1]  [  x + 2 6*x + 1 5*x + 4]
[4*x + 3   x + 6 3*x + 4], [4*x + 2 2*x + 3 4*x + 3]
)
sage: A == Q*B+R and R.degree() < 2
True
sage: A[:,:2].right_quo_rem(B)
Traceback (most recent call last):
...
ValueError: column dimension of self should be the column dimension
of the input matrix

sage: B = Matrix(pR, 3, 3,
....:     [[3,     3*x^3 + 3*x + 1, 4*x^2 + x + 4],
....:      [x + 3, 4*x^2 + 3*x,     3*x^2 + 4*x],
....:      [6,     6*x^3 + x + 4,   3*x^2 + 3*x + 2]])
sage: B.is_reduced(row_wise=False)
True
sage: Q, R = A.right_quo_rem(B); Q, R
(
[2*x^2 + 4*x + 6     3*x^2 + 5*x     6*x^2 + 3*x]
[6*x^2 + 4*x + 1   2*x^2 + x + 5 4*x^2 + 6*x + 1],

[              3               6         2*x + 3]
[              1 5*x^2 + 2*x + 3         6*x + 3]
)
sage: cdegR = R.column_degrees(); cdegB = B.column_degrees()
sage: A == Q*B+R and all([cdegR[i] < cdegB[i] for i in range(3)])
True

With a nonsingular but also non-reduced matrix, there exists a solution, but it might not be unique:

sage: B = Matrix(pR, 3, 3,
....:     [[              5,               0, 2*x + 6],
....:      [            4*x, 3*x^2 + 4*x + 5,   x + 1],
....:      [3*x^2 + 5*x + 2, 6*x^3 + 4*x + 6,       3]])
sage: B.det() != 0 and (not B.is_reduced(row_wise=False))
True
sage: Q, R = A.right_quo_rem(B); Q, R
(
[    6*x^2 + 3*x 4*x^2 + 3*x + 1         5*x + 1]
[  x^2 + 5*x + 5 5*x^2 + 3*x + 5           x + 2],

[      4*x + 5 x^2 + 2*x + 1             2]
[      6*x + 3     5*x^2 + 6             3]
)
sage: cdegR = R.column_degrees(); cdegB = B.column_degrees()
sage: A == Q*B+R and all(cdegR[i] < cdegB[i] for i in range(3))
True

sage: Q2 = Matrix(pR, 2, 3,
....:      [[6*x^2 + 3*x + 1, 4*x^2 + 3*x + 6, 5*x + 1],
....:       [  x^2 + 5*x + 3, 5*x^2 + 3*x + 2,   x + 2]])
sage: R2 = Matrix(pR, 2, 3,
....:      [[    5*x, 3*x + 4, 5],
....:       [4*x + 6,     5*x, 4]])
sage: A == Q2*B + R2
True

The same remark holds more generally for full column rank matrices: there exists a solution, but it might not be unique. However, for all other cases (rank-deficient matrix \(B\) or matrix \(B\) having strictly fewer rows than columns) there may be no solution:

sage: C = B.stack(B[1,:] + B[2,:])  # 4 x 3, full column rank
sage: Q, R = A.right_quo_rem(C); Q, R
(
[    6*x^2 + 3*x 4*x^2 + 3*x + 1         5*x + 1               0]
[  x^2 + 5*x + 5 5*x^2 + 3*x + 5           x + 2               0],

[      4*x + 5 x^2 + 2*x + 1             2]
[      6*x + 3     5*x^2 + 6             3]
)

sage: A.right_quo_rem(B[:2,:])  # matrix 2 x 3, full row rank               # needs sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: division of these matrices does not admit a remainder
with the required degree property
sage: D = copy(B); D[2,:] = B[0,:]+B[1,:]  # square, singular
sage: A.right_quo_rem(D)
Traceback (most recent call last):
...
ValueError: division of these matrices does not admit a remainder
with the required degree property

In the latter case (rank-deficient or strictly fewer rows than columns, with no solution to \(A = XB\)), there might stil be a quotient and remainder, in which case this method will find it via normal form computation:

sage: B = Matrix(pR, 1, 2, [[x, x]])
sage: A = Matrix(pR, 1, 2, [[x, x+2]])
sage: A.right_quo_rem(B)
([1], [0 2])
sage: A == 1*B + Matrix([[0,2]])
True

See also

left_quo_rem() , reduce() .

row_degrees(shifts=None)#

Return the (shifted) row degrees of this matrix.

