Matrix Spaces¶

You can create any space \(\text{Mat}_{n\times m}(R)\) of either dense or sparse matrices with given number of rows and columns over any commutative or noncommutative ring.

EXAMPLES:

Sage

sage: MS = MatrixSpace(QQ, 6,6, sparse=True); MS
Full MatrixSpace of 6 by 6 sparse matrices over Rational Field
sage: MS.base_ring()
Rational Field
sage: MS = MatrixSpace(ZZ, 3,5, sparse=False); MS
Full MatrixSpace of 3 by 5 dense matrices over Integer Ring

Python

>>> from sage.all import *
>>> MS = MatrixSpace(QQ, Integer(6),Integer(6), sparse=True); MS
Full MatrixSpace of 6 by 6 sparse matrices over Rational Field
>>> MS.base_ring()
Rational Field
>>> MS = MatrixSpace(ZZ, Integer(3),Integer(5), sparse=False); MS
Full MatrixSpace of 3 by 5 dense matrices over Integer Ring

Sage Live

MS = MatrixSpace(QQ, 6,6, sparse=True); MS
MS.base_ring()
MS = MatrixSpace(ZZ, 3,5, sparse=False); MS

class sage.matrix.matrix_space.MatrixSpace(base_ring, nrows, ncols, sparse, implementation)[source]¶

Bases: UniqueRepresentation, Parent

The space of matrices of given size and base ring.

INPUT:

base_ring – a ring
nrows or row_keys – nonnegative integer; the number of rows, or a finite family of arbitrary objects that index the rows of the matrix
ncols or column_keys – nonnegative integer (default: nrows); the number of columns, or a finite family of arbitrary objects that index the columns of the matrix
sparse – boolean (default: False); whether or not matrices are given a sparse representation
implementation – (optional) string or matrix class; a possible implementation. Depending on the base ring, the string can be
- 'generic' – on any base rings
- 'flint' – for integers and rationals
- 'meataxe' – finite fields using the optional package meataxe: Library for computing with modular representations
- 'm4ri' – for characteristic 2 using the m4ri: fast arithmetic with dense matrices over GF(2) library
- 'linbox-float' – for integer mod rings up to \(2^8 = 256\)
- 'linbox-double' – for integer mod rings up to \(floor(2^26*sqrt(2) + 1/2) = 94906266\)
- 'numpy' – for real and complex floating point numbers

OUTPUT: a matrix space or, more generally, a homspace between free modules

This factory function creates instances of various specialized classes depending on the input. Not all combinations of options are implemented.

If the parameters row_keys or column_keys are provided, they must be finite families of objects. In this case, instances of CombinatorialFreeModule are created via the factory function FreeModule(). Then the homspace between these modules is returned.

EXAMPLES:

Sage

sage: MatrixSpace(QQ, 2)
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MatrixSpace(ZZ, 3, 2)
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
sage: MatrixSpace(ZZ, 3, sparse=False)
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

sage: MatrixSpace(ZZ, 10, 5)
Full MatrixSpace of 10 by 5 dense matrices over Integer Ring
sage: MatrixSpace(ZZ, 10, 5).category()
Category of infinite enumerated finite dimensional modules with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
sage: MatrixSpace(ZZ, 10, 10).category()
Category of infinite enumerated finite dimensional algebras with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
sage: MatrixSpace(QQ, 10).category()
Category of infinite finite dimensional algebras with basis over
 (number fields and quotient fields and metric spaces)

Python

>>> from sage.all import *
>>> MatrixSpace(QQ, Integer(2))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
>>> MatrixSpace(ZZ, Integer(3), Integer(2))
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
>>> MatrixSpace(ZZ, Integer(3), sparse=False)
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

>>> MatrixSpace(ZZ, Integer(10), Integer(5))
Full MatrixSpace of 10 by 5 dense matrices over Integer Ring
>>> MatrixSpace(ZZ, Integer(10), Integer(5)).category()
Category of infinite enumerated finite dimensional modules with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
>>> MatrixSpace(ZZ, Integer(10), Integer(10)).category()
Category of infinite enumerated finite dimensional algebras with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
>>> MatrixSpace(QQ, Integer(10)).category()
Category of infinite finite dimensional algebras with basis over
 (number fields and quotient fields and metric spaces)

Sage Live

MatrixSpace(QQ, 2)
MatrixSpace(ZZ, 3, 2)
MatrixSpace(ZZ, 3, sparse=False)
MatrixSpace(ZZ, 10, 5)
MatrixSpace(ZZ, 10, 5).category()
MatrixSpace(ZZ, 10, 10).category()
MatrixSpace(QQ, 10).category()

Some examples of square 2 by 2 rational matrices:

Sage

sage: MS = MatrixSpace(QQ, 2)
sage: MS.dimension()
4
sage: MS.dims()
(2, 2)
sage: B = MS.basis()
sage: list(B)
[
[1 0]  [0 1]  [0 0]  [0 0]
[0 0], [0 0], [1 0], [0 1]
]
sage: B[0,0]
[1 0]
[0 0]
sage: B[0,1]
[0 1]
[0 0]
sage: B[1,0]
[0 0]
[1 0]
sage: B[1,1]
[0 0]
[0 1]
sage: A = MS.matrix([1,2,3,4]); A
[1 2]
[3 4]

Python

>>> from sage.all import *
>>> MS = MatrixSpace(QQ, Integer(2))
>>> MS.dimension()
4
>>> MS.dims()
(2, 2)
>>> B = MS.basis()
>>> list(B)
[
[1 0]  [0 1]  [0 0]  [0 0]
[0 0], [0 0], [1 0], [0 1]
]
>>> B[Integer(0),Integer(0)]
[1 0]
[0 0]
>>> B[Integer(0),Integer(1)]
[0 1]
[0 0]
>>> B[Integer(1),Integer(0)]
[0 0]
[1 0]
>>> B[Integer(1),Integer(1)]
[0 0]
[0 1]
>>> A = MS.matrix([Integer(1),Integer(2),Integer(3),Integer(4)]); A
[1 2]
[3 4]

Sage Live

MS = MatrixSpace(QQ, 2)
MS.dimension()
MS.dims()
B = MS.basis()
list(B)
B[0,0]
B[0,1]
B[1,0]
B[1,1]
A = MS.matrix([1,2,3,4]); A

The above matrix A can be multiplied by a 2 by 3 integer matrix:

Sage

sage: MS2 = MatrixSpace(ZZ, 2, 3)
sage: B = MS2.matrix([1,2,3,4,5,6])
sage: A * B
[ 9 12 15]
[19 26 33]

Python

>>> from sage.all import *
>>> MS2 = MatrixSpace(ZZ, Integer(2), Integer(3))
>>> B = MS2.matrix([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> A * B
[ 9 12 15]
[19 26 33]

Sage Live

MS2 = MatrixSpace(ZZ, 2, 3)
B = MS2.matrix([1,2,3,4,5,6])
A * B

Using row_keys and column_keys:

Sage

sage: MS = MatrixSpace(ZZ, ['u', 'v'], ['a', 'b', 'c']); MS
Set of Morphisms
 from Free module generated by {'a', 'b', 'c'} over Integer Ring
   to Free module generated by {'u', 'v'} over Integer Ring
   in Category of finite dimensional modules with basis over Integer Ring

Python

>>> from sage.all import *
>>> MS = MatrixSpace(ZZ, ['u', 'v'], ['a', 'b', 'c']); MS
Set of Morphisms
 from Free module generated by {'a', 'b', 'c'} over Integer Ring
   to Free module generated by {'u', 'v'} over Integer Ring
   in Category of finite dimensional modules with basis over Integer Ring

Sage Live

MS = MatrixSpace(ZZ, ['u', 'v'], ['a', 'b', 'c']); MS

Check categories:

