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: 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

TESTS:

sage: matrix(RR,2,2,sparse=True)
[0.000000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000]
sage: matrix(GF(11),2,2,sparse=True)
[0 0]
[0 0]
sage.matrix.matrix_space.MatrixSpace(base_ring, nrows, ncols=None, sparse=False)

Create with the command

MatrixSpace(base_ring , nrows [, ncols] [, sparse])

The default value of the optional argument sparse is False. The default value of the optional argument ncols is nrows.

INPUT:

  • base_ring - a ring
  • nrows - int, the number of rows
  • ncols - (default nrows) int, the number of columns
  • sparse - (default false) whether or not matrices are given a sparse representation

OUTPUT: The unique space of all nrows x ncols matrices over base_ring.

EXAMPLES:

sage: MS = MatrixSpace(RationalField(),2)
sage: MS.base_ring()
Rational Field
sage: MS.dimension()
4
sage: MS.dims()
(2, 2)
sage: B = MS.basis()
sage: B
[
[1 0]
[0 0],
[0 1]
[0 0],
[0 0]
[1 0],
[0 0]
[0 1]
]
sage: B[0]
[1 0]
[0 0]
sage: B[1]
[0 1]
[0 0]
sage: B[2]
[0 0]
[1 0]
sage: B[3]
[0 0]
[0 1]
sage: A = MS.matrix([1,2,3,4])
sage: A
[1 2]
[3 4]
sage: MS2 = MatrixSpace(RationalField(),2,3)
sage: B = MS2.matrix([1,2,3,4,5,6])
sage: A*B
[ 9 12 15]
[19 26 33]
sage: M = MatrixSpace(ZZ, 10)
sage: M
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring
sage: loads(M.dumps()) == M
True
class sage.matrix.matrix_space.MatrixSpace_generic(base_ring, nrows, ncols=None, sparse=False)

The space of all nrows x ncols matrices over base_ring.

EXAMPLES:

sage: MatrixSpace(ZZ,10,5)
Full MatrixSpace of 10 by 5 dense matrices over Integer Ring
sage: MatrixSpace(ZZ,10,2^31)
...                                                                  # 32-bit
ValueError: number of rows and columns must be less than 2^31 (on a 32-bit computer -- use a 64-bit computer for matrices with up to 2^63-1 rows and columns)           # 32-bit
Full MatrixSpace of 10 by 2147483648 dense matrices over Integer Ring   # 64-bit
__call__(entries=0, coerce=True, copy=True, rows=None)

EXAMPLES:

sage: k = GF(7); G = MatrixGroup([matrix(k,2,[1,1,0,1]), matrix(k,2,[1,0,0,2])])
sage: g = G.0
sage: MatrixSpace(k,2)(g)
[1 1]
[0 1]
sage: MS = MatrixSpace(ZZ,2,4)
sage: M2 = MS(range(8)); M2
[0 1 2 3]
[4 5 6 7]
sage: M2.columns()
[(0, 4), (1, 5), (2, 6), (3, 7)]
sage: MS(M2.columns())
[0 1 2 3]
[4 5 6 7]
sage: M2 == MS(M2.columns())
True
sage: M2 == MS(M2.rows())
True
sage: MS = MatrixSpace(ZZ,2,4, sparse=True)
sage: M2 = MS(range(8)); M2
[0 1 2 3]
[4 5 6 7]
sage: M2.columns()
[(0, 4), (1, 5), (2, 6), (3, 7)]
sage: MS(M2.columns())
[0 1 2 3]
[4 5 6 7]
sage: M2 == MS(M2.columns())
True
sage: M2 == MS(M2.rows())
True
sage: MS = MatrixSpace(ZZ,2,2)
sage: MS([1,2,3,4])
[1 2]
[3 4]
sage: MS([1,2,3,4], rows=True)
[1 2]
[3 4]
sage: MS([1,2,3,4], rows=False)
[1 3]
[2 4]
sage: MS = MatrixSpace(ZZ,2,2, sparse=True)
sage: MS([1,2,3,4])
[1 2]
[3 4]
sage: MS([1,2,3,4], rows=True)
[1 2]
[3 4]
sage: MS([1,2,3,4], rows=False)
[1 3]
[2 4]

sage: MS = MatrixSpace(ZZ, 2)
sage: g = Gamma0(5)([1,1,0,1])
sage: MS(g)
[1 1]
[0 1]
__cmp__(other)

Compare this matrix space with other. Sparse and dense matrix spaces with otherwise the same parameters are considered equal.