For a given polynomial matrix \(M = (M_{i,j})_{i,j}\) with \(m\) rows and \(n\) columns, its row degrees is the tuple \((d_1,\ldots,d_m)\) where \(d_i = \max_j(\deg(M_{i,j}))\) for \(1\leq i \leq m\). Thus, \(d_i=-1\) if the \(i\)-th row of \(M\) is zero, and \(d_i \geq 0\) otherwise.

For given shifts \(s_1,\ldots,s_n \in \ZZ\), the shifted row degrees of \(M\) is \((d_1,\ldots,d_m)\) where \(d_i = \max_j(\deg(M_{i,j})+s_j)\). Here, if the \(i\)-th row of \(M\) is zero then \(d_i =\min(s_1,\ldots,s_n)-1\); otherwise, \(d_i\) is larger than this value.

INPUT:

shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].

OUTPUT: a list of integers.

REFERENCES:

[Wol1974] (Section 2.5, without shifts), and [VBB1992] (Section 3).
Up to changes of signs, shifted row degrees coincide with the notion of defect commonly used in the rational approximation literature (see for example [Bec1992] ).

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0]])
sage: M.row_degrees()
[1, 3]

sage: M.row_degrees(shifts=[0,1,2])
[2, 3]

A zero row in a polynomial matrix can be identified in the (shifted) row degrees as the entries equal to min(shifts)-1:

sage: M = Matrix(pR, [[3*x+1, 0, 1], [x^3+3, 0, 0], [0, 0, 0]])
sage: M.row_degrees()
[1, 3, -1]

sage: M.row_degrees(shifts=[-2,1,2])
[2, 1, -3]

The row degrees of an empty matrix (\(0\times n\) or \(m\times 0\)) is not defined:

sage: M = Matrix(pR, 0, 3)
sage: M.row_degrees()
Traceback (most recent call last):
...
ValueError: empty matrix does not have row degrees

sage: M = Matrix(pR, 3, 0)
sage: M.row_degrees()
Traceback (most recent call last):
...
ValueError: empty matrix does not have row degrees

shift(d, row_wise=True)#

Return the matrix which is obtained from this matrix after shifting all its entries as specified by \(d\).

if \(d\) is an integer, the shift is by \(d\) for all entries;
if \(d\) is a list \((d_1,\ldots,d_m)\) and row_wise is True, all entries of the \(i`th row are shifted by `d_i\) for each \(i\);
if \(d\) is a list \((d_1,\ldots,d_n)\) and row_wise is False, all entries of the \(j`th column are shifted by `d_j\) for each \(j\).

Shifting by \(d\) means multiplying by the variable to the power \(d\); if \(d\) is negative then terms of negative degree after shifting are discarded.

INPUT:

d – a list of integers, or an integer,
row_wise – (optional, default: True) boolean, if True (resp. False) then \(d\) should be a list of length equal to the row (resp. column) dimension of this matrix.

OUTPUT: a polynomial matrix.

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.shift(-2)
[  x + 5       0       0       0]
[      6       0       0       0]
[2*x + 4       0       2       1]

Row-wise and column-wise shifting are available:

sage: M.shift([-1,2,-2])
[      x^2 + 5*x + 5                   0                   6                   0]
[6*x^4 + 3*x^3 + x^2                 x^2               2*x^2                   0]
[            2*x + 4                   0                   2                   1]

sage: M.shift([-1,1,0,0], row_wise=False)
[  x^2 + 5*x + 5             5*x         6*x + 4               0]
[        6*x + 3               x               2               0]
[2*x^2 + 4*x + 6       5*x^2 + x 2*x^2 + 5*x + 5   x^2 + 5*x + 6]

sage: M.shift([-d for d in M.row_degrees()]) == M.leading_matrix()
True

Length of input shift degree list is checked:

sage: M.shift([1,3,1,4])
Traceback (most recent call last):
...
ValueError: length of input shift list should be the row
dimension of the input matrix

sage: M.shift([5,2,-1], row_wise=False)
Traceback (most recent call last):
...
ValueError: length of input shift list should be the column
dimension of the input matrix

solve_left_series_trunc(B, d)#

Try to find a solution \(X\) to the equation \(X A = B\), at precision d.

If self is a matrix \(A\), then this function returns a vector or matrix \(X\) such that \(X A = B \bmod x^d\). If \(B\) is a vector then \(X\) is a vector, and if \(B\) is a matrix then \(X\) is a matrix.

