# Base class for matrices, part 1#

For design documentation see sage.matrix.docs.

class sage.matrix.matrix1.Matrix[source]#

Bases: Matrix

augment(right, subdivide=False)[source]#

Returns a new matrix formed by appending the matrix (or vector) right on the right side of self.

INPUT:

• right – a matrix, vector or free module element, whose dimensions are compatible with self.

• subdivide – (default: False); request the resulting matrix to have a new subdivision, separating self from right.

OUTPUT:

A new matrix formed by appending right onto the right side of self. If right is a vector (or free module element) then in this context it is appropriate to consider it as a column vector. (The code first converts a vector to a 1-column matrix.)

If subdivide is True then any column subdivisions for the two matrices are preserved, and a new subdivision is added between self and right. If the row divisions are identical, then they are preserved, otherwise they are discarded. When subdivide is False there is no subdivision information in the result.

Warning

If subdivide is True then unequal row subdivisions will be discarded, since it would be ambiguous how to interpret them. If the subdivision behavior is not what you need, you can manage subdivisions yourself with methods like get_subdivisions() and subdivide(). You might also find block_matrix() or block_diagonal_matrix() useful and simpler in some instances.

EXAMPLES:

Augmenting with a matrix.

sage: A = matrix(QQ, 3, range(12))
sage: B = matrix(QQ, 3, range(9))
sage: A.augment(B)
[ 0  1  2  3  0  1  2]
[ 4  5  6  7  3  4  5]
[ 8  9 10 11  6  7  8]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(3), range(Integer(12)))
>>> B = matrix(QQ, Integer(3), range(Integer(9)))
>>> A.augment(B)
[ 0  1  2  3  0  1  2]
[ 4  5  6  7  3  4  5]
[ 8  9 10 11  6  7  8]


Augmenting with a vector.

sage: A = matrix(QQ, 2, [0, 2, 4, 6, 8, 10])
sage: v = vector(QQ, 2, [100, 200])
sage: A.augment(v)
[  0   2   4 100]
[  6   8  10 200]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), [Integer(0), Integer(2), Integer(4), Integer(6), Integer(8), Integer(10)])
>>> v = vector(QQ, Integer(2), [Integer(100), Integer(200)])
>>> A.augment(v)
[  0   2   4 100]
[  6   8  10 200]


Errors are raised if the sizes are incompatible.

sage: A = matrix(RR, [[1, 2],[3, 4]])
sage: B = matrix(RR, [[10, 20], [30, 40], [50, 60]])
sage: A.augment(B)
Traceback (most recent call last):
...
TypeError: number of rows must be the same, 2 != 3

sage: v = vector(RR, [100, 200, 300])
sage: A.augment(v)
Traceback (most recent call last):
...
TypeError: number of rows must be the same, 2 != 3

>>> from sage.all import *
>>> A = matrix(RR, [[Integer(1), Integer(2)],[Integer(3), Integer(4)]])
>>> B = matrix(RR, [[Integer(10), Integer(20)], [Integer(30), Integer(40)], [Integer(50), Integer(60)]])
>>> A.augment(B)
Traceback (most recent call last):
...
TypeError: number of rows must be the same, 2 != 3

>>> v = vector(RR, [Integer(100), Integer(200), Integer(300)])
>>> A.augment(v)
Traceback (most recent call last):
...
TypeError: number of rows must be the same, 2 != 3


Setting subdivide to True will, in its simplest form, add a subdivision between self and right.

sage: A = matrix(QQ, 3, range(12))
sage: B = matrix(QQ, 3, range(15))
sage: A.augment(B, subdivide=True)
[ 0  1  2  3| 0  1  2  3  4]
[ 4  5  6  7| 5  6  7  8  9]
[ 8  9 10 11|10 11 12 13 14]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(3), range(Integer(12)))
>>> B = matrix(QQ, Integer(3), range(Integer(15)))
>>> A.augment(B, subdivide=True)
[ 0  1  2  3| 0  1  2  3  4]
[ 4  5  6  7| 5  6  7  8  9]
[ 8  9 10 11|10 11 12 13 14]


Column subdivisions are preserved by augmentation, and enriched, if subdivisions are requested. (So multiple augmentations can be recorded.)

sage: A = matrix(QQ, 3, range(6))
sage: A.subdivide(None, [1])
sage: B = matrix(QQ, 3, range(9))
sage: B.subdivide(None, [2])
sage: A.augment(B, subdivide=True)
[0|1|0 1|2]
[2|3|3 4|5]
[4|5|6 7|8]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(3), range(Integer(6)))
>>> A.subdivide(None, [Integer(1)])
>>> B = matrix(QQ, Integer(3), range(Integer(9)))
>>> B.subdivide(None, [Integer(2)])
>>> A.augment(B, subdivide=True)
[0|1|0 1|2]
[2|3|3 4|5]
[4|5|6 7|8]


Row subdivisions can be preserved, but only if they are identical. Otherwise, this information is discarded and must be managed separately.

sage: A = matrix(QQ, 3, range(6))
sage: A.subdivide([1,3], None)
sage: B = matrix(QQ, 3, range(9))
sage: B.subdivide([1,3], None)
sage: A.augment(B, subdivide=True)
[0 1|0 1 2]
[---+-----]
[2 3|3 4 5]
[4 5|6 7 8]
[---+-----]

sage: A.subdivide([1,2], None)
sage: A.augment(B, subdivide=True)
[0 1|0 1 2]
[2 3|3 4 5]
[4 5|6 7 8]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(3), range(Integer(6)))
>>> A.subdivide([Integer(1),Integer(3)], None)
>>> B = matrix(QQ, Integer(3), range(Integer(9)))
>>> B.subdivide([Integer(1),Integer(3)], None)
>>> A.augment(B, subdivide=True)
[0 1|0 1 2]
[---+-----]
[2 3|3 4 5]
[4 5|6 7 8]
[---+-----]

>>> A.subdivide([Integer(1),Integer(2)], None)
>>> A.augment(B, subdivide=True)
[0 1|0 1 2]
[2 3|3 4 5]
[4 5|6 7 8]


The result retains the base ring of self by coercing the elements of right into the base ring of self.

sage: A = matrix(QQ, 2, [1,2])
sage: B = matrix(RR, 2, [sin(1.1), sin(2.2)])
sage: C = A.augment(B); C                                                   # needs sage.symbolic
[                  1 183017397/205358938]
[                  2 106580492/131825561]
sage: C.parent()                                                            # needs sage.symbolic
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

sage: D = B.augment(A); D
[0.89120736006...  1.00000000000000]
[0.80849640381...  2.00000000000000]
sage: D.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), [Integer(1),Integer(2)])
>>> B = matrix(RR, Integer(2), [sin(RealNumber('1.1')), sin(RealNumber('2.2'))])
>>> C = A.augment(B); C                                                   # needs sage.symbolic
[                  1 183017397/205358938]
[                  2 106580492/131825561]
>>> C.parent()                                                            # needs sage.symbolic
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> D = B.augment(A); D
[0.89120736006...  1.00000000000000]
[0.80849640381...  2.00000000000000]
>>> D.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision


Sometimes it is not possible to coerce into the base ring of self. A solution is to change the base ring of self to a more expansive ring. Here we mix the rationals with a ring of polynomials with rational coefficients.

sage: R.<y> = PolynomialRing(QQ)
sage: A = matrix(QQ, 1, [1,2])
sage: B = matrix(R, 1, [y, y^2])

sage: C = B.augment(A); C
[  y y^2   1   2]
sage: C.parent()
Full MatrixSpace of 1 by 4 dense matrices over
Univariate Polynomial Ring in y over Rational Field

sage: D = A.augment(B)
Traceback (most recent call last):
...
TypeError: y is not a constant polynomial

sage: E = A.change_ring(R)
sage: F = E.augment(B); F
[  1   2   y y^2]
sage: F.parent()
Full MatrixSpace of 1 by 4 dense matrices over
Univariate Polynomial Ring in y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('y',)); (y,) = R._first_ngens(1)
>>> A = matrix(QQ, Integer(1), [Integer(1),Integer(2)])
>>> B = matrix(R, Integer(1), [y, y**Integer(2)])

