# Sparse matrices over $$\ZZ/n\ZZ$$ for $$n$$ small¶

This is a compiled implementation of sparse matrices over $$\ZZ/n\ZZ$$ for $$n$$ small.

TODO: - move vectors into a Cython vector class - add _add_ and _mul_ methods.

EXAMPLES:

sage: a = matrix(Integers(37),3,3,range(9),sparse=True); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: type(a)
<type 'sage.matrix.matrix_modn_sparse.Matrix_modn_sparse'>
sage: parent(a)
Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37
sage: a^2
[15 18 21]
[ 5 17 29]
[32 16  0]
sage: a+a
[ 0  2  4]
[ 6  8 10]
[12 14 16]
sage: b = a.new_matrix(2,3,range(6)); b
[0 1 2]
[3 4 5]
sage: a*b
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37' and 'Full MatrixSpace of 2 by 3 sparse matrices over Ring of integers modulo 37'
sage: b*a
[15 18 21]
[ 5 17 29]
sage: TestSuite(a).run()
sage: TestSuite(b).run()
sage: a.echelonize(); a
[ 1  0 36]
[ 0  1  2]
[ 0  0  0]
sage: b.echelonize(); b
[ 1  0 36]
[ 0  1  2]
sage: a.pivots()
(0, 1)
sage: b.pivots()
(0, 1)
sage: a.rank()
2
sage: b.rank()
2
sage: a[2,2] = 5
sage: a.rank()
3
TESTS:
sage: matrix(Integers(37),0,0,sparse=True).inverse() []
class sage.matrix.matrix_modn_sparse.Matrix_modn_sparse

Create a sparse matrix modulo n.

INPUT:

• parent - a matrix space
• entries
• a Python list of triples (i,j,x), where 0 <= i < nrows, 0 <= j < ncols, and x is coercible to an int. The i,j entry of self is set to x. The x’s can be 0.
• Alternatively, entries can be a list of all the entries of the sparse matrix (so they would be mostly 0).
• copy - ignored
• coerce - ignored
density()

Return the density of self, i.e., the ratio of the number of nonzero entries of self to the total size of self.

EXAMPLES:

sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8],sparse=True)
sage: A.density()
2/3

Notice that the density parameter does not ensure the density of a matrix; it is only an upper bound.

sage: A = random_matrix(GF(127),200,200,density=0.3, sparse=True)
sage: A.density()
2073/8000
lift()

Return lift of this matrix to a sparse matrix over the integers.

EXAMPLES:
sage: a = matrix(GF(7),2,3,[1..6], sparse=True) sage: a.lift() [1 2 3] [4 5 6] sage: a.lift().parent() Full MatrixSpace of 2 by 3 sparse matrices over Integer Ring

Subdivisions are preserved when lifting:

sage: a.subdivide([], [1,1]); a
[1||2 3]
[4||5 6]
sage: a.lift()
[1||2 3]
[4||5 6]
matrix_from_columns(cols)

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

EXAMPLES:

sage: M = MatrixSpace(GF(127),3,3,sparse=True)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.matrix_from_columns([2,1])
[2 1]
[5 4]
[8 7]
matrix_from_rows(rows)

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

INPUT:

• rows - list or tuple of row indices

EXAMPLE:

sage: M = MatrixSpace(GF(127),3,3,sparse=True)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.matrix_from_rows([2,1])
[6 7 8]
[3 4 5]
p
rank(gauss=False)

Compute the rank of self.

INPUT:

• gauss - if True LinBox’ Gaussian elimination is used. If False ‘Symbolic Reordering’ as implemented in LinBox is used. If ‘native’ the native Sage implementation is used. (default: False)

EXAMPLE:

sage: A = random_matrix(GF(127),200,200,density=0.01,sparse=True)
sage: r1 = A.rank(gauss=False)
sage: r2 = A.rank(gauss=True)
sage: r3 = A.rank(gauss='native')
sage: r1 == r2 == r3
True
sage: r1
155

ALGORITHM: Uses LinBox or native implementation.

REFERENCES:

Note

For very sparse matrices Gaussian elimination is faster because it barly has anything to do. If the fill in needs to be considered, ‘Symbolic Reordering’ is usually much faster.

swap_rows(r1, r2)
transpose()

Return the transpose of self.

EXAMPLE:

sage: A = matrix(GF(127),3,3,[0,1,0,2,0,0,3,0,0],sparse=True)
sage: A
[0 1 0]
[2 0 0]
[3 0 0]
sage: A.transpose()
[0 2 3]
[1 0 0]
[0 0 0]

.T is a convenient shortcut for the transpose:

sage: A.T
[0 2 3]
[1 0 0]
[0 0 0]
visualize_structure(filename=None, maxsize=512)

Write a PNG image to ‘filename’ which visualizes self by putting black pixels in those positions which have nonzero entries.

White pixels are put at positions with zero entries. If ‘maxsize’ is given, then the maximal dimension in either x or y direction is set to ‘maxsize’ depending on which is bigger. If the image is scaled, the darkness of the pixel reflects how many of the represented entries are nonzero. So if e.g. one image pixel actually represents a 2x2 submatrix, the dot is darker the more of the four values are nonzero.

INPUT:

• filename - either a path or None in which case a filename in the current directory is chosen automatically (default:None)
• maxsize - maximal dimension in either x or y direction of the resulting image. If None or a maxsize larger than max(self.nrows(),self.ncols()) is given the image will have the same pixelsize as the matrix dimensions (default: 512)

EXAMPLES:

sage: M = Matrix(GF(7), [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,1,0]], sparse=True); M
[0 0 0 1 0 0 0 0]
[0 1 0 0 0 0 1 0]
sage: M.visualize_structure()

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Dense matrices over $$\ZZ/n\ZZ$$ for $$n$$ small

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Symbolic matrices
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