Raises ValueError if d is not strictly positive, or if there is a dimension mismatch between \(A\) and \(B\), or if there is no solution to the given matrix equation at the specified precision.

INPUT:

B – a polynomial matrix or polynomial vector.
d – a positive integer.

OUTPUT:

A solution to the matrix equation, returned as polynomial matrix of degree less than d if B is a polynomial matrix; a polynomial vector of degree less than d if \(B\) is a polynomial vector.

ALGORITHM:

If \(A\) is square with invertible constant term, then the unique solution is found by calling inverse_series_trunc() and using the Dixon & Moenck-Carter iteration. Otherwise, a left minimal approximant basis of a matrix formed by \(A\) and \(B\) is computed, for an appropriate shift which ensures that this basis reveals either a solution \(X\) or the fact that no such solution exists.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: A = Matrix(pR, 3, 3,
....:            [[4*x+5,           5*x^2 + x + 1, 4*x^2 + 4],
....:             [6*x^2 + 6*x + 6, 4*x^2 + 5*x,   4*x^2 + x + 3],
....:             [3*x^2 + 2,       4*x + 1,       x^2 + 3*x]])
sage: A.is_square() and A.constant_matrix().is_invertible()
True
sage: B = vector([2*x^2 + 6*x + 6, 0, x + 6])
sage: X = A.solve_left_series_trunc(B, 4); X
(3*x^3 + 3*x^2 + 2*x + 4, 4*x^3 + x^2 + 2*x + 6, 6*x^3 + x + 3)
sage: B == X*A % x**4
True

sage: B = Matrix(pR, 2, 3,
....:            [[3*x, x^2 + x + 2, x^2 + 2*x + 3],
....:             [  0,   6*x^2 + 1,             1]])
sage: A.solve_left_series_trunc(B, 3)
[6*x^2 + 2*x + 2         4*x + 3     2*x^2 + 3*x]
[3*x^2 + 4*x + 5       4*x^2 + 3   x^2 + 6*x + 3]
sage: X = A.solve_left_series_trunc(B, 37); B == X*A % x**37
True

Dimensions of input are checked:

sage: A.solve_left_series_trunc(B[:,:2], 3)
Traceback (most recent call last):
...
ValueError: number of columns of self must equal number of columns of right-hand side

Raises an exception when no solution:

sage: A[2:,:].solve_left_series_trunc(B, 4)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions

sage: Ax = x*A; C = vector(pR, [1,1,1])
sage: Ax.solve_left_series_trunc(C, 5)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions

Supports rectangular and rank-deficient cases:

sage: A[:,:2].solve_left_series_trunc(B[:,:2], 4)
[5*x^2 + 2*x + 5         5*x + 5         2*x + 4]
[5*x^3 + 2*x + 1 2*x^2 + 2*x + 5           4*x^2]

sage: V = Matrix([[3*x^2 + 4*x + 1, 4*x]])
sage: A[:2,:].solve_left_series_trunc(V*A[:2,:], 4) == V
True

sage: A[1,:] = (x+1) * A[0,:]; A[2,:] = (x+5) * A[0,:]
sage: B = (3*x^3+x^2+2)*A[0,:]
sage: A.solve_left_series_trunc(B, 6)
[4*x^2 + 6*x + 2       3*x^2 + x               0]

See also

solve_right_series_trunc() .

solve_right_series_trunc(B, d)#

Try to find a solution \(X\) to the equation \(A X = B\), at precision d.

If self is a matrix \(A\), then this function returns a vector or matrix \(X\) such that \(A X = B \bmod x^d\). If \(B\) is a vector then \(X\) is a vector, and if \(B\) is a matrix then \(X\) is a matrix.

Raises ValueError if d is not strictly positive, or if there is a dimension mismatch between \(A\) and \(B\), or if there is no solution to the given matrix equation at the specified precision.

INPUT:

B – a polynomial matrix or polynomial vector.
d – a positive integer.

OUTPUT:

A solution to the matrix equation, returned as polynomial matrix of degree less than d if B is a polynomial matrix; a polynomial vector of degree less than d if \(B\) is a polynomial vector.