>>> C = B.augment(A); C
[  y y^2   1   2]
>>> C.parent()
Full MatrixSpace of 1 by 4 dense matrices over
Univariate Polynomial Ring in y over Rational Field

>>> D = A.augment(B)
Traceback (most recent call last):
...
TypeError: y is not a constant polynomial

>>> E = A.change_ring(R)
>>> F = E.augment(B); F
[  1   2   y y^2]
>>> F.parent()
Full MatrixSpace of 1 by 4 dense matrices over
Univariate Polynomial Ring in y over Rational Field


AUTHORS:

• Naqi Jaffery (2006-01-24): examples

• Rob Beezer (2010-12-07): vector argument, docstring, subdivisions

block_sum(other)[source]#

Return the block matrix that has self and other on the diagonal:

[ self     0 ]
[    0 other ]


EXAMPLES:

sage: A = matrix(QQ[['t']], 2, range(1, 5))
sage: A.block_sum(100*A)
[  1   2   0   0]
[  3   4   0   0]
[  0   0 100 200]
[  0   0 300 400]

>>> from sage.all import *
>>> A = matrix(QQ[['t']], Integer(2), range(Integer(1), Integer(5)))
>>> A.block_sum(Integer(100)*A)
[  1   2   0   0]
[  3   4   0   0]
[  0   0 100 200]
[  0   0 300 400]

column(i, from_list=False)[source]#

Return the i’th column of this matrix as a vector.

This column is a dense vector if and only if the matrix is a dense matrix.

INPUT:

• i – integer

• from_list – bool (default: False); if true, returns the i’th element of self.columns() (see columns()), which may be faster, but requires building a list of all columns the first time it is called after an entry of the matrix is changed.

EXAMPLES:

sage: a = matrix(2, 3, range(6)); a
[0 1 2]
[3 4 5]
sage: a.column(1)
(1, 4)

>>> from sage.all import *
>>> a = matrix(Integer(2), Integer(3), range(Integer(6))); a
[0 1 2]
[3 4 5]
>>> a.column(Integer(1))
(1, 4)


If the column is negative, it wraps around, just like with list indexing, e.g., -1 gives the right-most column:

sage: a.column(-1)
(2, 5)

>>> from sage.all import *
>>> a.column(-Integer(1))
(2, 5)

column_ambient_module(base_ring=None, sparse=None)[source]#

Return the free module that contains the columns of the matrix.

EXAMPLES:

sage: M = matrix(Zmod(5), 2, 3)
sage: M.column_ambient_module()
Vector space of dimension 2 over Ring of integers modulo 5
sage: M.column(1).parent() == M.column_ambient_module()
True

sage: M = Matrix(ZZ, 3, 4)
sage: M.column_ambient_module()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
sage: M.column_ambient_module(QQ)
Vector space of dimension 3 over Rational Field

sage: M = Matrix(QQ, 4, 5)
sage: M.column_ambient_module()
Vector space of dimension 4 over Rational Field
sage: M.column_ambient_module(ZZ)
Ambient free module of rank 4 over the principal ideal domain Integer Ring

>>> from sage.all import *
>>> M = matrix(Zmod(Integer(5)), Integer(2), Integer(3))
>>> M.column_ambient_module()
Vector space of dimension 2 over Ring of integers modulo 5
>>> M.column(Integer(1)).parent() == M.column_ambient_module()
True

>>> M = Matrix(ZZ, Integer(3), Integer(4))
>>> M.column_ambient_module()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
>>> M.column_ambient_module(QQ)
Vector space of dimension 3 over Rational Field

>>> M = Matrix(QQ, Integer(4), Integer(5))
>>> M.column_ambient_module()
Vector space of dimension 4 over Rational Field
>>> M.column_ambient_module(ZZ)
Ambient free module of rank 4 over the principal ideal domain Integer Ring

columns(copy=True)[source]#

Return a list of the columns of self.

INPUT:

• copy – (default: True) if True, return a copy of the list of columns which is safe to change.

If self is a sparse matrix, columns are returned as sparse vectors, otherwise returned vectors are dense.

EXAMPLES:

sage: matrix(3, [1..9]).columns()
[(1, 4, 7), (2, 5, 8), (3, 6, 9)]
sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).columns()                     # needs sage.symbolic
[(1.41421356237310, 2.71828182845905), (3.14159265358979, 0.000000000000000)]
sage: matrix(RR, 0, 2, []).columns()
[(), ()]
sage: matrix(RR, 2, 0, []).columns()
[]
sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)})         # needs sage.symbolic
sage: parent(m.columns()[0])                                                # needs sage.symbolic
Sparse vector space of dimension 3 over Real Field with 53 bits of precision

>>> from sage.all import *
>>> matrix(Integer(3), (ellipsis_range(Integer(1),Ellipsis,Integer(9)))).columns()
[(1, 4, 7), (2, 5, 8), (3, 6, 9)]
>>> matrix(RR, Integer(2), [sqrt(Integer(2)), pi, exp(Integer(1)), Integer(0)]).columns()                     # needs sage.symbolic
[(1.41421356237310, 2.71828182845905), (3.14159265358979, 0.000000000000000)]
>>> matrix(RR, Integer(0), Integer(2), []).columns()
[(), ()]
>>> matrix(RR, Integer(2), Integer(0), []).columns()
[]
>>> m = matrix(RR, Integer(3), Integer(3), {(Integer(1),Integer(2)): pi, (Integer(2), Integer(2)): -Integer(1), (Integer(0),Integer(1)): sqrt(Integer(2))})         # needs sage.symbolic
>>> parent(m.columns()[Integer(0)])                                                # needs sage.symbolic
Sparse vector space of dimension 3 over Real Field with 53 bits of precision


Sparse matrices produce sparse columns.

sage: A = matrix(QQ, 2, range(4), sparse=True)
sage: v = A.columns()[0]
sage: v.is_sparse()
True

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), range(Integer(4)), sparse=True)
>>> v = A.columns()[Integer(0)]
>>> v.is_sparse()
True

delete_columns(dcols, check=True)[source]#

Return the matrix constructed from deleting the columns with indices in the dcols list.

INPUT:

• dcols – list of indices of columns to be deleted from self.

• check – checks whether any index in dcols is out of range. Defaults to True.

EXAMPLES:

sage: A = Matrix(3, 4, range(12)); A
[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
sage: A.delete_columns([0,2])
[ 1  3]
[ 5  7]
[ 9 11]

>>> from sage.all import *
>>> A = Matrix(Integer(3), Integer(4), range(Integer(12))); A
[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
>>> A.delete_columns([Integer(0),Integer(2)])
[ 1  3]
[ 5  7]
[ 9 11]


dcols can be a tuple. But only the underlying set of indices matters.

sage: A.delete_columns((2,0,2))
[ 1  3]
[ 5  7]
[ 9 11]

>>> from sage.all import *
>>> A.delete_columns((Integer(2),Integer(0),Integer(2)))
[ 1  3]
[ 5  7]
[ 9 11]


The default is to check whether any index in dcols is out of range.

sage: A.delete_columns([-1,2,4])
Traceback (most recent call last):
...
IndexError: [-1, 4] contains invalid indices
sage: A.delete_columns([-1,2,4], check=False)
[ 0  1  3]
[ 4  5  7]
[ 8  9 11]

>>> from sage.all import *
>>> A.delete_columns([-Integer(1),Integer(2),Integer(4)])
Traceback (most recent call last):
...
IndexError: [-1, 4] contains invalid indices
>>> A.delete_columns([-Integer(1),Integer(2),Integer(4)], check=False)
[ 0  1  3]
[ 4  5  7]
[ 8  9 11]


AUTHORS:

• Wai Yan Pong (2012-03-05)

delete_rows(drows, check=True)[source]#

Return the matrix constructed from deleting the rows with indices in the drows list.

INPUT:

• drows – list of indices of rows to be deleted from self.

