Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
density()
Return the density of the matrix.
By density we understand the ratio of the number of nonzero positions and the 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([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] # long time
# long time (4s on sage.ma
285