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

class sage.matrix.matrix_polynomial_dense.Matrix_polynomial_dense¶

Bases: sage.matrix.matrix_generic_dense.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 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. 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 the description of leading_matrix()). There is a unique minimal basis, called the Popov basis of the module, which satisfies some additional normalization condition (see the description of row_degrees()).

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 $1 \leq j \leq n$. Then, minimal bases and Popov bases are defined similarly, with respect to this notion of degree.

Another important canonical basis is the Hermite basis, which is a lower 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 $(0,d,2d,\ldots,(n-1)d)$.

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().

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.

The Hermite form is also normalized, i.e., the pivot polynomials are monic.

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() .

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 $(0,d,2d,\ldots,(n-1)d)$, 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 True if and only if 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(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: (x^8*Matrix.identity(pR, 5)).is_minimal_approximant_basis(\
                                                        pmat, 8)
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: Matrix(pR, [x^8]).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 True if and only if 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).

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).
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

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]

reduced_form(transformation=None, shifts=None, row_wise=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.

A row reduced form is non-canonical and a given matrix has many row reduced forms; this method returns just one.

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).

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
[0]
[0]
[x]

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: 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

See also

is_reduced() , weak_popov_form() .

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

weak_popov_form(transformation=False, shifts=None)¶

Return a (row-wise) weak Popov form of this matrix.

A polynomial matrix is said to be in (row-wise) weak Popov form if the (shifted) leading positions of its nonzero rows are pairwise distinct. The leading position of a row is the right-most position whose entry has the maximal degree in the row, see leading_positions(). See the class description for an introduction to shifts.

The weak Popov form is non-canonical, so an input matrix have many weak Popov forms (for any given shifts).

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 $UW$, where $W$ is the output matrix.
shifts – (optional, default: None) list of integers; None is interpreted as shifts=[0,...,0].

OUTPUT:

A polynomial matrix $W$ which is a weak Popov form of self if transformation=False; otherwise two polynomial matrices $W, U$ such that $UA = W$ and $W$ is in weak Popov form and $U$ is unimodular where $A$ is self.

ALGORITHM:

This method implements the Mulders-Storjohann algorithm of [MS2003].

EXAMPLES:

sage: PF.<x> = GF(11)[]
sage: A = matrix(PF,[[1,  3*x^9 + x^2 + 1 ],
....:                [0,         x^11 + x ]])
sage: W, U = A.weak_popov_form(transformation=True); W
[              8*x^7 + 3*x^5 + 1    9*x^6 + 3*x^5 + 2*x^4 + x^2 + 1]
[                          7*x^2                  7*x^4 + 7*x^2 + x]
sage: U * A == W
True
sage: W.is_weak_popov()
True
sage: U.is_invertible()
True

Demonstrating shifts:

sage: A.weak_popov_form(shifts=[2, 0])
[              8*x^7 + 1 8*x^7 + 9*x^6 + x^2 + 1]
[                  7*x^2       7*x^4 + 7*x^2 + x]
sage: A.weak_popov_form(shifts=[10, 0]) == A
True

A zero matrix will return itself:

sage: Z = matrix(PF,2,2)
sage: Z.weak_popov_form()
[0 0]
[0 0]

Dense matrices over univariate polynomials over fields¶

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