If other is not a matrix space, return something arbitrary but deterministic. Otherwise, compare based on base ring, then on number of rows and columns.

EXAMPLES:

sage: Mat(ZZ,1000) == Mat(QQ,1000)
False
sage: Mat(ZZ,10) == Mat(ZZ,10)
True
sage: Mat(ZZ,10, sparse=False) == Mat(ZZ,10, sparse=True)
True
__init__(base_ring, nrows, ncols=None, sparse=False)
__iter__()

Returns a generator object which iterates through the elements of self. The order in which the elements are generated is based on a ‘weight’ of a matrix which is the number of iterations on the base ring that are required to reach that matrix.

The ordering is similar to a degree negative lexicographic order in monomials in a multivariate polynomial ring.

EXAMPLES: Consider the case of 2 x 2 matrices over GF(5).

sage: list( GF(5) )
[0, 1, 2, 3, 4]
sage: MS = MatrixSpace(GF(5), 2, 2)
sage: l = list(MS)

Then, consider the following matrices:

sage: A = MS([2,1,0,1]); A
[2 1]
[0 1]
sage: B = MS([1,2,1,0]); B
[1 2]
[1 0]
sage: C = MS([1,2,0,0]); C
[1 2]
[0 0]

A appears before B since the weight of one of A’s entries exceeds the weight of the corresponding entry in B earliest in the list.

sage: l.index(A)
41
sage: l.index(B)
46

However, A would come after the matrix C since C has a lower weight than A.

sage: l.index(A)
41
sage: l.index(C)
19

The weights of matrices over other base rings are not as obvious. For example, the weight of

sage: MS = MatrixSpace(ZZ, 2, 2)
sage: MS([-1,0,0,0])
[-1  0]
[ 0  0]

is 2 since

sage: i = iter(ZZ)
sage: i.next()
0
sage: i.next()
1
sage: i.next()
-1

Some more examples:

sage: MS = MatrixSpace(GF(2),2)
sage: a = list(MS)
sage: len(a)
16
sage: for m in a: print m, '\n-'
[0 0]
[0 0] 
-
[1 0]
[0 0] 
-
[0 1]
[0 0] 
-
[0 0]
[1 0] 
-
[0 0]
[0 1] 
-
[1 1]
[0 0] 
-           
[1 0]
[1 0] 
-
[1 0]
[0 1] 
-
[0 1]
[1 0] 
-
[0 1]
[0 1] 
-
[0 0]
[1 1] 
-
[1 1]
[1 0] 
-
[1 1]
[0 1] 
-
[1 0]
[1 1] 
-
[0 1]
[1 1] 
-
[1 1]
[1 1] 
-
sage: MS = MatrixSpace(GF(2),2, 3)
sage: a = list(MS)
sage: len(a)
64
sage: a[0]
[0 0 0]
[0 0 0]
sage: MS = MatrixSpace(ZZ, 2, 3)
sage: i = iter(MS)
sage: a = [ i.next() for _ in range(6) ]
sage: a[0]
[0 0 0]
[0 0 0]
sage: a[4]
[0 0 0]
[1 0 0]

For degenerate cases, where either the number of rows or columns (or both) are zero, then the single element of the space is returned.

sage: list( MatrixSpace(GF(2), 2, 0) )
[[]]
sage: list( MatrixSpace(GF(2), 0, 2) )
[[]]
sage: list( MatrixSpace(GF(2), 0, 0) )
[[]]

If the base ring does not support iteration (for example, with the reals), then the matrix space over that ring does not support iteration either.

sage: MS = MatrixSpace(RR, 2)
sage: a = list(MS)
...
NotImplementedError: object does not support iteration
__reduce__()

EXAMPLES:

sage: A = Mat(ZZ,5,7,sparse=True)
sage: A
Full MatrixSpace of 5 by 7 sparse matrices over Integer Ring
sage: loads(dumps(A)) == A
True
_coerce_impl(x)