• check – (boolean, default: True); whether to check if any index in drows is out of range.

EXAMPLES:

sage: A = Matrix(4, 3, range(12)); A
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]
[ 9 10 11]
sage: A.delete_rows([0,2])
[ 3  4  5]
[ 9 10 11]

>>> from sage.all import *
>>> A = Matrix(Integer(4), Integer(3), range(Integer(12))); A
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]
[ 9 10 11]
>>> A.delete_rows([Integer(0),Integer(2)])
[ 3  4  5]
[ 9 10 11]


drows can be a tuple. But only the underlying set of indices matters.

sage: A.delete_rows((2,0,2))
[ 3  4  5]
[ 9 10 11]

>>> from sage.all import *
>>> A.delete_rows((Integer(2),Integer(0),Integer(2)))
[ 3  4  5]
[ 9 10 11]


The default is to check whether the any index in drows is out of range.

sage: A.delete_rows([-1,2,4])
Traceback (most recent call last):
...
IndexError: [-1, 4] contains invalid indices
sage: A.delete_rows([-1,2,4], check=False)
[ 0  1  2]
[ 3  4  5]
[ 9 10 11]

>>> from sage.all import *
>>> A.delete_rows([-Integer(1),Integer(2),Integer(4)])
Traceback (most recent call last):
...
IndexError: [-1, 4] contains invalid indices
>>> A.delete_rows([-Integer(1),Integer(2),Integer(4)], check=False)
[ 0  1  2]
[ 3  4  5]
[ 9 10 11]

dense_columns(copy=True)[source]#

Return list of the dense columns of self.

INPUT:

• copy – (default: True) if True, return a copy so you can modify it safely

EXAMPLES:

An example over the integers:

sage: a = matrix(3, 3, range(9)); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: a.dense_columns()
[(0, 3, 6), (1, 4, 7), (2, 5, 8)]

>>> from sage.all import *
>>> a = matrix(Integer(3), Integer(3), range(Integer(9))); a
[0 1 2]
[3 4 5]
[6 7 8]
>>> a.dense_columns()
[(0, 3, 6), (1, 4, 7), (2, 5, 8)]


We do an example over a polynomial ring:

sage: R.<x> = QQ[]
sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5]); a
[      x     x^2]
[  2/3*x x^5 + 1]
sage: a.dense_columns()
[(x, 2/3*x), (x^2, x^5 + 1)]
sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5], sparse=True)
sage: c = a.dense_columns(); c
[(x, 2/3*x), (x^2, x^5 + 1)]
sage: parent(c[1])
Ambient free module of rank 2 over the principal ideal domain
Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> a = matrix(R, Integer(2), [x,x**Integer(2), Integer(2)/Integer(3)*x,Integer(1)+x**Integer(5)]); a
[      x     x^2]
[  2/3*x x^5 + 1]
>>> a.dense_columns()
[(x, 2/3*x), (x^2, x^5 + 1)]
>>> a = matrix(R, Integer(2), [x,x**Integer(2), Integer(2)/Integer(3)*x,Integer(1)+x**Integer(5)], sparse=True)
>>> c = a.dense_columns(); c
[(x, 2/3*x), (x^2, x^5 + 1)]
>>> parent(c[Integer(1)])
Ambient free module of rank 2 over the principal ideal domain
Univariate Polynomial Ring in x over Rational Field

dense_matrix()[source]#

If this matrix is sparse, return a dense matrix with the same entries. If this matrix is dense, return this matrix (not a copy).

Note

The definition of “dense” and “sparse” in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.

EXAMPLES:

sage: A = MatrixSpace(QQ,2, sparse=True)([1,2,0,1])
sage: A.is_sparse()
True
sage: B = A.dense_matrix()
sage: B.is_sparse()
False
sage: A == B
True
sage: B.dense_matrix() is B
True
sage: A*B
[1 4]
[0 1]
sage: A.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> from sage.all import *
>>> A = MatrixSpace(QQ,Integer(2), sparse=True)([Integer(1),Integer(2),Integer(0),Integer(1)])
>>> A.is_sparse()
True
>>> B = A.dense_matrix()
>>> B.is_sparse()
False
>>> A == B
True
>>> B.dense_matrix() is B
True
>>> A*B
[1 4]
[0 1]
>>> A.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
>>> B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field


In Sage, the product of a sparse and a dense matrix is always dense:

sage: (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> from sage.all import *
>>> (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
>>> (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

dense_rows(copy=True)[source]#

Return list of the dense rows of self.

INPUT:

• copy – (default: True) if True, return a copy so you can modify it safely (note that the individual vectors in the copy should not be modified since they are mutable!)

EXAMPLES:

sage: m = matrix(3, range(9)); m
[0 1 2]
[3 4 5]
[6 7 8]
sage: v = m.dense_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
sage: v is m.dense_rows()
False
sage: m.dense_rows(copy=False) is m.dense_rows(copy=False)
True
sage: m[0,0] = 10
sage: m.dense_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]

>>> from sage.all import *
>>> m = matrix(Integer(3), range(Integer(9))); m
[0 1 2]
[3 4 5]
[6 7 8]
>>> v = m.dense_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
>>> v is m.dense_rows()
False
>>> m.dense_rows(copy=False) is m.dense_rows(copy=False)
True
>>> m[Integer(0),Integer(0)] = Integer(10)
>>> m.dense_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]

lift()[source]#

Return lift of self to the covering ring of the base ring R, which is by definition the ring returned by calling cover_ring() on R, or just R itself if the cover_ring method is not defined.

EXAMPLES:

sage: M = Matrix(Integers(7), 2, 2, [5, 9, 13, 15]); M
[5 2]
[6 1]
sage: M.lift()
[5 2]
[6 1]
sage: parent(M.lift())
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

>>> from sage.all import *
>>> M = Matrix(Integers(Integer(7)), Integer(2), Integer(2), [Integer(5), Integer(9), Integer(13), Integer(15)]); M
[5 2]
[6 1]
>>> M.lift()
[5 2]
[6 1]
>>> parent(M.lift())
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring


The field QQ doesn’t have a cover_ring method:

sage: hasattr(QQ, 'cover_ring')
False

>>> from sage.all import *
>>> hasattr(QQ, 'cover_ring')
False


So lifting a matrix over QQ gives back the same exact matrix.

sage: B = matrix(QQ, 2, [1..4])
sage: B.lift()
[1 2]
[3 4]
sage: B.lift() is B
True

>>> from sage.all import *
>>> B = matrix(QQ, Integer(2), (ellipsis_range(Integer(1),Ellipsis,Integer(4))))
>>> B.lift()
[1 2]
[3 4]
>>> B.lift() is B
True

lift_centered()[source]#

Apply the lift_centered method to every entry of self.

OUTPUT:

If self is a matrix over the Integers mod $$n$$, this method returns the unique matrix $$m$$ such that $$m$$ is congruent to self mod $$n$$ and for every entry $$m[i,j]$$ we have $$-n/2 < m[i,j] \leq n/2$$. If the coefficient ring does not have a cover_ring method, return self.

EXAMPLES:

sage: M = Matrix(Integers(8), 2, 4, range(8)); M
[0 1 2 3]
[4 5 6 7]
sage: L = M.lift_centered(); L
[ 0  1  2  3]
[ 4 -3 -2 -1]
sage: parent(L)
Full MatrixSpace of 2 by 4 dense matrices over Integer Ring

>>> from sage.all import *
>>> M = Matrix(Integers(Integer(8)), Integer(2), Integer(4), range(Integer(8))); M
[0 1 2 3]
[4 5 6 7]
>>> L = M.lift_centered(); L
[ 0  1  2  3]
[ 4 -3 -2 -1]
>>> parent(L)
Full MatrixSpace of 2 by 4 dense matrices over Integer Ring


The returned matrix is congruent to M modulo 8.:

sage: L.mod(8)
[0 1 2 3]
[4 5 6 7]

>>> from sage.all import *
>>> L.mod(Integer(8))
[0 1 2 3]
[4 5 6 7]


The field QQ doesn’t have a cover_ring method:

sage: hasattr(QQ, 'cover_ring')
False

>>> from sage.all import *
>>> hasattr(QQ, 'cover_ring')
False


So lifting a matrix over QQ gives back the same exact matrix.

sage: B = matrix(QQ, 2, [1..4])
sage: B.lift_centered()
[1 2]
[3 4]
sage: B.lift_centered() is B
True

>>> from sage.all import *
>>> B = matrix(QQ, Integer(2), (ellipsis_range(Integer(1),Ellipsis,Integer(4))))
>>> B.lift_centered()
[1 2]
[3 4]
>>> B.lift_centered() is B
True

matrix_from_columns(columns)[source]#

Return the matrix constructed from self using columns with indices in the columns list.

