Sage Reference Manual: Matrices and Spaces of Matrices

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2TESTS:sage: MS1.one()[1,2] = 3Traceback (most recent call last):...ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change arandom_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 inthe random matrix; if set to None, all entries of the matrix are randomized, allowing for any elementof the underlying ring, but if set to a float, a proportion of entries is selected and randomized tonon-zero elements of the ring•*args, **kwds - remaining parameters, which may be passed to the random_element function ofthe base ring. (“may be”, since this function calls the randomize function on the zero matrix, whichneed not call the random_element function of the base ring at all in general.)OUTPUT:•MatrixNote: This method will randomize a proportion of roughly density entries in a newly allocated zeromatrix.By default, if the user sets the value of density explicitly, this method will enforce that these entries areset to non-zero values. However, if the test for equality with zero in the base ring is too expensive, the usercan 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 optionnonzero=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 thenumber of rows. It will be sparse or dense as this matrix space is sparse or dense.EXAMPLES:sage: M = Mat(ZZ,20,5,sparse=False); M.row_space()Ambient free module of rank 5 over the principal ideal domain Integer Ring11