Sage

sage: MatrixSpace(ZZ, 10, 5)
Full MatrixSpace of 10 by 5 dense matrices over Integer Ring
sage: MatrixSpace(ZZ, 10, 5).category()
Category of infinite enumerated finite dimensional modules with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
sage: MatrixSpace(ZZ, 10, 10).category()
Category of infinite enumerated finite dimensional algebras with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
sage: MatrixSpace(QQ, 10).category()
Category of infinite finite dimensional algebras with basis over
 (number fields and quotient fields and metric spaces)

Python

>>> from sage.all import *
>>> MatrixSpace(ZZ, Integer(10), Integer(5))
Full MatrixSpace of 10 by 5 dense matrices over Integer Ring
>>> MatrixSpace(ZZ, Integer(10), Integer(5)).category()
Category of infinite enumerated finite dimensional modules with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
>>> MatrixSpace(ZZ, Integer(10), Integer(10)).category()
Category of infinite enumerated finite dimensional algebras with basis over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
>>> MatrixSpace(QQ, Integer(10)).category()
Category of infinite finite dimensional algebras with basis over
 (number fields and quotient fields and metric spaces)

Sage Live

MatrixSpace(ZZ, 10, 5)
MatrixSpace(ZZ, 10, 5).category()
MatrixSpace(ZZ, 10, 10).category()
MatrixSpace(QQ, 10).category()

base_extend(R)[source]¶

Return base extension of this matrix space to R.

INPUT:

R – ring

OUTPUT: a matrix space

EXAMPLES:

Sage

sage: Mat(ZZ, 3, 5).base_extend(QQ)
Full MatrixSpace of 3 by 5 dense matrices over Rational Field
sage: Mat(QQ, 3, 5).base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no base extension defined

Python

>>> from sage.all import *
>>> Mat(ZZ, Integer(3), Integer(5)).base_extend(QQ)
Full MatrixSpace of 3 by 5 dense matrices over Rational Field
>>> Mat(QQ, Integer(3), Integer(5)).base_extend(GF(Integer(7)))
Traceback (most recent call last):
...
TypeError: no base extension defined

Sage Live

Mat(ZZ, 3, 5).base_extend(QQ)
Mat(QQ, 3, 5).base_extend(GF(7))

basis()[source]¶

Return a basis for this matrix space.

Warning

This will of course compute every generator of this matrix space. So for large dimensions, this could take a long time, waste a massive amount of memory (for dense matrices), and is likely not very useful. Don’t use this on large matrix spaces.

EXAMPLES:

Sage

sage: list(Mat(ZZ,2,2).basis())
[
[1 0]  [0 1]  [0 0]  [0 0]
[0 0], [0 0], [1 0], [0 1]
]

Python

>>> from sage.all import *
>>> list(Mat(ZZ,Integer(2),Integer(2)).basis())
[
[1 0]  [0 1]  [0 0]  [0 0]
[0 0], [0 0], [1 0], [0 1]
]

Sage Live

list(Mat(ZZ,2,2).basis())

cached_method(f, name=None, key=None, do_pickle=None)[source]¶

A decorator for cached methods.

EXAMPLES:

In the following examples, one can see how a cached method works in application. Below, we demonstrate what is done behind the scenes:

Sage

sage: class C:
....:     @cached_method
....:     def __hash__(self):
....:         print("compute hash")
....:         return int(5)
....:     @cached_method
....:     def f(self, x):
....:         print("computing cached method")
....:         return x*2
sage: c = C()
sage: type(C.__hash__)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>
sage: hash(c)
compute hash
5

Python

>>> from sage.all import *
>>> class C:
...     @cached_method
...     def __hash__(self):
...         print("compute hash")
...         return int(Integer(5))
...     @cached_method
...     def f(self, x):
...         print("computing cached method")
...         return x*Integer(2)
>>> c = C()
>>> type(C.__hash__)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>
>>> hash(c)
compute hash
5

Sage Live

class C:
    @cached_method
    def __hash__(self):
        print("compute hash")
        return int(5)
    @cached_method
    def f(self, x):
        print("computing cached method")
        return x*2
c = C()
type(C.__hash__)
hash(c)

When calling a cached method for the second time with the same arguments, the value is gotten from the cache, so that a new computation is not needed:

Sage

sage: hash(c)
5
sage: c.f(4)
computing cached method
8
sage: c.f(4) is c.f(4)
True

Python

>>> from sage.all import *
>>> hash(c)
5
>>> c.f(Integer(4))
computing cached method
8
>>> c.f(Integer(4)) is c.f(Integer(4))
True

Sage Live

hash(c)
c.f(4)
c.f(4) is c.f(4)

Different instances have distinct caches:

Sage

sage: d = C()
sage: d.f(4) is c.f(4)
computing cached method
False
sage: d.f.clear_cache()
sage: c.f(4)
8
sage: d.f(4)
computing cached method
8

Python

>>> from sage.all import *
>>> d = C()
>>> d.f(Integer(4)) is c.f(Integer(4))
computing cached method
False
>>> d.f.clear_cache()
>>> c.f(Integer(4))
8
>>> d.f(Integer(4))
computing cached method
8

Sage Live

d = C()
d.f(4) is c.f(4)
d.f.clear_cache()
c.f(4)
d.f(4)

Using cached methods for the hash and other special methods was implemented in Issue #12601, by means of CachedSpecialMethod. We show that it is used behind the scenes:

Sage

sage: cached_method(c.__hash__)
<sage.misc.cachefunc.CachedSpecialMethod object at ...>
sage: cached_method(c.f)
<sage.misc.cachefunc.CachedMethod object at ...>

Python

>>> from sage.all import *
>>> cached_method(c.__hash__)
<sage.misc.cachefunc.CachedSpecialMethod object at ...>
>>> cached_method(c.f)
<sage.misc.cachefunc.CachedMethod object at ...>

Sage Live

cached_method(c.__hash__)
cached_method(c.f)

The parameter do_pickle can be used if the contents of the cache should be stored in a pickle of the cached method. This can be dangerous with special methods such as __hash__:

Sage

sage: class C:
....:     @cached_method(do_pickle=True)
....:     def __hash__(self):
....:         return id(self)

sage: import __main__
sage: __main__.C = C
sage: c = C()
sage: hash(c)  # random output
sage: d = loads(dumps(c))
sage: hash(d) == hash(c)
True

Python

>>> from sage.all import *
>>> class C:
...     @cached_method(do_pickle=True)
...     def __hash__(self):
...         return id(self)

>>> import __main__
>>> __main__.C = C
>>> c = C()
>>> hash(c)  # random output
>>> d = loads(dumps(c))
>>> hash(d) == hash(c)
True

Sage Live

class C:
    @cached_method(do_pickle=True)
    def __hash__(self):
        return id(self)
import __main__
__main__.C = C
c = C()
hash(c)  # random output
d = loads(dumps(c))
hash(d) == hash(c)

However, the contents of a method’s cache are not pickled unless do_pickle is set:

Sage

sage: class C:
....:     @cached_method
....:     def __hash__(self):
....:         return id(self)

sage: __main__.C = C
sage: c = C()
sage: hash(c)  # random output
sage: d = loads(dumps(c))
sage: hash(d) == hash(c)
False

Python

>>> from sage.all import *
>>> class C:
...     @cached_method
...     def __hash__(self):
...         return id(self)

>>> __main__.C = C
>>> c = C()
>>> hash(c)  # random output
>>> d = loads(dumps(c))
>>> hash(d) == hash(c)
False

Sage Live

class C:
    @cached_method
    def __hash__(self):
        return id(self)
__main__.C = C
c = C()
hash(c)  # random output
d = loads(dumps(c))
hash(d) == hash(c)

cardinality()[source]¶

Return the number of elements in self.