EXAMPLES:

sage: M = MatrixSpace(Integers(8), 3, 3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_columns([2,1])
[2 1]
[5 4]
[0 7]

>>> from sage.all import *
>>> M = MatrixSpace(Integers(Integer(8)), Integer(3), Integer(3))
>>> A = M(range(Integer(9))); A
[0 1 2]
[3 4 5]
[6 7 0]
>>> A.matrix_from_columns([Integer(2),Integer(1)])
[2 1]
[5 4]
[0 7]

matrix_from_rows(rows)[source]#

Return the matrix constructed from self using rows with indices in the rows list.

EXAMPLES:

sage: M = MatrixSpace(Integers(8), 3, 3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_rows([2,1])
[6 7 0]
[3 4 5]

>>> from sage.all import *
>>> M = MatrixSpace(Integers(Integer(8)), Integer(3), Integer(3))
>>> A = M(range(Integer(9))); A
[0 1 2]
[3 4 5]
[6 7 0]
>>> A.matrix_from_rows([Integer(2),Integer(1)])
[6 7 0]
[3 4 5]

matrix_from_rows_and_columns(rows, columns)[source]#

Return the matrix constructed from self from the given rows and columns.

EXAMPLES:

sage: M = MatrixSpace(Integers(8), 3, 3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_rows_and_columns([1], [0,2])
[3 5]
sage: A.matrix_from_rows_and_columns([1,2], [1,2])
[4 5]
[7 0]

>>> from sage.all import *
>>> M = MatrixSpace(Integers(Integer(8)), Integer(3), Integer(3))
>>> A = M(range(Integer(9))); A
[0 1 2]
[3 4 5]
[6 7 0]
>>> A.matrix_from_rows_and_columns([Integer(1)], [Integer(0),Integer(2)])
[3 5]
>>> A.matrix_from_rows_and_columns([Integer(1),Integer(2)], [Integer(1),Integer(2)])
[4 5]
[7 0]


Note that row and column indices can be reordered or repeated:

sage: A.matrix_from_rows_and_columns([2,1], [2,1])
[0 7]
[5 4]

>>> from sage.all import *
>>> A.matrix_from_rows_and_columns([Integer(2),Integer(1)], [Integer(2),Integer(1)])
[0 7]
[5 4]


For example here we take from row 1 columns 2 then 0 twice, and do this 3 times:

sage: A.matrix_from_rows_and_columns([1,1,1], [2,0,0])
[5 3 3]
[5 3 3]
[5 3 3]

>>> from sage.all import *
>>> A.matrix_from_rows_and_columns([Integer(1),Integer(1),Integer(1)], [Integer(2),Integer(0),Integer(0)])
[5 3 3]
[5 3 3]
[5 3 3]


AUTHORS:

• Jaap Spies (2006-02-18)

• Didier Deshommes: some Pyrex speedups implemented

matrix_over_field()[source]#

Return copy of this matrix, but with entries viewed as elements of the fraction field of the base ring (assuming it is defined).

EXAMPLES:

sage: A = MatrixSpace(IntegerRing(),2)([1,2,3,4])
sage: B = A.matrix_over_field()
sage: B
[1 2]
[3 4]
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> from sage.all import *
>>> A = MatrixSpace(IntegerRing(),Integer(2))([Integer(1),Integer(2),Integer(3),Integer(4)])
>>> B = A.matrix_over_field()
>>> B
[1 2]
[3 4]
>>> B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

matrix_space(nrows=None, ncols=None, sparse=None)[source]#

Return the ambient matrix space of self.

INPUT:

• nrows, ncols – (optional) number of rows and columns in returned matrix space.

• sparse – whether the returned matrix space uses sparse or dense matrices.

EXAMPLES:

sage: m = matrix(3, [1..9])
sage: m.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
sage: m.matrix_space(ncols=2)
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
sage: m.matrix_space(1)
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m.matrix_space(1, 2, True)
Full MatrixSpace of 1 by 2 sparse matrices over Integer Ring

sage: M = MatrixSpace(QQ, 3, implementation='generic')
sage: m = M.an_element()
sage: m.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
(using Matrix_generic_dense)
sage: m.matrix_space(nrows=2, ncols=12)
Full MatrixSpace of 2 by 12 dense matrices over Rational Field
(using Matrix_generic_dense)
sage: m.matrix_space(nrows=2, sparse=True)
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(Integer(3), (ellipsis_range(Integer(1),Ellipsis,Integer(9))))
>>> m.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
>>> m.matrix_space(ncols=Integer(2))
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
>>> m.matrix_space(Integer(1))
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
>>> m.matrix_space(Integer(1), Integer(2), True)
Full MatrixSpace of 1 by 2 sparse matrices over Integer Ring

>>> M = MatrixSpace(QQ, Integer(3), implementation='generic')
>>> m = M.an_element()
>>> m.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
(using Matrix_generic_dense)
>>> m.matrix_space(nrows=Integer(2), ncols=Integer(12))
Full MatrixSpace of 2 by 12 dense matrices over Rational Field
(using Matrix_generic_dense)
>>> m.matrix_space(nrows=Integer(2), sparse=True)
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

new_matrix(nrows=None, ncols=None, entries=None, coerce=True, copy=True, sparse=None)[source]#

Create a matrix in the parent of this matrix with the given number of rows, columns, etc. The default parameters are the same as for self.

INPUT:

These three variables get sent to matrix_space():

• nrows, ncols – number of rows and columns in returned matrix. If not specified, defaults to None and will give a matrix of the same size as self.

• sparse – whether returned matrix is sparse or not. Defaults to same value as self.

The remaining three variables (coerce, entries, and copy) are used by sage.matrix.matrix_space.MatrixSpace() to construct the new matrix.

Warning

This function called with no arguments returns the zero matrix of the same dimension and sparseness of self.

EXAMPLES:

sage: A = matrix(ZZ,2,2,[1,2,3,4]); A
[1 2]
[3 4]
sage: A.new_matrix()
[0 0]
[0 0]
sage: A.new_matrix(1,1)
[0]
sage: A.new_matrix(3,3).parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> A = matrix(ZZ,Integer(2),Integer(2),[Integer(1),Integer(2),Integer(3),Integer(4)]); A
[1 2]
[3 4]
>>> A.new_matrix()
[0 0]
[0 0]
>>> A.new_matrix(Integer(1),Integer(1))
[0]
>>> A.new_matrix(Integer(3),Integer(3)).parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

sage: A = matrix(RR,2,3,[1.1,2.2,3.3,4.4,5.5,6.6]); A
[1.10000000000000 2.20000000000000 3.30000000000000]
[4.40000000000000 5.50000000000000 6.60000000000000]
sage: A.new_matrix()
[0.000000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000 0.000000000000000]
sage: A.new_matrix().parent()
Full MatrixSpace of 2 by 3 dense matrices
over Real Field with 53 bits of precision

>>> from sage.all import *
>>> A = matrix(RR,Integer(2),Integer(3),[RealNumber('1.1'),RealNumber('2.2'),RealNumber('3.3'),RealNumber('4.4'),RealNumber('5.5'),RealNumber('6.6')]); A
[1.10000000000000 2.20000000000000 3.30000000000000]
[4.40000000000000 5.50000000000000 6.60000000000000]
>>> A.new_matrix()
[0.000000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000 0.000000000000000]
>>> A.new_matrix().parent()
Full MatrixSpace of 2 by 3 dense matrices
over Real Field with 53 bits of precision

sage: M = MatrixSpace(ZZ, 2, 3, implementation='generic')
sage: m = M.an_element()
sage: m.new_matrix().parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
(using Matrix_generic_dense)
sage: m.new_matrix(3,3).parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
(using Matrix_generic_dense)
sage: m.new_matrix(3,3, sparse=True).parent()
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring

>>> from sage.all import *
>>> M = MatrixSpace(ZZ, Integer(2), Integer(3), implementation='generic')
>>> m = M.an_element()
>>> m.new_matrix().parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
(using Matrix_generic_dense)
>>> m.new_matrix(Integer(3),Integer(3)).parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
(using Matrix_generic_dense)
>>> m.new_matrix(Integer(3),Integer(3), sparse=True).parent()
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring

numpy(dtype=None)[source]#

Return the Numpy matrix associated to this matrix.

