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 class sage.matrix.matrix_rational_dense.Matrix_rational_dense Bases: sage.matrix.matrix_dense.Matrix_dense antitranspose() Returns the antitranspose of self, without changing self. EXAMPLES: sage: A = matrix(QQ,2,3,range(6)) sage: type(A) sage: A.antitranspose() [5 2] [4 1] [3 0] sage: A [0 1 2] [3 4 5] sage: A.subdivide(1,2); A [0 1|2] [---+-] [3 4|5] sage: A.antitranspose() [5|2] [-+-] [4|1] [3|0] change_ring(R) Create the matrix over R with entries the entries of self coerced into R. EXAMPLES: sage: a = matrix(QQ,2,[1/2,-1,2,3]) sage: a.change_ring(GF(3)) [2 2] [2 0] sage: a.change_ring(ZZ) Traceback (most recent call last): ... TypeError: matrix has denominators so can’t change to ZZ. sage: b = a.change_ring(QQ[’x’]); b [1/2 -1] [ 2 3] sage: b.parent() Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational TESTS: Make sure that subdivisions are preserved when changing rings: sage: a = matrix(QQ, 3, range(9)) sage: a.subdivide(2,1); a [0|1 2] [3|4 5] [-+---] [6|7 8] sage: a.change_ring(ZZ).change_ring(QQ) [0|1 2] [3|4 5] 340 Chapter 18. Dense matrices over the rational field
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [-+---] [6|7 8] sage: a.change_ring(GF(3)) [0|1 2] [0|1 2] [-+---] [0|1 2] charpoly(var=’x’, algorithm=’linbox’) Return the characteristic polynomial of this matrix. INPUT: •var - ‘x’ (string) •algorithm - ‘linbox’ (default) or ‘generic’ OUTPUT: a polynomial over the rational numbers. EXAMPLES: sage: a = matrix(QQ, 3, [4/3, 2/5, 1/5, 4, -3/2, 0, 0, -2/3, 3/4]) sage: f = a.charpoly(); f x^3 - 7/12*x^2 - 149/40*x + 97/30 sage: f(a) [0 0 0] [0 0 0] [0 0 0] TESTS: The cached polynomial should be independent of the var argument (trac ticket #12292). We check (indirectly) that the second call uses the cached value by noting that its result is not cached: sage: M = MatrixSpace(QQ, 2) sage: A = M(range(0, 2^2)) sage: type(A) sage: A.charpoly(’x’) x^2 - 3*x - 2 sage: A.charpoly(’y’) y^2 - 3*y - 2 sage: A._cache[’charpoly_linbox’] x^2 - 3*x - 2 column(i, from_list=False) Return the i-th column of this matrix as a dense vector. INPUT: • i - integer • from_list - ignored EXAMPLES: sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(1) (-2/3, 4/9) sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(1,from_list=True) (-2/3, 4/9) sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(-1) (-2/3, 4/9) 341
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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