EXAMPLES:

sage: MS1 = MatrixSpace(QQ,3)
sage: MS2 = MatrixSpace(ZZ,4,5,true)
sage: A = MS1(range(9))
sage: D = MS2(range(20))
sage: MS1._coerce_(A)
[0 1 2]
[3 4 5]
[6 7 8]
sage: MS2._coerce_(D)
[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]
[15 16 17 18 19]
_get_matrix_class()

Returns the class of self

EXAMPLES:

sage: MS1 = MatrixSpace(QQ,4)
sage: MS2 = MatrixSpace(ZZ,4,5,true)
sage: MS1._get_matrix_class()
<type 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
sage: MS2._get_matrix_class()
<type 'sage.matrix.matrix_integer_sparse.Matrix_integer_sparse'>
sage: type(matrix(SR, 2, 2, 0))
<type 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense'>
_latex_()

Returns the latex representation of a MatrixSpace

EXAMPLES:

sage: MS3 = MatrixSpace(QQ,6,6,true)
sage: latex(MS3)
\mathrm{Mat}_{6\times 6}(\Bold{Q})
_magma_init_(magma)

EXAMPLES: We first coerce a square matrix.

sage: magma(MatrixSpace(QQ,3))                      # optional - magma
Full Matrix Algebra of degree 3 over Rational Field
sage: magma(MatrixSpace(Integers(8),2,3))           # optional - magma
Full RMatrixSpace of 2 by 3 matrices over IntegerRing(8)
_repr_()

Returns the string representation of a MatrixSpace

EXAMPLES:

sage: MS = MatrixSpace(ZZ,2,4,true)
sage: repr(MS)
'Full MatrixSpace of 2 by 4 sparse matrices over Integer Ring'
sage: MS
Full MatrixSpace of 2 by 4 sparse matrices over Integer Ring
base_extend(R)

Return base extension of this matrix space to R.

INPUT:

  • R - ring

OUTPUT: a matrix space

EXAMPLES:

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))
...
TypeError: no base extension defined
basis()

Returns a basis for this matrix space.

Warning

This will of course compute every generator of this matrix space. So for large matrices, 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: Mat(ZZ,2,2).basis()
[
[1 0]
[0 0],
[0 1]
[0 0],
[0 0]
[1 0],
[0 0]
[0 1]
]
change_ring(R)

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

INPUT:

  • R - ring

OUTPUT: a matrix space

EXAMPLES:

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

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

EXAMPLES:

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
dimension()

Returns (m rows) * (n cols) of self as Integer

EXAMPLES:

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

Returns (m row, n col) representation of self dimension

EXAMPLES:

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

Return the n-th generator of this matrix space.

This doesn’t compute all basis matrices, so it is reasonably intelligent.

EXAMPLES:

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]
get_action_impl(S, op, self_on_left)
identity_matrix()

Create an identity matrix in self. (Must be a space of square matrices).

EXAMPLES:

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()
...
TypeError: self must be a space of square matrices
is_dense()

Returns True if matrices in self are dense and False otherwise.

EXAMPLES:

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

EXAMPLES:

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

Returns True if matrices in self are sparse and False otherwise.

EXAMPLES:

sage: Mat(GF(2011),10000).is_sparse()
False
sage: Mat(GF(2011),10000,sparse=True).is_sparse()
True
matrix(x=0, coerce=True, copy=True, rows=True)

Create a matrix in self. The entries can be specified either as a single list of length nrows*ncols, or as a list of lists.

EXAMPLES:

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]
sage: M.matrix([1,2,3,4],rows=False)
[1 3]
[2 4]
matrix_space(nrows=None, ncols=None, sparse=False)

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

EXAMPLES:

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
ncols()

Return the number of columns of matrices in this space.

EXAMPLES:

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

Return the number of generators of this matrix space, which is the number of entries in the matrices in this space.

EXAMPLES:

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

Return the number of rows of matrices in this space.

EXAMPLES:

sage: M = Mat(ZZ,200000,500000)
sage: M.nrows()
200000
random_element(density=1, *args, **kwds)

INPUT:

  • density - integer (default: 1) rough measure of the proportion of nonzero entries in the random matrix
  • *args, **kwds - rest of parameters 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.)