EXAMPLES:

Sage

sage: MatrixSpace(GF(3), 2, 3).cardinality()
729
sage: MatrixSpace(ZZ, 2).cardinality()
+Infinity
sage: MatrixSpace(ZZ, 0, 3).cardinality()
1

Python

>>> from sage.all import *
>>> MatrixSpace(GF(Integer(3)), Integer(2), Integer(3)).cardinality()
729
>>> MatrixSpace(ZZ, Integer(2)).cardinality()
+Infinity
>>> MatrixSpace(ZZ, Integer(0), Integer(3)).cardinality()
1

Sage Live

MatrixSpace(GF(3), 2, 3).cardinality()
MatrixSpace(ZZ, 2).cardinality()
MatrixSpace(ZZ, 0, 3).cardinality()

change_ring(R)[source]¶

Return matrix space over R with otherwise same parameters as self.

INPUT:

R – ring

OUTPUT: a matrix space

EXAMPLES:

Sage

sage: Mat(QQ, 3, 5).change_ring(GF(7))
Full MatrixSpace of 3 by 5 dense matrices
 over Finite Field of size 7

Python

>>> from sage.all import *
>>> Mat(QQ, Integer(3), Integer(5)).change_ring(GF(Integer(7)))
Full MatrixSpace of 3 by 5 dense matrices
 over Finite Field of size 7

Sage Live

Mat(QQ, 3, 5).change_ring(GF(7))

characteristic()[source]¶

Return the characteristic.

EXAMPLES:

Sage

sage: MatrixSpace(ZZ, 2).characteristic()
0
sage: MatrixSpace(GF(9), 0).characteristic()                                # needs sage.rings.finite_rings
3

Python

>>> from sage.all import *
>>> MatrixSpace(ZZ, Integer(2)).characteristic()
0
>>> MatrixSpace(GF(Integer(9)), Integer(0)).characteristic()                                # needs sage.rings.finite_rings
3

Sage Live

MatrixSpace(ZZ, 2).characteristic()
MatrixSpace(GF(9), 0).characteristic()                                # needs sage.rings.finite_rings

column_space()[source]¶

Return the module spanned by all columns of matrices in this matrix space. This is a free module of rank the number of columns. It will be sparse or dense as this matrix space is sparse or dense.

EXAMPLES:

Sage

sage: M = Mat(GF(9,'a'), 20, 5, sparse=True); M.column_space()              # needs sage.rings.finite_rings
Sparse vector space of dimension 20 over Finite Field in a of size 3^2

Python

>>> from sage.all import *
>>> M = Mat(GF(Integer(9),'a'), Integer(20), Integer(5), sparse=True); M.column_space()              # needs sage.rings.finite_rings
Sparse vector space of dimension 20 over Finite Field in a of size 3^2

Sage Live

M = Mat(GF(9,'a'), 20, 5, sparse=True); M.column_space()              # needs sage.rings.finite_rings

construction()[source]¶

EXAMPLES:

Sage

sage: A = matrix(ZZ, 2, [1..4], sparse=True)
sage: A.parent().construction()
(MatrixFunctor, Integer Ring)
sage: A.parent().construction()[0](QQ['x'])
Full MatrixSpace of 2 by 2 sparse matrices over
 Univariate Polynomial Ring in x over Rational Field
sage: parent(A/2)
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

Python

>>> from sage.all import *
>>> A = matrix(ZZ, Integer(2), (ellipsis_range(Integer(1),Ellipsis,Integer(4))), sparse=True)
>>> A.parent().construction()
(MatrixFunctor, Integer Ring)
>>> A.parent().construction()[Integer(0)](QQ['x'])
Full MatrixSpace of 2 by 2 sparse matrices over
 Univariate Polynomial Ring in x over Rational Field
>>> parent(A/Integer(2))
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

Sage Live

A = matrix(ZZ, 2, [1..4], sparse=True)
A.parent().construction()
A.parent().construction()[0](QQ['x'])
parent(A/2)

diagonal_matrix(entries)[source]¶

Create a diagonal matrix in self using the specified elements.

INPUT:

entries – the elements to use as the diagonal entries

self must be a space of square matrices. The length of entries must be less than or equal to the matrix dimensions. If the length of entries is less than the matrix dimensions, entries is padded with zeroes at the end.

EXAMPLES:

Sage

sage: MS1 = MatrixSpace(ZZ,4)
sage: MS2 = MatrixSpace(QQ,3,4)
sage: I = MS1.diagonal_matrix([1, 2, 3, 4])
sage: I
[1 0 0 0]
[0 2 0 0]
[0 0 3 0]
[0 0 0 4]
sage: MS2.diagonal_matrix([1, 2])
Traceback (most recent call last):
...
TypeError: diagonal matrix must be square
sage: MS1.diagonal_matrix([1, 2, 3, 4, 5])
Traceback (most recent call last):
...
ValueError: number of diagonal matrix entries (5) exceeds the matrix size (4)
sage: MS1.diagonal_matrix([1/2, 2, 3, 4])
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

Python

>>> from sage.all import *
>>> MS1 = MatrixSpace(ZZ,Integer(4))
>>> MS2 = MatrixSpace(QQ,Integer(3),Integer(4))
>>> I = MS1.diagonal_matrix([Integer(1), Integer(2), Integer(3), Integer(4)])
>>> I
[1 0 0 0]
[0 2 0 0]
[0 0 3 0]
[0 0 0 4]
>>> MS2.diagonal_matrix([Integer(1), Integer(2)])
Traceback (most recent call last):
...
TypeError: diagonal matrix must be square
>>> MS1.diagonal_matrix([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5)])
Traceback (most recent call last):
...
ValueError: number of diagonal matrix entries (5) exceeds the matrix size (4)
>>> MS1.diagonal_matrix([Integer(1)/Integer(2), Integer(2), Integer(3), Integer(4)])
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

Sage Live

MS1 = MatrixSpace(ZZ,4)
MS2 = MatrixSpace(QQ,3,4)
I = MS1.diagonal_matrix([1, 2, 3, 4])
I
MS2.diagonal_matrix([1, 2])
MS1.diagonal_matrix([1, 2, 3, 4, 5])
MS1.diagonal_matrix([1/2, 2, 3, 4])

Check different implementations:

Sage

sage: M1 = MatrixSpace(ZZ, 2, implementation='flint')                       # needs sage.libs.linbox
sage: M2 = MatrixSpace(ZZ, 2, implementation='generic')

sage: type(M1.diagonal_matrix([1, 2]))                                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: type(M2.diagonal_matrix([1, 2]))
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

Python

>>> from sage.all import *
>>> M1 = MatrixSpace(ZZ, Integer(2), implementation='flint')                       # needs sage.libs.linbox
>>> M2 = MatrixSpace(ZZ, Integer(2), implementation='generic')

>>> type(M1.diagonal_matrix([Integer(1), Integer(2)]))                                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
>>> type(M2.diagonal_matrix([Integer(1), Integer(2)]))
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

Sage Live

M1 = MatrixSpace(ZZ, 2, implementation='flint')                       # needs sage.libs.linbox
M2 = MatrixSpace(ZZ, 2, implementation='generic')
type(M1.diagonal_matrix([1, 2]))                                      # needs sage.libs.linbox
type(M2.diagonal_matrix([1, 2]))

dimension()[source]¶

Return (m rows) * (n cols) of self as Integer.

EXAMPLES:

Sage

sage: MS = MatrixSpace(ZZ,4,6)
sage: u = MS.dimension()
sage: u - 24 == 0
True

Python

>>> from sage.all import *
>>> MS = MatrixSpace(ZZ,Integer(4),Integer(6))
>>> u = MS.dimension()
>>> u - Integer(24) == Integer(0)
True

Sage Live

MS = MatrixSpace(ZZ,4,6)
u = MS.dimension()
u - 24 == 0

dims()[source]¶

Return (m row, n col) representation of self dimension.