ALGORITHM:

If \(A\) is square with invertible constant term, then the unique solution is found by calling inverse_series_trunc() and using the Dixon & Moenck-Carter iteration. Otherwise, a right minimal approximant basis of a matrix formed by \(A\) and \(B\) is computed, for an appropriate shift which ensures that this basis reveals either a solution \(X\) or the fact that no such solution exists.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: A = Matrix(pR, 3, 3,
....:     [[4*x+5,           5*x^2 + x + 1, 4*x^2 + 4],
....:      [6*x^2 + 6*x + 6, 4*x^2 + 5*x,   4*x^2 + x + 3],
....:      [3*x^2 + 2,       4*x + 1,       x^2 + 3*x]])
sage: A.is_square() and A.constant_matrix().is_invertible()
True
sage: B = vector([2*x^2 + 6*x + 6, 0, x + 6])
sage: X = A.solve_right_series_trunc(B, 4); X
(2*x^3 + x^2, 5*x^3 + x^2 + 5*x + 6, 4*x^3 + 6*x^2 + 4*x)
sage: B == A*X % x**4
True
sage: B = Matrix(pR, 3, 2,
....:            [[5*x^2 + 6*x + 3, 4*x^2 + 6*x + 4],
....:             [  x^2 + 4*x + 2,         5*x + 2],
....:             [        5*x + 3,               0]])
sage: A.solve_right_series_trunc(B, 3)
[  3*x^2 + x + 1 5*x^2 + 4*x + 3]
[6*x^2 + 3*x + 1         4*x + 1]
[      6*x^2 + 1   2*x^2 + x + 4]
sage: X = A.solve_right_series_trunc(B, 37); B == A*X % x**37
True

Dimensions of input are checked:

sage: A.solve_right_series_trunc(B[:2,:], 3)
Traceback (most recent call last):
...
ValueError: number of rows of self must equal number of rows of right-hand side

Raises an exception when no solution:

sage: A[:,2:].solve_right_series_trunc(B, 4)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions

sage: Ax = x*A; C = vector(pR, [1,1,1])
sage: Ax.solve_right_series_trunc(C, 5)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions

Supports rectangular and rank-deficient cases:

sage: A[:2,:].solve_right_series_trunc(B[:2,:],4)
[    5*x^2 + 4*x           x + 4]
[  x^2 + 3*x + 5 3*x^2 + 4*x + 4]
[        5*x + 3         3*x + 2]

sage: V = Matrix([[2*x^2 + 5*x + 1], [3*x^2+4]])
sage: A[:,:2].solve_right_series_trunc(A[:,:2]*V, 4) == V
True

sage: A[:,1] = (x+1) * A[:,0]; A[:,2] = (x+5) * A[:,0]
sage: B = (3*x^3+x^2+2)*A[:,0]
sage: A.solve_right_series_trunc(B, 6)
[4*x^2 + 6*x + 2]
[      3*x^2 + x]
[              0]

See also

solve_left_series_trunc() .

truncate(d, row_wise=True)#

Return the matrix which is obtained from this matrix after truncating all its entries according to precisions specified by \(d\).

if \(d\) is an integer, the truncation is at precision \(d\) for all entries;
if \(d\) is a list \((d_1,\ldots,d_m)\) and row_wise is True, all entries of the \(i`th row are truncated at precision `d_i\) for each \(i\);
if \(d\) is a list \((d_1,\ldots,d_n)\) and row_wise is False, all entries of the \(j`th column are truncated at precision `d_j\) for each \(j\).

Here the convention for univariate polynomials is to take zero for the truncation for a negative \(d\).

INPUT:

d – a list of integers, or an integer,
row_wise – (optional, default: True) boolean, if True (resp. False) then \(d\) should be a list of length equal to the row (resp. column) dimension of this matrix.

OUTPUT: a polynomial matrix.

EXAMPLES:

sage: pR.<x> = GF(7)[]

sage: M = Matrix([
....:    [  x^3+5*x^2+5*x+1,       5,       6*x+4,         0],
....:    [      6*x^2+3*x+1,       1,           2,         0],
....:    [2*x^3+4*x^2+6*x+4, 5*x + 1, 2*x^2+5*x+5, x^2+5*x+6]
....:     ])
sage: M.truncate(2)
[5*x + 1       5 6*x + 4       0]
[3*x + 1       1       2       0]
[6*x + 4 5*x + 1 5*x + 5 5*x + 6]
sage: M.truncate(1) == M.constant_matrix()
True

Row-wise and column-wise truncation are available:

sage: M.truncate([3,2,1])
[5*x^2 + 5*x + 1               5         6*x + 4               0]
[        3*x + 1               1               2               0]
[              4               1               5               6]

sage: M.truncate([2,1,1,2], row_wise=False)
[5*x + 1       5       4       0]
[3*x + 1       1       2       0]
[6*x + 4       1       5 5*x + 6]

Length of list of truncation orders is checked:

sage: M.truncate([2,1,1,2])
Traceback (most recent call last):
...
ValueError: length of input precision list should be the row
dimension of the input matrix

sage: M.truncate([3,2,1], row_wise=False)
Traceback (most recent call last):
...
ValueError: length of input precision list should be the column
dimension of the input matrix

weak_popov_form(transformation=False, shifts=None, row_wise=True, ordered=False, include_zero_vectors=True)#

Return a (shifted) (ordered) weak Popov form of this matrix.

