Base class for sparse matrices

class sage.matrix.matrix_sparse.Matrix_sparse

Bases: sage.matrix.matrix.Matrix

antitranspose()

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:

sparse – False to make the output a dense matrix; default True

• phi - arbitrary Python function or callable object
• R - (optional) ring

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 didn’t 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, [0] * 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

TESTS:

sage: m = matrix([], sparse=True)
sage: m.apply_map(lambda x: x*x) == m
True

sage: m.apply_map(lambda x: x*x, sparse=False).parent()
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring

Check that we don’t unnecessarily apply phi to 0 in the sparse case:

sage: m = matrix(QQ, 2, 2, range(1, 5), sparse=True)
sage: m.apply_map(lambda x: 1/x)
[  1 1/2]
[1/3 1/4]

Test subdivisions when phi maps 0 to non-zero:

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+1)
[1|1]
[4|1]

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

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

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

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

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([1], [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)
sage: B=A[50000:,:50000]                                   
sage: len(B.nonzero_positions())
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])
...
TypeError: rows must be a list of integers
sage: A.matrix_from_rows_and_columns([1,2],3)  
...
TypeError: columns must be a list of integers

AUTHORS:

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

transpose()

Returns 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])
sage: B = A.transpose()
sage: print B
[1 3]
[2 4]
sage: print A
[1 2]
[3 4]