EXAMPLES:

Sage

sage: MS = MatrixSpace(ZZ,4,6)
sage: MS.dims()
(4, 6)

Python

>>> from sage.all import *
>>> MS = MatrixSpace(ZZ,Integer(4),Integer(6))
>>> MS.dims()
(4, 6)

Sage Live

MS = MatrixSpace(ZZ,4,6)
MS.dims()

from_vector(vector, order=None, coerce=True)[source]¶

Build an element of self from a vector.

EXAMPLES:

Sage

sage: A = matrix([[1,2,3], [4,5,6]])
sage: v = vector(A); v
(1, 2, 3, 4, 5, 6)
sage: MS = A.parent()
sage: MS.from_vector(v)
[1 2 3]
[4 5 6]
sage: order = [(1,2), (1,0), (0,1), (0,2), (0,0), (1,1)]
sage: MS.from_vector(v, order=order)
[5 3 4]
[2 6 1]

Python

>>> from sage.all import *
>>> A = matrix([[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]])
>>> v = vector(A); v
(1, 2, 3, 4, 5, 6)
>>> MS = A.parent()
>>> MS.from_vector(v)
[1 2 3]
[4 5 6]
>>> order = [(Integer(1),Integer(2)), (Integer(1),Integer(0)), (Integer(0),Integer(1)), (Integer(0),Integer(2)), (Integer(0),Integer(0)), (Integer(1),Integer(1))]
>>> MS.from_vector(v, order=order)
[5 3 4]
[2 6 1]

Sage Live

A = matrix([[1,2,3], [4,5,6]])
v = vector(A); v
MS = A.parent()
MS.from_vector(v)
order = [(1,2), (1,0), (0,1), (0,2), (0,0), (1,1)]
MS.from_vector(v, order=order)

gen(n)[source]¶

Return the \(n\)-th generator of this matrix space.

This does not compute all basis matrices, so it is reasonably intelligent.

EXAMPLES:

Sage

sage: M = Mat(GF(7), 10000, 5); M.ngens()
50000
sage: a = M.10
sage: a[:4]
[0 0 0 0 0]
[0 0 0 0 0]
[1 0 0 0 0]
[0 0 0 0 0]

Python

>>> from sage.all import *
>>> M = Mat(GF(Integer(7)), Integer(10000), Integer(5)); M.ngens()
50000
>>> a = M.gen(10)
>>> a[:Integer(4)]
[0 0 0 0 0]
[0 0 0 0 0]
[1 0 0 0 0]
[0 0 0 0 0]

Sage Live

M = Mat(GF(7), 10000, 5); M.ngens()
a = M.10
a[:4]

identity_matrix()[source]¶

Return the identity matrix in self.

self must be a space of square matrices. The returned matrix is immutable. Please use copy if you want a modified copy.

EXAMPLES:

Sage

sage: MS1 = MatrixSpace(ZZ,4)
sage: MS2 = MatrixSpace(QQ,3,4)
sage: I = MS1.identity_matrix()
sage: I
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: Er = MS2.identity_matrix()
Traceback (most recent call last):
...
TypeError: identity matrix must be square

Python

>>> from sage.all import *
>>> MS1 = MatrixSpace(ZZ,Integer(4))
>>> MS2 = MatrixSpace(QQ,Integer(3),Integer(4))
>>> I = MS1.identity_matrix()
>>> I
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> Er = MS2.identity_matrix()
Traceback (most recent call last):
...
TypeError: identity matrix must be square

Sage Live

MS1 = MatrixSpace(ZZ,4)
MS2 = MatrixSpace(QQ,3,4)
I = MS1.identity_matrix()
I
Er = MS2.identity_matrix()

is_dense()[source]¶

Return whether matrices in self are dense.

EXAMPLES:

Sage

sage: Mat(RDF,2,3).is_sparse()
False
sage: Mat(RR,123456,22,sparse=True).is_sparse()
True

Python

>>> from sage.all import *
>>> Mat(RDF,Integer(2),Integer(3)).is_sparse()
False
>>> Mat(RR,Integer(123456),Integer(22),sparse=True).is_sparse()
True

Sage Live

Mat(RDF,2,3).is_sparse()
Mat(RR,123456,22,sparse=True).is_sparse()

is_finite()[source]¶

Return whether this matrix space is finite.

EXAMPLES:

Sage

sage: MatrixSpace(GF(101), 10000).is_finite()
True
sage: MatrixSpace(QQ, 2).is_finite()
False

Python

>>> from sage.all import *
>>> MatrixSpace(GF(Integer(101)), Integer(10000)).is_finite()
True
>>> MatrixSpace(QQ, Integer(2)).is_finite()
False

Sage Live

MatrixSpace(GF(101), 10000).is_finite()
MatrixSpace(QQ, 2).is_finite()

is_sparse()[source]¶

Return whether matrices in self are sparse.

EXAMPLES:

Sage

sage: Mat(GF(2011), 10000).is_sparse()                                      # needs sage.rings.finite_rings
False
sage: Mat(GF(2011), 10000, sparse=True).is_sparse()                         # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> Mat(GF(Integer(2011)), Integer(10000)).is_sparse()                                      # needs sage.rings.finite_rings
False
>>> Mat(GF(Integer(2011)), Integer(10000), sparse=True).is_sparse()                         # needs sage.rings.finite_rings
True

Sage Live

Mat(GF(2011), 10000).is_sparse()                                      # needs sage.rings.finite_rings
Mat(GF(2011), 10000, sparse=True).is_sparse()                         # needs sage.rings.finite_rings

matrix(x=None, **kwds)[source]¶

Create a matrix in self.

INPUT:

x – data to construct a new matrix from. See matrix()
coerce – boolean (default: True); if False, assume without checking that the values in x lie in the base ring

OUTPUT: a matrix in self

EXAMPLES:

Sage

sage: M = MatrixSpace(ZZ, 2)
sage: M.matrix([[1,0],[0,-1]])
[ 1  0]
[ 0 -1]
sage: M.matrix([1,0,0,-1])
[ 1  0]
[ 0 -1]
sage: M.matrix([1,2,3,4])
[1 2]
[3 4]

Python

>>> from sage.all import *
>>> M = MatrixSpace(ZZ, Integer(2))
>>> M.matrix([[Integer(1),Integer(0)],[Integer(0),-Integer(1)]])
[ 1  0]
[ 0 -1]
>>> M.matrix([Integer(1),Integer(0),Integer(0),-Integer(1)])
[ 1  0]
[ 0 -1]
>>> M.matrix([Integer(1),Integer(2),Integer(3),Integer(4)])
[1 2]
[3 4]

Sage Live

M = MatrixSpace(ZZ, 2)
M.matrix([[1,0],[0,-1]])
M.matrix([1,0,0,-1])
M.matrix([1,2,3,4])

Note that the last “flip” cannot be performed if x is a matrix, no matter what is rows (it used to be possible but was fixed by Issue #10793):

Sage

sage: projection = matrix(ZZ,[[1,0,0],[0,1,0]])
sage: projection
[1 0 0]
[0 1 0]
sage: projection.parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: M = MatrixSpace(ZZ, 3 , 2)
sage: M
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
sage: M(projection)
Traceback (most recent call last):
...
ValueError: inconsistent number of rows: should be 3 but got 2

Python

>>> from sage.all import *
>>> projection = matrix(ZZ,[[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)]])
>>> projection
[1 0 0]
[0 1 0]
>>> projection.parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
>>> M = MatrixSpace(ZZ, Integer(3) , Integer(2))
>>> M
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
>>> M(projection)
Traceback (most recent call last):
...
ValueError: inconsistent number of rows: should be 3 but got 2

Sage Live

projection = matrix(ZZ,[[1,0,0],[0,1,0]])
projection
projection.parent()
M = MatrixSpace(ZZ, 3 , 2)
M
M(projection)

If you really want to make from a matrix another matrix of different dimensions, use either transpose method or explicit conversion to a list:

Sage

sage: M(projection.list())
[1 0]
[0 0]
[1 0]

Python

>>> from sage.all import *
>>> M(projection.list())
[1 0]
[0 0]
[1 0]

Sage Live

M(projection.list())

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

Return the matrix space with given number of rows, columns and sparsity over the same base ring as self, and defaults the same as self.