EXAMPLES:

sage: Mat(ZZ,2,5).random_element()
[ -8   2   0   0   1]
[ -1   2   1 -95  -1]
sage: Mat(QQ,2,5).random_element(density=0.5)
[  2   0   0   0   1]
[  0   0   0 1/2   0]
sage: Mat(QQ,3,sparse=True).random_element()
[  -1  1/3    1]
[   0   -1    0]
[  -1    1 -1/4]
sage: Mat(GF(9,'a'),3,sparse=True).random_element()
[      1       2       1]
[2*a + 1       a       2]
[      2 2*a + 2       1]
row_space()

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: M = Mat(ZZ,20,5,sparse=False); M.row_space()
Ambient free module of rank 5 over the principal ideal domain Integer Ring
zero_matrix()
Return the zero matrix.
sage.matrix.matrix_space.dict_to_list(entries, nrows, ncols)

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

EXAMPLES:

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]
sage.matrix.matrix_space.is_MatrixSpace(x)

Returns True if self is an instance of MatrixSpace returns false if self is not an instance of MatrixSpace

EXAMPLES:

sage: from sage.matrix.matrix_space import is_MatrixSpace
sage: MS = MatrixSpace(QQ,2)
sage: A = MS.random_element()
sage: is_MatrixSpace(MS)
True
sage: is_MatrixSpace(A)
False
sage: is_MatrixSpace(5)
False
sage.matrix.matrix_space.list_to_dict(entries, nrows, ncols, rows=True)

Given a list of entries, create a dictionary whose keys are coordinate tuples and values are the entries.

EXAMPLES:

sage: from sage.matrix.matrix_space import list_to_dict
sage: d = list_to_dict([1,2,3,4],2,2)
sage: d[(0,1)]
2
sage: d = list_to_dict([1,2,3,4],2,2,rows=False)
sage: d[(0,1)]
3
sage.matrix.matrix_space.test_trivial_matrices_inverse(ring, sparse=True, checkrank=True)

Tests inversion, determinant and is_inverstible for trivial matrices.

This function is a helper to check that the inversion of trivial matrices (of size 0x0, nx0, 0xn or 1x1) is handled consistently by the various implementation of matrices. The coherency is checked through a bunch of assertions. If an inconsistency is found, an AssertionError is raised which should make clear what is the problem.

INPUT:

  • ring - a ring
  • sparse - a boolean
  • checkrank - a boolean

OUTPUT:

  • nothing if everything is correct otherwise raise an AssertionError
The methods determinant, is_invertible, rank and inverse are checked for
  • the 0x0 empty identity matrix
  • the 0x3 and 3x0 matrices
  • the 1x1 null matrix [0]
  • the 1x1 identity matrix [1]

If checkrank is False then the rank is not checked. This is used the check matrix over ring where echelon form is not implemented.

TODO: must be adapted to category check framework when ready (see trac #5274).

TESTS:

sage: from sage.matrix.matrix_space import test_trivial_matrices_inverse as tinv
sage: tinv(ZZ, sparse=True)
sage: tinv(ZZ, sparse=False)
sage: tinv(QQ, sparse=True)
sage: tinv(QQ, sparse=False)
sage: tinv(GF(11), sparse=True)
sage: tinv(GF(11), sparse=False)
sage: tinv(GF(2), sparse=True)
sage: tinv(GF(2), sparse=False)
sage: tinv(SR, sparse=True)
sage: tinv(SR, sparse=False)
sage: tinv(RDF, sparse=True)
sage: tinv(RDF, sparse=False)
sage: tinv(CDF, sparse=True)
sage: tinv(CDF, sparse=False)
sage: tinv(CyclotomicField(7), sparse=True)
sage: tinv(CyclotomicField(7), sparse=False)
sage: tinv(QQ['x,y'], sparse=True)

TODO: As soon as rank of dense matrix over QQ[‘x,y’] is implemented, please remove the following test and the checkrank=False in the next one:

sage: MatrixSpace(QQ[‘x,y’], 3, 3, sparse=False)(1).rank() Traceback (most recent call last): ... RuntimeError: BUG: matrix pivots should have been set but weren’t, matrix parent = ‘Full MatrixSpace of 3 by 3 dense matrices over Multivariate Polynomial Ring in x, y over Rational Field’

sage: tinv(QQ[‘x,y’], sparse=False, checkrank=False)

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