INPUT:

• dtype – The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence.

EXAMPLES:

sage: # needs numpy
sage: a = matrix(3, range(12))
sage: a.numpy()
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]])
sage: a.numpy('f')
array([[  0.,   1.,   2.,   3.],
[  4.,   5.,   6.,   7.],
[  8.,   9.,  10.,  11.]], dtype=float32)
sage: a.numpy('d')
array([[  0.,   1.,   2.,   3.],
[  4.,   5.,   6.,   7.],
[  8.,   9.,  10.,  11.]])
sage: a.numpy('B')
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]], dtype=uint8)

>>> from sage.all import *
>>> # needs numpy
>>> a = matrix(Integer(3), range(Integer(12)))
>>> a.numpy()
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]])
>>> a.numpy('f')
array([[  0.,   1.,   2.,   3.],
[  4.,   5.,   6.,   7.],
[  8.,   9.,  10.,  11.]], dtype=float32)
>>> a.numpy('d')
array([[  0.,   1.,   2.,   3.],
[  4.,   5.,   6.,   7.],
[  8.,   9.,  10.,  11.]])
>>> a.numpy('B')
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]], dtype=uint8)


Type numpy.typecodes for a list of the possible typecodes:

sage: import numpy                                                          # needs numpy
sage: sorted(numpy.typecodes.items())                                       # needs numpy
[('All', '?bhilqpBHILQPefdgFDGSUVOMm'), ('AllFloat', 'efdgFDG'),
('AllInteger', 'bBhHiIlLqQpP'), ('Character', 'c'), ('Complex', 'FDG'),
('Datetime', 'Mm'), ('Float', 'efdg'), ('Integer', 'bhilqp'),
('UnsignedInteger', 'BHILQP')]

>>> from sage.all import *
>>> import numpy                                                          # needs numpy
>>> sorted(numpy.typecodes.items())                                       # needs numpy
[('All', '?bhilqpBHILQPefdgFDGSUVOMm'), ('AllFloat', 'efdgFDG'),
('AllInteger', 'bBhHiIlLqQpP'), ('Character', 'c'), ('Complex', 'FDG'),
('Datetime', 'Mm'), ('Float', 'efdg'), ('Integer', 'bhilqp'),
('UnsignedInteger', 'BHILQP')]


Alternatively, numpy automatically calls this function (via the magic __array__() method) to convert Sage matrices to numpy arrays:

sage: # needs numpy
sage: import numpy
sage: b = numpy.array(a); b
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]])
sage: b.dtype
dtype('int32')  # 32-bit
dtype('int64')  # 64-bit
sage: b.shape
(3, 4)

>>> from sage.all import *
>>> # needs numpy
>>> import numpy
>>> b = numpy.array(a); b
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11]])
>>> b.dtype
dtype('int32')  # 32-bit
dtype('int64')  # 64-bit
>>> b.shape
(3, 4)

row(i, from_list=False)[source]#

Return the i’th row of this matrix as a vector.

This row is a dense vector if and only if the matrix is a dense matrix.

INPUT:

• i – integer

• from_list – bool (default: False); if true, returns the i’th element of self.rows() (see rows()), which may be faster, but requires building a list of all rows the first time it is called after an entry of the matrix is changed.

EXAMPLES:

sage: a = matrix(2, 3, range(6)); a
[0 1 2]
[3 4 5]
sage: a.row(0)
(0, 1, 2)
sage: a.row(1)
(3, 4, 5)
sage: a.row(-1)  # last row
(3, 4, 5)

>>> from sage.all import *
>>> a = matrix(Integer(2), Integer(3), range(Integer(6))); a
[0 1 2]
[3 4 5]
>>> a.row(Integer(0))
(0, 1, 2)
>>> a.row(Integer(1))
(3, 4, 5)
>>> a.row(-Integer(1))  # last row
(3, 4, 5)

row_ambient_module(base_ring=None, sparse=None)[source]#

Return the free module that contains the rows of the matrix.

EXAMPLES:

sage: M = matrix(Zmod(5), 2, 3)
sage: M.row_ambient_module()
Vector space of dimension 3 over Ring of integers modulo 5
sage: M.row(1).parent() == M.row_ambient_module()
True

sage: M = Matrix(ZZ, 3, 4)
sage: M.row_ambient_module()
Ambient free module of rank 4 over the principal ideal domain Integer Ring
sage: M.row_ambient_module(QQ)
Vector space of dimension 4 over Rational Field

sage: M = Matrix(QQ, 4, 5)
sage: M.row_ambient_module()
Vector space of dimension 5 over Rational Field
sage: M.row_ambient_module(ZZ)
Ambient free module of rank 5 over the principal ideal domain Integer Ring

>>> from sage.all import *
>>> M = matrix(Zmod(Integer(5)), Integer(2), Integer(3))
>>> M.row_ambient_module()
Vector space of dimension 3 over Ring of integers modulo 5
>>> M.row(Integer(1)).parent() == M.row_ambient_module()
True

>>> M = Matrix(ZZ, Integer(3), Integer(4))
>>> M.row_ambient_module()
Ambient free module of rank 4 over the principal ideal domain Integer Ring
>>> M.row_ambient_module(QQ)
Vector space of dimension 4 over Rational Field

>>> M = Matrix(QQ, Integer(4), Integer(5))
>>> M.row_ambient_module()
Vector space of dimension 5 over Rational Field
>>> M.row_ambient_module(ZZ)
Ambient free module of rank 5 over the principal ideal domain Integer Ring

rows(copy=True)[source]#

Return a list of the rows of self.

INPUT:

• copy – (default: True) if True, return a copy of the list of rows which is safe to change.

If self is a sparse matrix, rows are returned as sparse vectors, otherwise returned vectors are dense.

EXAMPLES:

sage: matrix(3, [1..9]).rows()
[(1, 2, 3), (4, 5, 6), (7, 8, 9)]
sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).rows()                        # needs sage.symbolic
[(1.41421356237310, 3.14159265358979), (2.71828182845905, 0.000000000000000)]
sage: matrix(RR, 0, 2, []).rows()
[]
sage: matrix(RR, 2, 0, []).rows()
[(), ()]
sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)})         # needs sage.symbolic
sage: parent(m.rows()[0])                                                   # needs sage.symbolic
Sparse vector space of dimension 3 over Real Field with 53 bits of precision

>>> from sage.all import *
>>> matrix(Integer(3), (ellipsis_range(Integer(1),Ellipsis,Integer(9)))).rows()
[(1, 2, 3), (4, 5, 6), (7, 8, 9)]
>>> matrix(RR, Integer(2), [sqrt(Integer(2)), pi, exp(Integer(1)), Integer(0)]).rows()                        # needs sage.symbolic
[(1.41421356237310, 3.14159265358979), (2.71828182845905, 0.000000000000000)]
>>> matrix(RR, Integer(0), Integer(2), []).rows()
[]
>>> matrix(RR, Integer(2), Integer(0), []).rows()
[(), ()]
>>> m = matrix(RR, Integer(3), Integer(3), {(Integer(1),Integer(2)): pi, (Integer(2), Integer(2)): -Integer(1), (Integer(0),Integer(1)): sqrt(Integer(2))})         # needs sage.symbolic
>>> parent(m.rows()[Integer(0)])                                                   # needs sage.symbolic
Sparse vector space of dimension 3 over Real Field with 53 bits of precision


Sparse matrices produce sparse rows.

sage: A = matrix(QQ, 2, range(4), sparse=True)
sage: v = A.rows()[0]
sage: v.is_sparse()
True

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), range(Integer(4)), sparse=True)
>>> v = A.rows()[Integer(0)]
>>> v.is_sparse()
True

set_column(col, v)[source]#

Sets the entries of column col to the entries of v.