EXAMPLES:

Sage

sage: M = Mat(GF(7), 100, 200)
sage: M.matrix_space(5000)
Full MatrixSpace of 5000 by 200 dense matrices over Finite Field of size 7
sage: M.matrix_space(ncols=5000)
Full MatrixSpace of 100 by 5000 dense matrices over Finite Field of size 7
sage: M.matrix_space(sparse=True)
Full MatrixSpace of 100 by 200 sparse matrices over Finite Field of size 7

Python

>>> from sage.all import *
>>> M = Mat(GF(Integer(7)), Integer(100), Integer(200))
>>> M.matrix_space(Integer(5000))
Full MatrixSpace of 5000 by 200 dense matrices over Finite Field of size 7
>>> M.matrix_space(ncols=Integer(5000))
Full MatrixSpace of 100 by 5000 dense matrices over Finite Field of size 7
>>> M.matrix_space(sparse=True)
Full MatrixSpace of 100 by 200 sparse matrices over Finite Field of size 7

Sage Live

M = Mat(GF(7), 100, 200)
M.matrix_space(5000)
M.matrix_space(ncols=5000)
M.matrix_space(sparse=True)

ncols()[source]¶

Return the number of columns of matrices in this space.

EXAMPLES:

Sage

sage: M = Mat(ZZ['x'], 200000, 500000, sparse=True)
sage: M.ncols()
500000

Python

>>> from sage.all import *
>>> M = Mat(ZZ['x'], Integer(200000), Integer(500000), sparse=True)
>>> M.ncols()
500000

Sage Live

M = Mat(ZZ['x'], 200000, 500000, sparse=True)
M.ncols()

ngens()[source]¶

Return the number of generators of this matrix space.

This is the number of entries in the matrices in this space.

EXAMPLES:

Sage

sage: M = Mat(GF(7), 100, 200); M.ngens()
20000

Python

>>> from sage.all import *
>>> M = Mat(GF(Integer(7)), Integer(100), Integer(200)); M.ngens()
20000

Sage Live

M = Mat(GF(7), 100, 200); M.ngens()

nrows()[source]¶

Return the number of rows of matrices in this space.

EXAMPLES:

Sage

sage: M = Mat(ZZ, 200000, 500000)
sage: M.nrows()
200000

Python

>>> from sage.all import *
>>> M = Mat(ZZ, Integer(200000), Integer(500000))
>>> M.nrows()
200000

Sage Live

M = Mat(ZZ, 200000, 500000)
M.nrows()

one()[source]¶

Return the identity matrix in self.

self must be a space of square matrices. The returned matrix is immutable. Please use copy if you want a modified copy.

EXAMPLES:

Sage

sage: MS1 = MatrixSpace(ZZ,4)
sage: MS2 = MatrixSpace(QQ,3,4)
sage: I = MS1.identity_matrix()
sage: I
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: Er = MS2.identity_matrix()
Traceback (most recent call last):
...
TypeError: identity matrix must be square

Python

>>> from sage.all import *
>>> MS1 = MatrixSpace(ZZ,Integer(4))
>>> MS2 = MatrixSpace(QQ,Integer(3),Integer(4))
>>> I = MS1.identity_matrix()
>>> I
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> Er = MS2.identity_matrix()
Traceback (most recent call last):
...
TypeError: identity matrix must be square

Sage Live

MS1 = MatrixSpace(ZZ,4)
MS2 = MatrixSpace(QQ,3,4)
I = MS1.identity_matrix()
I
Er = MS2.identity_matrix()

random_element(density=None, *args, **kwds)[source]¶

Return a random element from this matrix space.

INPUT:

density – float or None (default: None); rough measure of the proportion of nonzero entries in the random matrix; if set to None, all entries of the matrix are randomized, allowing for any element of the underlying ring, but if set to a float, a proportion of entries is selected and randomized to nonzero elements of the ring
*args, **kwds – remaining parameters, which may be passed to the random_element function of the base ring. (“may be”, since this function calls the randomize function on the zero matrix, which need not call the random_element function of the base ring at all in general.)

OUTPUT: Matrix

Note

This method will randomize a proportion of roughly density entries in a newly allocated zero matrix.

By default, if the user sets the value of density explicitly, this method will enforce that these entries are set to nonzero values. However, if the test for equality with zero in the base ring is too expensive, the user can override this behaviour by passing the argument nonzero=False to this method.

Otherwise, if the user does not set the value of density, the default value is taken to be 1, and the option nonzero=False is passed to the randomize method.

EXAMPLES:

Sage

sage: M = Mat(ZZ, 2, 5).random_element()
sage: TestSuite(M).run()

sage: M = Mat(QQ, 2, 5).random_element(density=0.5)
sage: TestSuite(M).run()

sage: M = Mat(QQ, 3, sparse=True).random_element()
sage: TestSuite(M).run()                                                    # needs sage.libs.pari

sage: M = Mat(GF(9,'a'), 3, sparse=True).random_element()                   # needs sage.rings.finite_rings
sage: TestSuite(M).run()                                                    # needs sage.rings.finite_rings

Python

>>> from sage.all import *
>>> M = Mat(ZZ, Integer(2), Integer(5)).random_element()
>>> TestSuite(M).run()

>>> M = Mat(QQ, Integer(2), Integer(5)).random_element(density=RealNumber('0.5'))
>>> TestSuite(M).run()

>>> M = Mat(QQ, Integer(3), sparse=True).random_element()
>>> TestSuite(M).run()                                                    # needs sage.libs.pari

>>> M = Mat(GF(Integer(9),'a'), Integer(3), sparse=True).random_element()                   # needs sage.rings.finite_rings
>>> TestSuite(M).run()                                                    # needs sage.rings.finite_rings

Sage Live

M = Mat(ZZ, 2, 5).random_element()
TestSuite(M).run()
M = Mat(QQ, 2, 5).random_element(density=0.5)
TestSuite(M).run()
M = Mat(QQ, 3, sparse=True).random_element()
TestSuite(M).run()                                                    # needs sage.libs.pari
M = Mat(GF(9,'a'), 3, sparse=True).random_element()                   # needs sage.rings.finite_rings
TestSuite(M).run()                                                    # needs sage.rings.finite_rings

row_space()[source]¶

Return the module spanned by all rows of matrices in this matrix space. This is a free module of rank the number of rows. It will be sparse or dense as this matrix space is sparse or dense.

EXAMPLES:

Sage

sage: M = Mat(ZZ,20,5,sparse=False); M.row_space()
Ambient free module of rank 5 over the principal ideal domain Integer Ring

Python

>>> from sage.all import *
>>> M = Mat(ZZ,Integer(20),Integer(5),sparse=False); M.row_space()
Ambient free module of rank 5 over the principal ideal domain Integer Ring

Sage Live

M = Mat(ZZ,20,5,sparse=False); M.row_space()

some_elements()[source]¶

Return some elements of this matrix space.

See TestSuite for a typical use case.

