# Base class for sparse matrices¶

class sage.matrix.matrix_sparse.Matrix_sparse
antitranspose()

Return the antitranspose of self, without changing self.

This is the mirror image along the other diagonal.

EXAMPLES:

sage: M = MatrixSpace(QQ, 2, sparse=True)
sage: A = M([1,2,3,4]); A
[1 2]
[3 4]
sage: A.antitranspose()
[4 2]
[3 1]

apply_map(phi, R=None, sparse=True)

Apply the given map phi (an arbitrary Python function or callable object) to this matrix.

If R is not given, automatically determine the base ring of the resulting matrix.

INPUT:

• phi – arbitrary Python function or callable object
• R – (optional) ring
• sparse – (optional, default True) whether to return a sparse or a dense matrix

OUTPUT: a matrix over R

EXAMPLES:

sage: m = matrix(ZZ, 10000, {(1,2): 17}, sparse=True)
sage: k.<a> = GF(9)
sage: f = lambda x: k(x)
sage: n = m.apply_map(f)
sage: n.parent()
Full MatrixSpace of 10000 by 10000 sparse matrices over Finite Field in a of size 3^2
sage: n[1,2]
2


An example where the codomain is explicitly specified.

sage: n = m.apply_map(lambda x:x%3, GF(3))
sage: n.parent()
Full MatrixSpace of 10000 by 10000 sparse matrices over Finite Field of size 3
sage: n[1,2]
2


If we did not specify the codomain, the resulting matrix in the above case ends up over $$\ZZ$$ again:

sage: n = m.apply_map(lambda x:x%3)
sage: n.parent()
Full MatrixSpace of 10000 by 10000 sparse matrices over Integer Ring
sage: n[1,2]
2


If self is subdivided, the result will be as well:

sage: m = matrix(2, 2, [0, 0, 3, 0])
sage: m.subdivide(None, 1); m
[0|0]
[3|0]
sage: m.apply_map(lambda x: x*x)
[0|0]
[9|0]


If the map sends zero to a non-zero value, then it may be useful to get the result as a dense matrix.

sage: m = matrix(ZZ, 3, 3,  * 7 + [1,2], sparse=True); m
[0 0 0]
[0 0 0]
[0 1 2]
sage: parent(m)
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring
sage: n = m.apply_map(lambda x: x+polygen(QQ), sparse=False); n
[    x     x     x]
[    x     x     x]
[    x x + 1 x + 2]
sage: parent(n)
Full MatrixSpace of 3 by 3 dense matrices over Univariate Polynomial Ring in x over Rational Field

apply_morphism(phi)

Apply the morphism phi to the coefficients of this sparse matrix.

The resulting matrix is over the codomain of phi.

INPUT:

• phi – a morphism, so phi is callable and
phi.domain() and phi.codomain() are defined. The codomain must be a ring.

OUTPUT: a matrix over the codomain of phi

EXAMPLES:

sage: m = matrix(ZZ, 3, range(9), sparse=True)
sage: phi = ZZ.hom(GF(5))
sage: m.apply_morphism(phi)
[0 1 2]
[3 4 0]
[1 2 3]
sage: m.apply_morphism(phi).parent()
Full MatrixSpace of 3 by 3 sparse matrices over Finite Field of size 5

augment(right, subdivide=False)

Return the augmented matrix of the form:

[self | right].


EXAMPLES:

sage: M = MatrixSpace(QQ, 2, 2, sparse=True)
sage: A = M([1,2, 3,4])
sage: A
[1 2]
[3 4]
sage: N = MatrixSpace(QQ, 2, 1, sparse=True)
sage: B = N([9,8])
sage: B


sage: A.augment(B)
[1 2 9]
[3 4 8]
sage: B.augment(A)
[9 1 2]
[8 3 4]


A vector may be augmented to a matrix.

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


The subdivide option will add a natural subdivision between self and right. For more details about how subdivisions are managed when augmenting, see sage.matrix.matrix1.Matrix.augment().

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

change_ring(ring)

Return the matrix obtained by coercing the entries of this matrix into the given ring.

Always returns a copy (unless self is immutable, in which case returns self).

EXAMPLES:

sage: A = matrix(QQ['x,y'], 2, [0,-1,2*x,-2], sparse=True); A
[  0  -1]
[2*x  -2]
sage: A.change_ring(QQ['x,y,z'])
[  0  -1]
[2*x  -2]


Subdivisions are preserved when changing rings:

sage: A.subdivide(,[]); A
[  0  -1]
[2*x  -2]
[-------]
sage: A.change_ring(RR['x,y'])
[                 0  -1.00000000000000]
[2.00000000000000*x  -2.00000000000000]
[-------------------------------------]

charpoly(var='x', **kwds)

Return the characteristic polynomial of this matrix.

Note

the generic sparse charpoly implementation in Sage is to just compute the charpoly of the corresponding dense matrix, so this could use a lot of memory. In particular, for this matrix, the charpoly will be computed using a dense algorithm.

EXAMPLES:

sage: A = matrix(ZZ, 4, range(16), sparse=True)
sage: A.charpoly()
x^4 - 30*x^3 - 80*x^2
sage: A.charpoly('y')
y^4 - 30*y^3 - 80*y^2
sage: A.charpoly()
x^4 - 30*x^3 - 80*x^2

density()

Return the density of the matrix.

By density we understand the ratio of the number of nonzero positions and the number self.nrows() * self.ncols(), i.e. the number of possible nonzero positions.

EXAMPLES:

sage: a = matrix([[],[],[],[]], sparse=True); a.density()
0
sage: a = matrix(5000,5000,{(1,2): 1}); a.density()
1/25000000

determinant(**kwds)

Return the determinant of this matrix.

Note

the generic sparse determinant implementation in Sage is to just compute the determinant of the corresponding dense matrix, so this could use a lot of memory. In particular, for this matrix, the determinant will be computed using a dense algorithm.

EXAMPLES:

sage: A = matrix(ZZ, 4, range(16), sparse=True)
sage: B = A + identity_matrix(ZZ, 4, sparse=True)
sage: B.det()
-49

matrix_from_rows_and_columns(rows, columns)

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

EXAMPLES:

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


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]


We can efficiently extract large submatrices:

sage: A = random_matrix(ZZ, 100000, density=.00005, sparse=True)  # long time (4s on sage.math, 2012)
sage: B = A[50000:,:50000]        # long time
sage: len(B.nonzero_positions())  # long time
17550              # 32-bit
125449             # 64-bit


We must pass in a list of indices:

sage: A = random_matrix(ZZ,100,density=.02,sparse=True)
sage: A.matrix_from_rows_and_columns(1,[2,3])
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable

sage: A.matrix_from_rows_and_columns([1,2],3)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable


AUTHORS:

• Jaap Spies (2006-02-18)
• Didier Deshommes: some Pyrex speedups implemented
• Jason Grout: sparse matrix optimizations
transpose()

Return the transpose of self, without changing self.

EXAMPLES: We create a matrix, compute its transpose, and note that the original matrix is not changed.

sage: M = MatrixSpace(QQ, 2, sparse=True)
sage: A = M([1,2,3,4]); A
[1 2]
[3 4]
sage: B = A.transpose(); B
[1 3]
[2 4]


.T is a convenient shortcut for the transpose:

sage: A.T
[1 3]
[2 4]