INPUT:

• col – index of column to be set.

• v – a list or vector of the new entries.

OUTPUT:

Changes the matrix in-place, so there is no output.

EXAMPLES:

New entries may be contained in a vector.:

sage: A = matrix(QQ, 5, range(25))
sage: u = vector(QQ, [0, -1, -2, -3, -4])
sage: A.set_column(2, u)
sage: A
[ 0  1  0  3  4]
[ 5  6 -1  8  9]
[10 11 -2 13 14]
[15 16 -3 18 19]
[20 21 -4 23 24]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(5), range(Integer(25)))
>>> u = vector(QQ, [Integer(0), -Integer(1), -Integer(2), -Integer(3), -Integer(4)])
>>> A.set_column(Integer(2), u)
>>> A
[ 0  1  0  3  4]
[ 5  6 -1  8  9]
[10 11 -2 13 14]
[15 16 -3 18 19]
[20 21 -4 23 24]


New entries may be in any sort of list.:

sage: A = matrix([[1, 2], [3, 4]]); A
[1 2]
[3 4]
sage: A.set_column(0, [0, 0]); A
[0 2]
[0 4]
sage: A.set_column(1, (0, 0)); A
[0 0]
[0 0]

>>> from sage.all import *
>>> A = matrix([[Integer(1), Integer(2)], [Integer(3), Integer(4)]]); A
[1 2]
[3 4]
>>> A.set_column(Integer(0), [Integer(0), Integer(0)]); A
[0 2]
[0 4]
>>> A.set_column(Integer(1), (Integer(0), Integer(0))); A
[0 0]
[0 0]

set_row(row, v)[source]#

Sets the entries of row row to the entries of v.

INPUT:

• row – index of row to be set.

• v – a list or vector of the new entries.

OUTPUT:

Changes the matrix in-place, so there is no output.

EXAMPLES:

New entries may be contained in a vector.:

sage: A = matrix(QQ, 5, range(25))
sage: u = vector(QQ, [0, -1, -2, -3, -4])
sage: A.set_row(2, u)
sage: A
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[ 0 -1 -2 -3 -4]
[15 16 17 18 19]
[20 21 22 23 24]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(5), range(Integer(25)))
>>> u = vector(QQ, [Integer(0), -Integer(1), -Integer(2), -Integer(3), -Integer(4)])
>>> A.set_row(Integer(2), u)
>>> A
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[ 0 -1 -2 -3 -4]
[15 16 17 18 19]
[20 21 22 23 24]


New entries may be in any sort of list.:

sage: A = matrix([[1, 2], [3, 4]]); A
[1 2]
[3 4]
sage: A.set_row(0, [0, 0]); A
[0 0]
[3 4]
sage: A.set_row(1, (0, 0)); A
[0 0]
[0 0]

>>> from sage.all import *
>>> A = matrix([[Integer(1), Integer(2)], [Integer(3), Integer(4)]]); A
[1 2]
[3 4]
>>> A.set_row(Integer(0), [Integer(0), Integer(0)]); A
[0 0]
[3 4]
>>> A.set_row(Integer(1), (Integer(0), Integer(0))); A
[0 0]
[0 0]

sparse_columns(copy=True)[source]#

Return a list of the columns of self as sparse vectors (or free module elements).

INPUT:

• copy – (default: True) if True, return a copy so you can

modify it safely

EXAMPLES:

sage: a = matrix(2, 3, range(6)); a
[0 1 2]
[3 4 5]
sage: v = a.sparse_columns(); v
[(0, 3), (1, 4), (2, 5)]
sage: v[1].is_sparse()
True

>>> from sage.all import *
>>> a = matrix(Integer(2), Integer(3), range(Integer(6))); a
[0 1 2]
[3 4 5]
>>> v = a.sparse_columns(); v
[(0, 3), (1, 4), (2, 5)]
>>> v[Integer(1)].is_sparse()
True

sparse_matrix()[source]#

If this matrix is dense, return a sparse matrix with the same entries. If this matrix is sparse, return this matrix (not a copy).

Note

The definition of “dense” and “sparse” in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.

EXAMPLES:

sage: A = MatrixSpace(QQ,2, sparse=False)([1,2,0,1])
sage: A.is_sparse()
False
sage: B = A.sparse_matrix()
sage: B.is_sparse()
True
sage: A == B
True
sage: B.sparse_matrix() is B
True
sage: A*B
[1 4]
[0 1]
sage: A.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: B.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> from sage.all import *
>>> A = MatrixSpace(QQ,Integer(2), sparse=False)([Integer(1),Integer(2),Integer(0),Integer(1)])
>>> A.is_sparse()
False
>>> B = A.sparse_matrix()
>>> B.is_sparse()
True
>>> A == B
True
>>> B.sparse_matrix() is B
True
>>> A*B
[1 4]
[0 1]
>>> A.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
>>> B.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
>>> (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
>>> (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

sparse_rows(copy=True)[source]#

Return a list of the rows of self as sparse vectors (or free module elements).

INPUT:

• copy – (default: True) if True, return a copy so you can

modify it safely

EXAMPLES:

sage: m = Mat(ZZ, 3, 3, sparse=True)(range(9)); m
[0 1 2]
[3 4 5]
[6 7 8]
sage: v = m.sparse_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
sage: m.sparse_rows(copy=False) is m.sparse_rows(copy=False)
True
sage: v[1].is_sparse()
True
sage: m[0,0] = 10
sage: m.sparse_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]

>>> from sage.all import *
>>> m = Mat(ZZ, Integer(3), Integer(3), sparse=True)(range(Integer(9))); m
[0 1 2]
[3 4 5]
[6 7 8]
>>> v = m.sparse_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
>>> m.sparse_rows(copy=False) is m.sparse_rows(copy=False)
True
>>> v[Integer(1)].is_sparse()
True
>>> m[Integer(0),Integer(0)] = Integer(10)
>>> m.sparse_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]

stack(bottom, subdivide=False)[source]#

Return a new matrix formed by appending the matrix (or vector) bottom below self:

[  self  ]
[ bottom ]


INPUT:

• bottom – a matrix, vector or free module element, whose dimensions are compatible with self.

• subdivide – (default: False); request the resulting matrix to have a new subdivision, separating self from bottom.

OUTPUT:

A new matrix formed by appending bottom beneath self. If bottom is a vector (or free module element) then in this context it is appropriate to consider it as a row vector. (The code first converts a vector to a 1-row matrix.)

If subdivide is True then any row subdivisions for the two matrices are preserved, and a new subdivision is added between self and bottom. If the column divisions are identical, then they are preserved, otherwise they are discarded. When subdivide is False there is no subdivision information in the result.

Warning

If subdivide is True then unequal column subdivisions will be discarded, since it would be ambiguous how to interpret them. If the subdivision behavior is not what you need, you can manage subdivisions yourself with methods like subdivisions() and subdivide(). You might also find block_matrix() or block_diagonal_matrix() useful and simpler in some instances.