OUTPUT: an iterator

EXAMPLES:

Sage

sage: M = MatrixSpace(ZZ, 2, 2)
sage: tuple(M.some_elements())
(
[ 0  1]  [1 0]  [0 1]  [0 0]  [0 0]
[-1  2], [0 0], [0 0], [1 0], [0 1]
)
sage: M = MatrixSpace(QQ, 2, 3)
sage: tuple(M.some_elements())
(
[ 1/2 -1/2    2]  [1 0 0]  [0 1 0]  [0 0 1]  [0 0 0]  [0 0 0]  [0 0 0]
[  -2    0    1], [0 0 0], [0 0 0], [0 0 0], [1 0 0], [0 1 0], [0 0 1]
)
sage: M = MatrixSpace(SR, 2, 2)                                             # needs sage.symbolic
sage: tuple(M.some_elements())                                              # needs sage.symbolic
(
[some_variable some_variable]  [1 0]  [0 1]  [0 0]  [0 0]
[some_variable some_variable], [0 0], [0 0], [1 0], [0 1]
)

Python

>>> from sage.all import *
>>> M = MatrixSpace(ZZ, Integer(2), Integer(2))
>>> tuple(M.some_elements())
(
[ 0  1]  [1 0]  [0 1]  [0 0]  [0 0]
[-1  2], [0 0], [0 0], [1 0], [0 1]
)
>>> M = MatrixSpace(QQ, Integer(2), Integer(3))
>>> tuple(M.some_elements())
(
[ 1/2 -1/2    2]  [1 0 0]  [0 1 0]  [0 0 1]  [0 0 0]  [0 0 0]  [0 0 0]
[  -2    0    1], [0 0 0], [0 0 0], [0 0 0], [1 0 0], [0 1 0], [0 0 1]
)
>>> M = MatrixSpace(SR, Integer(2), Integer(2))                                             # needs sage.symbolic
>>> tuple(M.some_elements())                                              # needs sage.symbolic
(
[some_variable some_variable]  [1 0]  [0 1]  [0 0]  [0 0]
[some_variable some_variable], [0 0], [0 0], [1 0], [0 1]
)

Sage Live

M = MatrixSpace(ZZ, 2, 2)
tuple(M.some_elements())
M = MatrixSpace(QQ, 2, 3)
tuple(M.some_elements())
M = MatrixSpace(SR, 2, 2)                                             # needs sage.symbolic
tuple(M.some_elements())                                              # needs sage.symbolic

submodule(gens, check=True, already_echelonized=False, unitriangular=False, support_order=None, category=None, *args, **opts)[source]¶

The submodule spanned by a finite set of matrices.

INPUT:

gens – list or family of elements of self
check – boolean (default: True); whether to verify that the elements of gens are in self
already_echelonized – boolean (default: False); whether the elements of gens are already in (not necessarily reduced) echelon form
unitriangular – boolean (default: False); whether the lift morphism is unitriangular
support_order – (optional) either something that can be converted into a tuple or a key function

If already_echelonized is False, then the generators are put in reduced echelon form using echelonize(), and reindexed by \(0, 1, \ldots\).

Warning

At this point, this method only works for finite dimensional submodules and if matrices can be echelonized over the base ring.

If in addition unitriangular is True, then the generators are made such that the coefficients of the pivots are 1, so that lifting map is unitriangular.

The basis of the submodule uses the same index set as the generators, and the lifting map sends \(y_i\) to \(gens[i]\).

EXAMPLES:

Sage

sage: M = MatrixSpace(QQ, 2)
sage: mat = M.matrix([[1, 2], [3, 4]])
sage: X = M.submodule([mat], already_echelonized=True); X
Free module generated by {0} over Rational Field

sage: mat2 = M.matrix([[1, 0], [-3, 2]])
sage: X = M.submodule([mat, mat2])
sage: [X.lift(b) for b in X.basis()]
[
[ 1  0]  [0 1]
[-3  2], [3 1]
]

sage: A = matrix([[1, 1], [0, -1]])
sage: B = matrix([[0, 1], [0, 2]])
sage: X = M.submodule([A, B])
sage: Xp = M.submodule([A, B], support_order=[(0,1), (1,1), (0,0)])
sage: [X.lift(b) for b in X.basis()]
[
[ 1  0]  [0 1]
[ 0 -3], [0 2]
]
sage: [Xp.lift(b) for b in Xp.basis()]
[
[2/3   1]  [-1/3    0]
[  0   0], [   0    1]
]

Python

>>> from sage.all import *
>>> M = MatrixSpace(QQ, Integer(2))
>>> mat = M.matrix([[Integer(1), Integer(2)], [Integer(3), Integer(4)]])
>>> X = M.submodule([mat], already_echelonized=True); X
Free module generated by {0} over Rational Field

>>> mat2 = M.matrix([[Integer(1), Integer(0)], [-Integer(3), Integer(2)]])
>>> X = M.submodule([mat, mat2])
>>> [X.lift(b) for b in X.basis()]
[
[ 1  0]  [0 1]
[-3  2], [3 1]
]

>>> A = matrix([[Integer(1), Integer(1)], [Integer(0), -Integer(1)]])
>>> B = matrix([[Integer(0), Integer(1)], [Integer(0), Integer(2)]])
>>> X = M.submodule([A, B])
>>> Xp = M.submodule([A, B], support_order=[(Integer(0),Integer(1)), (Integer(1),Integer(1)), (Integer(0),Integer(0))])
>>> [X.lift(b) for b in X.basis()]
[
[ 1  0]  [0 1]
[ 0 -3], [0 2]
]
>>> [Xp.lift(b) for b in Xp.basis()]
[
[2/3   1]  [-1/3    0]
[  0   0], [   0    1]
]

Sage Live

M = MatrixSpace(QQ, 2)
mat = M.matrix([[1, 2], [3, 4]])
X = M.submodule([mat], already_echelonized=True); X
mat2 = M.matrix([[1, 0], [-3, 2]])
X = M.submodule([mat, mat2])
[X.lift(b) for b in X.basis()]
A = matrix([[1, 1], [0, -1]])
B = matrix([[0, 1], [0, 2]])
X = M.submodule([A, B])
Xp = M.submodule([A, B], support_order=[(0,1), (1,1), (0,0)])
[X.lift(b) for b in X.basis()]
[Xp.lift(b) for b in Xp.basis()]

transposed()[source]¶

The transposed matrix space, having the same base ring and sparseness, but number of columns and rows is swapped.

EXAMPLES:

Sage

sage: MS = MatrixSpace(GF(3), 7, 10)
sage: MS.transposed
Full MatrixSpace of 10 by 7 dense matrices over Finite Field of size 3
sage: MS = MatrixSpace(GF(3), 7, 7)
sage: MS.transposed is MS
True

sage: M = MatrixSpace(ZZ, 2, 3)
sage: M.transposed
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring

Python

>>> from sage.all import *
>>> MS = MatrixSpace(GF(Integer(3)), Integer(7), Integer(10))
>>> MS.transposed
Full MatrixSpace of 10 by 7 dense matrices over Finite Field of size 3
>>> MS = MatrixSpace(GF(Integer(3)), Integer(7), Integer(7))
>>> MS.transposed is MS
True

>>> M = MatrixSpace(ZZ, Integer(2), Integer(3))
>>> M.transposed
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring

Sage Live

MS = MatrixSpace(GF(3), 7, 10)
MS.transposed
MS = MatrixSpace(GF(3), 7, 7)
MS.transposed is MS
M = MatrixSpace(ZZ, 2, 3)
M.transposed

zero()[source]¶

Return the zero matrix in self.

self must be a space of square matrices. The returned matrix is immutable. Please use copy if you want a modified copy.

EXAMPLES:

Sage

sage: z = MatrixSpace(GF(7), 2, 4).zero_matrix(); z
[0 0 0 0]
[0 0 0 0]
sage: z.is_mutable()
False

Python

>>> from sage.all import *
>>> z = MatrixSpace(GF(Integer(7)), Integer(2), Integer(4)).zero_matrix(); z
[0 0 0 0]
[0 0 0 0]
>>> z.is_mutable()
False

Sage Live

z = MatrixSpace(GF(7), 2, 4).zero_matrix(); z
z.is_mutable()

zero_matrix()[source]¶

Return the zero matrix in self.

self must be a space of square matrices. The returned matrix is immutable. Please use copy if you want a modified copy.