See is_weak_popov() for a definition of weak Popov forms. If the input matrix is \(A\), a weak Popov form of \(A\) is any matrix \(P\) in weak Popov form and such that \(UA = P\) for some unimodular matrix \(U\). The latter matrix is called the transformation, and the first optional argument allows one to specify whether to return this transformation.

Sometimes, one forbids weak Popov forms to have zero rows (resp. columns) in the above definitions; an optional parameter allows one to adopt this more restrictive setting. If zero rows (resp. columns) are allowed, the convention here is to place them as the bottommost rows (resp. the rightmost columns) of the output weak Popov form.

Note that, if asking for the transformation and discarding zero vectors (i.e. transformation=True and include_zero_vectors=False), then the returned transformation is still the complete unimodular matrix, including its bottommost rows (resp. rightmost columns) which correspond to zero rows (resp. columns) of the complete weak Popov form. In fact, this bottom part of the transformation yields a basis of the left (resp. right) kernel of the input matrix.

INPUT:

transformation – (optional, default: False). If this is True, the transformation matrix \(U\) will be returned as well.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].
row_wise – (optional, default: True) boolean, True if working row-wise (see the class description).
ordered – (optional, default: False) boolean, True if seeking an ordered weak Popov form.
include_zero_vectors – (optional, default: True) boolean, False if zero rows (resp. zero columns) should be discarded from the (ordered) weak Popov forms.

OUTPUT:

A polynomial matrix which is a weak Popov form of self if transformation is False; otherwise two polynomial matrices which are a weak Popov form of self and the corresponding unimodular transformation.

ALGORITHM:

This method implements the Mulders-Storjohann algorithm of [MS2003], straightforwardly extended to the case of shifted forms.

EXAMPLES:

sage: pR.<x> = GF(7)[]
sage: M = Matrix(pR, [
....:    [      6*x+4,       5*x^3+5*x,       6*x^2+2*x+2],
....:    [4*x^2+5*x+2, x^4+5*x^2+2*x+4, 4*x^3+6*x^2+6*x+5]])
sage: P, U = M.weak_popov_form(transformation=True)
sage: P
[              4             x^2   6*x^2 + x + 2]
[              2 4*x^2 + 2*x + 4               5]
sage: U
[2*x^2 + 1       4*x]
[      4*x         1]
sage: P.is_weak_popov() and U.is_invertible() and U*M == P
True

Demonstrating the ordered option:

sage: P.leading_positions()
[2, 1]
sage: PP = M.weak_popov_form(ordered=True); PP
[              2 4*x^2 + 2*x + 4               5]
[              4             x^2   6*x^2 + x + 2]
sage: PP.leading_positions()
[1, 2]

Demonstrating shifts:

sage: P = M.weak_popov_form(shifts=[0,2,4]); P
[            6*x^2 + 6*x + 4 5*x^4 + 4*x^3 + 5*x^2 + 5*x                     2*x + 2]
[                          2             4*x^2 + 2*x + 4                           5]
sage: P == M.weak_popov_form(shifts=[-10,-8,-6])
True

Column-wise form is the row-wise form of the transpose:

sage: M.weak_popov_form() == M.T.weak_popov_form(row_wise=False).T
True

Zero vectors can be discarded:

sage: M.weak_popov_form(row_wise=False)
[x + 4     6     0]
[    5     1     0]

sage: # needs sage.combinat
sage: P, U = M.weak_popov_form(transformation=True,
....:                          row_wise=False,
....:                          include_zero_vectors=False)
sage: P
[x + 4     6]
[    5     1]
sage: U
[                5*x + 2         5*x^2 + 4*x + 4 3*x^3 + 3*x^2 + 2*x + 4]
[                      1                       1                 2*x + 1]
[                5*x + 5                       2                       6]
sage: M*U[:,:2] == P and (M*U[:,2]).is_zero()
True