EXAMPLES:

Stacking with a matrix.

sage: A = matrix(QQ, 4, 3, range(12))
sage: B = matrix(QQ, 3, 3, range(9))
sage: A.stack(B)
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]
[ 9 10 11]
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(4), Integer(3), range(Integer(12)))
>>> B = matrix(QQ, Integer(3), Integer(3), range(Integer(9)))
>>> A.stack(B)
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]
[ 9 10 11]
[ 0  1  2]
[ 3  4  5]
[ 6  7  8]


Stacking with a vector.

sage: A = matrix(QQ, 3, 2, [0, 2, 4, 6, 8, 10])
sage: v = vector(QQ, 2, [100, 200])
sage: A.stack(v)
[  0   2]
[  4   6]
[  8  10]
[100 200]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(3), Integer(2), [Integer(0), Integer(2), Integer(4), Integer(6), Integer(8), Integer(10)])
>>> v = vector(QQ, Integer(2), [Integer(100), Integer(200)])
>>> A.stack(v)
[  0   2]
[  4   6]
[  8  10]
[100 200]


Errors are raised if the sizes are incompatible.

sage: A = matrix(RR, [[1, 2],[3, 4]])
sage: B = matrix(RR, [[10, 20, 30], [40, 50, 60]])
sage: A.stack(B)
Traceback (most recent call last):
...
TypeError: number of columns must be the same, not 2 and 3

sage: v = vector(RR, [100, 200, 300])
sage: A.stack(v)
Traceback (most recent call last):
...
TypeError: number of columns must be the same, not 2 and 3

>>> from sage.all import *
>>> A = matrix(RR, [[Integer(1), Integer(2)],[Integer(3), Integer(4)]])
>>> B = matrix(RR, [[Integer(10), Integer(20), Integer(30)], [Integer(40), Integer(50), Integer(60)]])
>>> A.stack(B)
Traceback (most recent call last):
...
TypeError: number of columns must be the same, not 2 and 3

>>> v = vector(RR, [Integer(100), Integer(200), Integer(300)])
>>> A.stack(v)
Traceback (most recent call last):
...
TypeError: number of columns must be the same, not 2 and 3


Setting subdivide to True will, in its simplest form, add a subdivision between self and bottom.

sage: A = matrix(QQ, 2, 5, range(10))
sage: B = matrix(QQ, 3, 5, range(15))
sage: A.stack(B, subdivide=True)
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[--------------]
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), Integer(5), range(Integer(10)))
>>> B = matrix(QQ, Integer(3), Integer(5), range(Integer(15)))
>>> A.stack(B, subdivide=True)
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[--------------]
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]


Row subdivisions are preserved by stacking, and enriched, if subdivisions are requested. (So multiple stackings can be recorded.)

sage: A = matrix(QQ, 2, 4, range(8))
sage: A.subdivide([1], None)
sage: B = matrix(QQ, 3, 4, range(12))
sage: B.subdivide([2], None)
sage: A.stack(B, subdivide=True)
[ 0  1  2  3]
[-----------]
[ 4  5  6  7]
[-----------]
[ 0  1  2  3]
[ 4  5  6  7]
[-----------]
[ 8  9 10 11]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), Integer(4), range(Integer(8)))
>>> A.subdivide([Integer(1)], None)
>>> B = matrix(QQ, Integer(3), Integer(4), range(Integer(12)))
>>> B.subdivide([Integer(2)], None)
>>> A.stack(B, subdivide=True)
[ 0  1  2  3]
[-----------]
[ 4  5  6  7]
[-----------]
[ 0  1  2  3]
[ 4  5  6  7]
[-----------]
[ 8  9 10 11]


Column subdivisions can be preserved, but only if they are identical. Otherwise, this information is discarded and must be managed separately.

sage: A = matrix(QQ, 2, 5, range(10))
sage: A.subdivide(None, [2,4])
sage: B = matrix(QQ, 3, 5, range(15))
sage: B.subdivide(None, [2,4])
sage: A.stack(B, subdivide=True)
[ 0  1| 2  3| 4]
[ 5  6| 7  8| 9]
[-----+-----+--]
[ 0  1| 2  3| 4]
[ 5  6| 7  8| 9]
[10 11|12 13|14]

sage: A.subdivide(None, [1,2])
sage: A.stack(B, subdivide=True)
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[--------------]
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]

>>> from sage.all import *
>>> A = matrix(QQ, Integer(2), Integer(5), range(Integer(10)))
>>> A.subdivide(None, [Integer(2),Integer(4)])
>>> B = matrix(QQ, Integer(3), Integer(5), range(Integer(15)))
>>> B.subdivide(None, [Integer(2),Integer(4)])
>>> A.stack(B, subdivide=True)
[ 0  1| 2  3| 4]
[ 5  6| 7  8| 9]
[-----+-----+--]
[ 0  1| 2  3| 4]
[ 5  6| 7  8| 9]
[10 11|12 13|14]

>>> A.subdivide(None, [Integer(1),Integer(2)])
>>> A.stack(B, subdivide=True)
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[--------------]
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]


The base ring of the result is the common parent for the base rings of self and bottom. In particular, the parent for A.stack(B) and B.stack(A) should be equal:

sage: A = matrix(QQ, 1, 2, [1,2])
sage: B = matrix(RR, 1, 2, [sin(1.1), sin(2.2)])
sage: C = A.stack(B); C
[ 1.00000000000000  2.00000000000000]
[0.891207360061435 0.808496403819590]
sage: C.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision

sage: D = B.stack(A); D
[0.891207360061435 0.808496403819590]
[ 1.00000000000000  2.00000000000000]
sage: D.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision

>>> from sage.all import *
>>> A = matrix(QQ, Integer(1), Integer(2), [Integer(1),Integer(2)])
>>> B = matrix(RR, Integer(1), Integer(2), [sin(RealNumber('1.1')), sin(RealNumber('2.2'))])
>>> C = A.stack(B); C
[ 1.00000000000000  2.00000000000000]
[0.891207360061435 0.808496403819590]
>>> C.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision

>>> D = B.stack(A); D
[0.891207360061435 0.808496403819590]
[ 1.00000000000000  2.00000000000000]
>>> D.parent()
Full MatrixSpace of 2 by 2 dense matrices
over Real Field with 53 bits of precision

sage: R.<y> = PolynomialRing(ZZ)
sage: A = matrix(QQ, 1, 2, [1, 2/3])
sage: B = matrix(R, 1, 2, [y, y^2])

sage: C = A.stack(B); C
[  1 2/3]
[  y y^2]
sage: C.parent()
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, names=('y',)); (y,) = R._first_ngens(1)
>>> A = matrix(QQ, Integer(1), Integer(2), [Integer(1), Integer(2)/Integer(3)])
>>> B = matrix(R, Integer(1), Integer(2), [y, y**Integer(2)])

>>> C = A.stack(B); C
[  1 2/3]
[  y y^2]
>>> C.parent()
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Rational Field


Stacking a dense matrix atop a sparse one returns a sparse matrix:

sage: M = Matrix(ZZ, 2, 3, range(6), sparse=False)
sage: N = diagonal_matrix([10,11,12], sparse=True)
sage: P = M.stack(N); P
[ 0  1  2]
[ 3  4  5]
[10  0  0]
[ 0 11  0]
[ 0  0 12]
sage: P.is_sparse()
True
sage: P = N.stack(M); P
[10  0  0]
[ 0 11  0]
[ 0  0 12]
[ 0  1  2]
[ 3  4  5]
sage: P.is_sparse()
True

>>> from sage.all import *
>>> M = Matrix(ZZ, Integer(2), Integer(3), range(Integer(6)), sparse=False)
>>> N = diagonal_matrix([Integer(10),Integer(11),Integer(12)], sparse=True)
>>> P = M.stack(N); P
[ 0  1  2]
[ 3  4  5]
[10  0  0]
[ 0 11  0]
[ 0  0 12]
>>> P.is_sparse()
True
>>> P = N.stack(M); P
[10  0  0]
[ 0 11  0]
[ 0  0 12]
[ 0  1  2]
[ 3  4  5]
>>> P.is_sparse()
True


One can stack matrices over different rings (Issue #16399).

sage: M = Matrix(ZZ, 2, 3, range(6))
sage: N = Matrix(QQ, 1, 3, [10,11,12])
sage: M.stack(N)
[ 0  1  2]
[ 3  4  5]
[10 11 12]
sage: N.stack(M)
[10 11 12]
[ 0  1  2]
[ 3  4  5]

>>> from sage.all import *
>>> M = Matrix(ZZ, Integer(2), Integer(3), range(Integer(6)))
>>> N = Matrix(QQ, Integer(1), Integer(3), [Integer(10),Integer(11),Integer(12)])
>>> M.stack(N)
[ 0  1  2]
[ 3  4  5]
[10 11 12]
>>> N.stack(M)
[10 11 12]
[ 0  1  2]
[ 3  4  5]


AUTHORS:

• Rob Beezer (2011-03-19): rewritten to mirror code for augment()

• Jeroen Demeyer (2015-01-06): refactor, see Issue #16399. Put all boilerplate in one place (here) and put the actual type-dependent implementation in _stack_impl.

submatrix(row=0, col=0, nrows=-1, ncols=-1)[source]#

Return the matrix constructed from self using the specified range of rows and columns.