EXAMPLES:

Sage

sage: z = MatrixSpace(GF(7), 2, 4).zero_matrix(); z
[0 0 0 0]
[0 0 0 0]
sage: z.is_mutable()
False

Python

>>> from sage.all import *
>>> z = MatrixSpace(GF(Integer(7)), Integer(2), Integer(4)).zero_matrix(); z
[0 0 0 0]
[0 0 0 0]
>>> z.is_mutable()
False

Sage Live

z = MatrixSpace(GF(7), 2, 4).zero_matrix(); z
z.is_mutable()

sage.matrix.matrix_space.dict_to_list(entries, nrows, ncols)[source]¶

Given a dictionary of coordinate tuples, return the list given by reading off the nrows*ncols matrix in row order.

EXAMPLES:

Sage

sage: from sage.matrix.matrix_space import dict_to_list
sage: d = {}
sage: d[(0,0)] = 1
sage: d[(1,1)] = 2
sage: dict_to_list(d, 2, 2)
[1, 0, 0, 2]
sage: dict_to_list(d, 2, 3)
[1, 0, 0, 0, 2, 0]

Python

>>> from sage.all import *
>>> from sage.matrix.matrix_space import dict_to_list
>>> d = {}
>>> d[(Integer(0),Integer(0))] = Integer(1)
>>> d[(Integer(1),Integer(1))] = Integer(2)
>>> dict_to_list(d, Integer(2), Integer(2))
[1, 0, 0, 2]
>>> dict_to_list(d, Integer(2), Integer(3))
[1, 0, 0, 0, 2, 0]

Sage Live

from sage.matrix.matrix_space import dict_to_list
d = {}
d[(0,0)] = 1
d[(1,1)] = 2
dict_to_list(d, 2, 2)
dict_to_list(d, 2, 3)

sage.matrix.matrix_space.get_matrix_class(R, nrows, ncols, sparse, implementation)[source]¶

Return a matrix class according to the input.

Note

This returns the base class without the category.

INPUT:

R – a base ring
nrows – number of rows
ncols – number of columns
sparse – boolean; whether the matrix class should be sparse
implementation – None or string or a matrix class; a possible implementation. See the documentation of the constructor of MatrixSpace.

EXAMPLES:

Sage

sage: from sage.matrix.matrix_space import get_matrix_class

sage: get_matrix_class(ZZ, 4, 5, False, None)                                   # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: get_matrix_class(ZZ, 4, 5, True, None)                                    # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_sparse.Matrix_integer_sparse'>

sage: get_matrix_class(ZZ, 3, 3, False, 'flint')                                # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: get_matrix_class(ZZ, 3, 3, False, 'gap')                                  # needs sage.libs.gap
<class 'sage.matrix.matrix_gap.Matrix_gap'>
sage: get_matrix_class(ZZ, 3, 3, False, 'generic')
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

sage: get_matrix_class(GF(2^15), 3, 3, False, None)                             # needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
sage: get_matrix_class(GF(2^17), 3, 3, False, None)                             # needs sage.rings.finite_rings
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

sage: get_matrix_class(GF(2), 2, 2, False, 'm4ri')                              # needs sage.libs.m4ri
<class 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'>
sage: get_matrix_class(GF(4), 2, 2, False, 'm4ri')                              # needs sage.libs.m4ri sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
sage: get_matrix_class(GF(7), 2, 2, False, 'linbox-float')                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_float.Matrix_modn_dense_float'>
sage: get_matrix_class(GF(7), 2, 2, False, 'linbox-double')                     # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_double.Matrix_modn_dense_double'>

sage: get_matrix_class(RDF, 2, 2, False, 'numpy')                               # needs numpy
<class 'sage.matrix.matrix_real_double_dense.Matrix_real_double_dense'>
sage: get_matrix_class(CDF, 2, 3, False, 'numpy')                               # needs numpy sage.rings.complex_double
<class 'sage.matrix.matrix_complex_double_dense.Matrix_complex_double_dense'>

sage: get_matrix_class(GF(25,'x'), 4, 4, False, 'meataxe')          # optional - meataxe, needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>
sage: get_matrix_class(IntegerModRing(3), 4, 4, False, 'meataxe')   # optional - meataxe
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>
sage: get_matrix_class(IntegerModRing(4), 4, 4, False, 'meataxe')
Traceback (most recent call last):
...
ValueError: 'meataxe' matrix can only deal with finite fields of order < 256
sage: get_matrix_class(GF(next_prime(255)), 4, 4, False, 'meataxe')             # needs sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: 'meataxe' matrix can only deal with finite fields of order < 256

sage: get_matrix_class(ZZ, 3, 5, False, 'crazy_matrix')
Traceback (most recent call last):
...
ValueError: unknown matrix implementation 'crazy_matrix' over Integer Ring
sage: get_matrix_class(GF(3), 2, 2, False, 'm4ri')
Traceback (most recent call last):
...
ValueError: 'm4ri' matrices are only available for fields of characteristic 2
and order <= 65536
sage: get_matrix_class(Zmod(2**30), 2, 2, False, 'linbox-float')                # needs sage.libs.linbox
Traceback (most recent call last):
...
ValueError: 'linbox-float' matrices can only deal with order < 256
sage: get_matrix_class(Zmod(2**30), 2, 2, False, 'linbox-double')               # needs sage.libs.linbox
Traceback (most recent call last):
...
ValueError: 'linbox-double' matrices can only deal with order < 94906266

sage: type(matrix(SR, 2, 2, 0))                                                 # needs sage.symbolic
<class 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense'>
sage: type(matrix(SR, 2, 2, 0, sparse=True))                                    # needs sage.symbolic
<class 'sage.matrix.matrix_symbolic_sparse.Matrix_symbolic_sparse'>
sage: type(matrix(GF(7), 2, range(4)))                                          # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_float.Matrix_modn_dense_float'>
sage: type(matrix(GF(16007), 2, range(4)))                                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_double.Matrix_modn_dense_double'>
sage: type(matrix(CBF, 2, range(4)))                                            # needs sage.libs.flint
<class 'sage.matrix.matrix_complex_ball_dense.Matrix_complex_ball_dense'>
sage: type(matrix(GF(2), 2, range(4)))                                          # needs sage.libs.m4ri
<class 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'>
sage: type(matrix(GF(64, 'z'), 2, range(4)))                                    # needs sage.libs.m4ri sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
sage: type(matrix(GF(125, 'z'), 2, range(4)))                       # optional - meataxe, needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>

Python

>>> from sage.all import *
>>> from sage.matrix.matrix_space import get_matrix_class

>>> get_matrix_class(ZZ, Integer(4), Integer(5), False, None)                                   # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
>>> get_matrix_class(ZZ, Integer(4), Integer(5), True, None)                                    # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_sparse.Matrix_integer_sparse'>

>>> get_matrix_class(ZZ, Integer(3), Integer(3), False, 'flint')                                # needs sage.libs.linbox
<class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
>>> get_matrix_class(ZZ, Integer(3), Integer(3), False, 'gap')                                  # needs sage.libs.gap
<class 'sage.matrix.matrix_gap.Matrix_gap'>
>>> get_matrix_class(ZZ, Integer(3), Integer(3), False, 'generic')
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

>>> get_matrix_class(GF(Integer(2)**Integer(15)), Integer(3), Integer(3), False, None)                             # needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
>>> get_matrix_class(GF(Integer(2)**Integer(17)), Integer(3), Integer(3), False, None)                             # needs sage.rings.finite_rings
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

>>> get_matrix_class(GF(Integer(2)), Integer(2), Integer(2), False, 'm4ri')                              # needs sage.libs.m4ri
<class 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'>
>>> get_matrix_class(GF(Integer(4)), Integer(2), Integer(2), False, 'm4ri')                              # needs sage.libs.m4ri sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
>>> get_matrix_class(GF(Integer(7)), Integer(2), Integer(2), False, 'linbox-float')                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_float.Matrix_modn_dense_float'>
>>> get_matrix_class(GF(Integer(7)), Integer(2), Integer(2), False, 'linbox-double')                     # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_double.Matrix_modn_dense_double'>

