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: MatrixSpace(Qp(3),1,1)([Qp(3).zero()]) [0] sage: MatrixSpace(Qp(3),1,1)([Qp(3)(4/3)]) [3^-1 + 1 + O(3^19)] matrix_space(nrows=None, ncols=None, sparse=False) Return the matrix space with given number of rows, columns and sparcity over the same base ring as self, and defaults the same as self. EXAMPLES: sage: M = Mat(GF(7),100,200) sage: M.matrix_space(5000) Full MatrixSpace of 5000 by 200 dense matrices over Finite Field of size 7 sage: M.matrix_space(ncols=5000) Full MatrixSpace of 100 by 5000 dense matrices over Finite Field of size 7 sage: M.matrix_space(sparse=True) Full MatrixSpace of 100 by 200 sparse matrices over Finite Field of size 7 ncols() Return the number of columns of matrices in this space. EXAMPLES: sage: M = Mat(ZZ[’x’],200000,500000,sparse=True) sage: M.ncols() 500000 ngens() Return the number of generators of this matrix space, which is the number of entries in the matrices in this space. EXAMPLES: sage: M = Mat(GF(7),100,200); M.ngens() 20000 nrows() Return the number of rows of matrices in this space. EXAMPLES: sage: M = Mat(ZZ,200000,500000) sage: M.nrows() 200000 one() Returns the identity matrix in self. self must be a space of square matrices. The returned matrix is immutable. Please use copy if you want a modified copy. EXAMPLES: sage: MS1 = MatrixSpace(ZZ,4) sage: MS2 = MatrixSpace(QQ,3,4) sage: I = MS1.identity_matrix() sage: I [1 0 0 0] [0 1 0 0] [0 0 1 0] 10 Chapter 1. Matrix Spaces
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [0 0 0 1] sage: Er = MS2.identity_matrix() Traceback (most recent call last): ... TypeError: self must be a space of square matrices TESTS: sage: MS1.one()[1,2] = 3 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a random_element(density=None, *args, **kwds) Returns a random element from this matrix space. INPUT: •density - float or None (default: None); rough measure of the proportion of nonzero entries in the random matrix; if set to None, all entries of the matrix are randomized, allowing for any element of the underlying ring, but if set to a float, a proportion of entries is selected and randomized to non-zero elements of the ring •*args, **kwds - remaining parameters, which may be passed to the random_element function of the base ring. (“may be”, since this function calls the randomize function on the zero matrix, which need not call the random_element function of the base ring at all in general.) OUTPUT: •Matrix NOTES: This method will randomize a proportion of roughly density entries in a newly allocated zero matrix. By default, if the user sets the value of density explicitly, this method will enforce that these entries are set to non-zero values. However, if the test for equality with zero in the base ring is too expensive, the user can override this behaviour by passing the argument nonzero=False to this method. Otherwise, if the user does not set the value of density, the default value is taken to be 1, and the option nonzero=False is passed to the randomize method. EXAMPLES: sage: Mat(ZZ,2,5).random_element() [ -8 2 0 0 1] [ -1 2 1 -95 -1] sage: Mat(QQ,2,5).random_element(density=0.5) [ 2 0 0 0 1] [ 0 0 0 -1 0] sage: Mat(QQ,3,sparse=True).random_element() [ -1 -1 -1] [ -3 -1/3 -1] [ 0 -1 1] sage: Mat(GF(9,’a’),3,sparse=True).random_element() [ a 2*a 1] [ 2 1 a + 2] [ 2*a 2 2] row_space() Return the module spanned by all rows of matrices in this matrix space. This is a free module of rank the number of rows. It will be sparse or dense as this matrix space is sparse or dense. 11
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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