INPUT:

• row, col – index of the starting row and column. Indices start at zero.

• nrows, ncols – (optional) number of rows and columns to take. If not provided, take all rows below and all columns to the right of the starting entry.

The functions matrix_from_rows(), matrix_from_columns(), and matrix_from_rows_and_columns() allow one to select arbitrary subsets of rows and/or columns.

EXAMPLES:

Take the $$3 \times 3$$ submatrix starting from entry (1,1) in a $$4 \times 4$$ matrix:

sage: m = matrix(4, [1..16])
sage: m.submatrix(1, 1)
[ 6  7  8]
[10 11 12]
[14 15 16]

>>> from sage.all import *
>>> m = matrix(Integer(4), (ellipsis_range(Integer(1),Ellipsis,Integer(16))))
>>> m.submatrix(Integer(1), Integer(1))
[ 6  7  8]
[10 11 12]
[14 15 16]


Same thing, except take only two rows:

sage: m.submatrix(1, 1, 2)
[ 6  7  8]
[10 11 12]

>>> from sage.all import *
>>> m.submatrix(Integer(1), Integer(1), Integer(2))
[ 6  7  8]
[10 11 12]


And now take only one column:

sage: m.submatrix(1, 1, 2, 1)
[ 6]
[10]

>>> from sage.all import *
>>> m.submatrix(Integer(1), Integer(1), Integer(2), Integer(1))
[ 6]
[10]


You can take zero rows or columns if you want:

sage: m.submatrix(1, 1, 0)
[]
sage: parent(m.submatrix(1, 1, 0))
Full MatrixSpace of 0 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> m.submatrix(Integer(1), Integer(1), Integer(0))
[]
>>> parent(m.submatrix(Integer(1), Integer(1), Integer(0)))
Full MatrixSpace of 0 by 3 dense matrices over Integer Ring

zero_pattern_matrix(ring=None)[source]#

Return a matrix that contains one for corresponding zero entries.

All other entries are zero.

INPUT:

• ring – (optional); base ring of the output; default is ZZ

OUTPUT:

A new dense matrix with same dimensions as self and with base ring ring.

EXAMPLES:

sage: M = Matrix(ZZ, 2, [1,2,-2,0])
sage: M.zero_pattern_matrix()
[0 0]
[0 1]

sage: M = Matrix(QQ, 2, [1,2/3,-2,0])
sage: M.zero_pattern_matrix()
[0 0]
[0 1]

>>> from sage.all import *
>>> M = Matrix(ZZ, Integer(2), [Integer(1),Integer(2),-Integer(2),Integer(0)])
>>> M.zero_pattern_matrix()
[0 0]
[0 1]

>>> M = Matrix(QQ, Integer(2), [Integer(1),Integer(2)/Integer(3),-Integer(2),Integer(0)])
>>> M.zero_pattern_matrix()
[0 0]
[0 1]


Default base ring for the output is ZZ:

sage: M.zero_pattern_matrix().base_ring()
Integer Ring

>>> from sage.all import *
>>> M.zero_pattern_matrix().base_ring()
Integer Ring


Specify a different base ring for the output:

sage: M.zero_pattern_matrix(GF(2)).base_ring()
Finite Field of size 2

>>> from sage.all import *
>>> M.zero_pattern_matrix(GF(Integer(2))).base_ring()
Finite Field of size 2


Examples for different base rings for self:

sage: M = Matrix(Zmod(8), 3, 2, [2, 3, 9, 8, 1, 0]); M
[2 3]
[1 0]
[1 0]
sage: M.zero_pattern_matrix()
[0 0]
[0 1]
[0 1]

>>> from sage.all import *
>>> M = Matrix(Zmod(Integer(8)), Integer(3), Integer(2), [Integer(2), Integer(3), Integer(9), Integer(8), Integer(1), Integer(0)]); M
[2 3]
[1 0]
[1 0]
>>> M.zero_pattern_matrix()
[0 0]
[0 1]
[0 1]

sage: W.<a> = CyclotomicField(100)                                          # needs sage.rings.number_field
sage: M = Matrix(2, 3, [a, a/2, 0, a^2, a^100-1, a^2 - a]); M               # needs sage.rings.number_field
[      a   1/2*a       0]
[    a^2       0 a^2 - a]
sage: M.zero_pattern_matrix()                                               # needs sage.rings.number_field
[0 0 1]
[0 1 0]

>>> from sage.all import *
>>> W = CyclotomicField(Integer(100), names=('a',)); (a,) = W._first_ngens(1)# needs sage.rings.number_field
>>> M = Matrix(Integer(2), Integer(3), [a, a/Integer(2), Integer(0), a**Integer(2), a**Integer(100)-Integer(1), a**Integer(2) - a]); M               # needs sage.rings.number_field
[      a   1/2*a       0]
[    a^2       0 a^2 - a]
>>> M.zero_pattern_matrix()                                               # needs sage.rings.number_field
[0 0 1]
[0 1 0]

sage: # needs sage.rings.finite_rings
sage: K.<a> = GF(2^4)
sage: l = [a^2 + 1, a^3 + 1, 0, 0, a, a^3 + a + 1, a + 1,
....:      a + 1, a^2, a^3 + a + 1, a^3 + a, a^3 + a]
sage: M = Matrix(K, 3, 4, l); M
[    a^2 + 1     a^3 + 1           0           0]
[          a a^3 + a + 1       a + 1       a + 1]
[        a^2 a^3 + a + 1     a^3 + a     a^3 + a]
sage: M.zero_pattern_matrix()
[0 0 1 1]
[0 0 0 0]
[0 0 0 0]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(2)**Integer(4), names=('a',)); (a,) = K._first_ngens(1)
>>> l = [a**Integer(2) + Integer(1), a**Integer(3) + Integer(1), Integer(0), Integer(0), a, a**Integer(3) + a + Integer(1), a + Integer(1),
...      a + Integer(1), a**Integer(2), a**Integer(3) + a + Integer(1), a**Integer(3) + a, a**Integer(3) + a]
>>> M = Matrix(K, Integer(3), Integer(4), l); M
[    a^2 + 1     a^3 + 1           0           0]
[          a a^3 + a + 1       a + 1       a + 1]
[        a^2 a^3 + a + 1     a^3 + a     a^3 + a]
>>> M.zero_pattern_matrix()
[0 0 1 1]
[0 0 0 0]
[0 0 0 0]

sage: # needs sage.rings.finite_rings
sage: K.<a> = GF(25)
sage: M = Matrix(K, 2, 3, [0, 2, 3, 5, a, a^2])
sage: M
[    0     2     3]
[    0     a a + 3]
sage: M.zero_pattern_matrix()
[1 0 0]
[1 0 0]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(25), names=('a',)); (a,) = K._first_ngens(1)
>>> M = Matrix(K, Integer(2), Integer(3), [Integer(0), Integer(2), Integer(3), Integer(5), a, a**Integer(2)])
>>> M
[    0     2     3]
[    0     a a + 3]
>>> M.zero_pattern_matrix()
[1 0 0]
[1 0 0]


Note

This method can be optimized by improving get_is_zero_unsafe() for derived matrix classes.