>>> get_matrix_class(RDF, Integer(2), Integer(2), False, 'numpy')                               # needs numpy
<class 'sage.matrix.matrix_real_double_dense.Matrix_real_double_dense'>
>>> get_matrix_class(CDF, Integer(2), Integer(3), False, 'numpy')                               # needs numpy sage.rings.complex_double
<class 'sage.matrix.matrix_complex_double_dense.Matrix_complex_double_dense'>

>>> get_matrix_class(GF(Integer(25),'x'), Integer(4), Integer(4), False, 'meataxe')          # optional - meataxe, needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>
>>> get_matrix_class(IntegerModRing(Integer(3)), Integer(4), Integer(4), False, 'meataxe')   # optional - meataxe
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>
>>> get_matrix_class(IntegerModRing(Integer(4)), Integer(4), Integer(4), False, 'meataxe')
Traceback (most recent call last):
...
ValueError: 'meataxe' matrix can only deal with finite fields of order < 256
>>> get_matrix_class(GF(next_prime(Integer(255))), Integer(4), Integer(4), False, 'meataxe')             # needs sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: 'meataxe' matrix can only deal with finite fields of order < 256

>>> get_matrix_class(ZZ, Integer(3), Integer(5), False, 'crazy_matrix')
Traceback (most recent call last):
...
ValueError: unknown matrix implementation 'crazy_matrix' over Integer Ring
>>> get_matrix_class(GF(Integer(3)), Integer(2), Integer(2), False, 'm4ri')
Traceback (most recent call last):
...
ValueError: 'm4ri' matrices are only available for fields of characteristic 2
and order <= 65536
>>> get_matrix_class(Zmod(Integer(2)**Integer(30)), Integer(2), Integer(2), False, 'linbox-float')                # needs sage.libs.linbox
Traceback (most recent call last):
...
ValueError: 'linbox-float' matrices can only deal with order < 256
>>> get_matrix_class(Zmod(Integer(2)**Integer(30)), Integer(2), Integer(2), False, 'linbox-double')               # needs sage.libs.linbox
Traceback (most recent call last):
...
ValueError: 'linbox-double' matrices can only deal with order < 94906266

>>> type(matrix(SR, Integer(2), Integer(2), Integer(0)))                                                 # needs sage.symbolic
<class 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense'>
>>> type(matrix(SR, Integer(2), Integer(2), Integer(0), sparse=True))                                    # needs sage.symbolic
<class 'sage.matrix.matrix_symbolic_sparse.Matrix_symbolic_sparse'>
>>> type(matrix(GF(Integer(7)), Integer(2), range(Integer(4))))                                          # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_float.Matrix_modn_dense_float'>
>>> type(matrix(GF(Integer(16007)), Integer(2), range(Integer(4))))                                      # needs sage.libs.linbox
<class 'sage.matrix.matrix_modn_dense_double.Matrix_modn_dense_double'>
>>> type(matrix(CBF, Integer(2), range(Integer(4))))                                            # needs sage.libs.flint
<class 'sage.matrix.matrix_complex_ball_dense.Matrix_complex_ball_dense'>
>>> type(matrix(GF(Integer(2)), Integer(2), range(Integer(4))))                                          # needs sage.libs.m4ri
<class 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'>
>>> type(matrix(GF(Integer(64), 'z'), Integer(2), range(Integer(4))))                                    # needs sage.libs.m4ri sage.rings.finite_rings
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
>>> type(matrix(GF(Integer(125), 'z'), Integer(2), range(Integer(4))))                       # optional - meataxe, needs sage.rings.finite_rings
<class 'sage.matrix.matrix_gfpn_dense.Matrix_gfpn_dense'>

Sage Live

from sage.matrix.matrix_space import get_matrix_class
get_matrix_class(ZZ, 4, 5, False, None)                                   # needs sage.libs.linbox
get_matrix_class(ZZ, 4, 5, True, None)                                    # needs sage.libs.linbox
get_matrix_class(ZZ, 3, 3, False, 'flint')                                # needs sage.libs.linbox
get_matrix_class(ZZ, 3, 3, False, 'gap')                                  # needs sage.libs.gap
get_matrix_class(ZZ, 3, 3, False, 'generic')
get_matrix_class(GF(2^15), 3, 3, False, None)                             # needs sage.rings.finite_rings
get_matrix_class(GF(2^17), 3, 3, False, None)                             # needs sage.rings.finite_rings
get_matrix_class(GF(2), 2, 2, False, 'm4ri')                              # needs sage.libs.m4ri
get_matrix_class(GF(4), 2, 2, False, 'm4ri')                              # needs sage.libs.m4ri sage.rings.finite_rings
get_matrix_class(GF(7), 2, 2, False, 'linbox-float')                      # needs sage.libs.linbox
get_matrix_class(GF(7), 2, 2, False, 'linbox-double')                     # needs sage.libs.linbox
get_matrix_class(RDF, 2, 2, False, 'numpy')                               # needs numpy
get_matrix_class(CDF, 2, 3, False, 'numpy')                               # needs numpy sage.rings.complex_double
get_matrix_class(GF(25,'x'), 4, 4, False, 'meataxe')          # optional - meataxe, needs sage.rings.finite_rings
get_matrix_class(IntegerModRing(3), 4, 4, False, 'meataxe')   # optional - meataxe
get_matrix_class(IntegerModRing(4), 4, 4, False, 'meataxe')
get_matrix_class(GF(next_prime(255)), 4, 4, False, 'meataxe')             # needs sage.rings.finite_rings
get_matrix_class(ZZ, 3, 5, False, 'crazy_matrix')
get_matrix_class(GF(3), 2, 2, False, 'm4ri')
get_matrix_class(Zmod(2**30), 2, 2, False, 'linbox-float')                # needs sage.libs.linbox
get_matrix_class(Zmod(2**30), 2, 2, False, 'linbox-double')               # needs sage.libs.linbox
type(matrix(SR, 2, 2, 0))                                                 # needs sage.symbolic
type(matrix(SR, 2, 2, 0, sparse=True))                                    # needs sage.symbolic
type(matrix(GF(7), 2, range(4)))                                          # needs sage.libs.linbox
type(matrix(GF(16007), 2, range(4)))                                      # needs sage.libs.linbox
type(matrix(CBF, 2, range(4)))                                            # needs sage.libs.flint
type(matrix(GF(2), 2, range(4)))                                          # needs sage.libs.m4ri
type(matrix(GF(64, 'z'), 2, range(4)))                                    # needs sage.libs.m4ri sage.rings.finite_rings
type(matrix(GF(125, 'z'), 2, range(4)))                       # optional - meataxe, needs sage.rings.finite_rings

sage.matrix.matrix_space.is_MatrixSpace(x)[source]¶

Return whether self is an instance of MatrixSpace.

EXAMPLES:

Sage

sage: from sage.matrix.matrix_space import is_MatrixSpace
sage: MS = MatrixSpace(QQ,2)
sage: A = MS.random_element()
sage: is_MatrixSpace(MS)
doctest:warning...
DeprecationWarning: the function is_MatrixSpace is deprecated;
use 'isinstance(..., MatrixSpace)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
sage: is_MatrixSpace(A)
False
sage: is_MatrixSpace(5)
False

Python

>>> from sage.all import *
>>> from sage.matrix.matrix_space import is_MatrixSpace
>>> MS = MatrixSpace(QQ,Integer(2))
>>> A = MS.random_element()
>>> is_MatrixSpace(MS)
doctest:warning...
DeprecationWarning: the function is_MatrixSpace is deprecated;
use 'isinstance(..., MatrixSpace)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
>>> is_MatrixSpace(A)
False
>>> is_MatrixSpace(Integer(5))
False

Sage Live

from sage.matrix.matrix_space import is_MatrixSpace
MS = MatrixSpace(QQ,2)
A = MS.random_element()
is_MatrixSpace(MS)
is_MatrixSpace(A)
is_MatrixSpace(5)