Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: c.sparse_rows() [(5, x, x^2), (x^3, x^4, x^5)] sage: d = c.dense_matrix(); d [ 5 x x^2] [x^3 x^4 x^5] sage: parent(d) Full MatrixSpace of 2 by 3 dense matrices over Univariate Polynomial Ring in x over Rational Field sage: c.sparse_matrix() is c True sage: c.is_sparse() True class sage.matrix.matrix_generic_sparse.Matrix_generic_sparse Bases: sage.matrix.matrix_sparse.Matrix_sparse The Matrix_generic_sparse class derives from Matrix, and defines functionality for sparse matrices over any base ring. A generic sparse matrix is represented using a dictionary with keys pairs (i, j) and values in the base ring. The values of the dictionary must never be zero. sage.matrix.matrix_generic_sparse.Matrix_sparse_from_rows(X) INPUT: •X - nonempty list of SparseVector rows OUTPUT: Sparse_matrix with those rows. EXAMPLES: sage: V = VectorSpace(QQ,20,sparse=True) sage: v = V(0) sage: v[9] = 4 sage: from sage.matrix.matrix_generic_sparse import Matrix_sparse_from_rows sage: Matrix_sparse_from_rows([v]) [0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0] sage: Matrix_sparse_from_rows([v, v, v, V(0)]) [0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 292 Chapter 13. Sparse Matrices over a general ring
CHAPTER FOURTEEN DENSE MATRICES OVER Z/NZ FOR N SMALL Dense matrices over Z/nZ for n small AUTHORS: • William Stein • Robert Bradshaw This is a compiled implementation of dense matrices over Z/nZ for n small. EXAMPLES: sage: a = matrix(Integers(37),3,range(9),sparse=False); a [0 1 2] [3 4 5] [6 7 8] sage: a.rank() 2 sage: type(a) sage: a[0,0] = 5 sage: a.rank() 3 sage: parent(a) Full MatrixSpace of 3 by 3 dense matrices over Ring of integers modulo 37 sage: a^2 [ 3 23 31] [20 17 29] [25 16 0] sage: a+a [10 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 dense matrices over Rin sage: b*a 293
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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