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: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(-2) (1/5, 3/4) decomposition(is_diagonalizable=False, dual=False, algorithm=’default’, height_guess=None, proof=None) Returns the decomposition of the free module on which this matrix A acts from the right (i.e., the action is x goes to x A), along with whether this matrix acts irreducibly on each factor. The factors are guaranteed to be sorted in the same way as the corresponding factors of the characteristic polynomial. Let A be the matrix acting from the on the vector space V of column vectors. Assume that A is square. This function computes maximal subspaces W_1, ..., W_n corresponding to Galois conjugacy classes of eigenvalues of A. More precisely, let f(X) be the characteristic polynomial of A. This function computes the subspace W i = ker(g ( A) n ), where g_i(X) is an irreducible factor of f(X) and g_i(X) exactly divides f(X). If the optional parameter is_diagonalizable is True, then we let W_i = ker(g(A)), since then we know that ker(g(A)) = ker(g(A) n ). If dual is True, also returns the corresponding decomposition of V under the action of the transpose of A. The factors are guaranteed to correspond. INPUT: •is_diagonalizable - ignored •dual - whether to also return decompositions for the dual •algorithm –‘default’: use default algorithm for computing Echelon forms –‘multimodular’: much better if the answers factors have small height •height_guess - positive integer; only used by the multimodular algorithm •proof - bool or None (default: None, see proof.linear_algebra or sage.structure.proof); only used by the multimodular algorithm. Note that the Sage global default is proof=True. Note: IMPORTANT: If you expect that the subspaces in the answer are spanned by vectors with small height coordinates, use algorithm=’multimodular’ and height_guess=1; this is potentially much faster than the default. If you know for a fact the answer will be very small, use algorithm=’multimodular’, height_guess=bound on height, proof=False. You can get very very fast decomposition with proof=False. EXAMPLES: sage: a = matrix(QQ,3,[1..9]) sage: a.decomposition() [ (Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 -2 1], True), (Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2], True) ] denominator() Return the denominator of this matrix. OUTPUT: a Sage Integer 342 Chapter 18. Dense matrices over the rational field
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 EXAMPLES: sage: b = matrix(QQ,2,range(6)); b[0,0]=-5007/293; b [-5007/293 1 2] [ 3 4 5] sage: b.denominator() 293 determinant(algorithm=’default’, proof=None) Return the determinant of this matrix. INPUT: •proof - bool or None; if None use proof.linear_algebra(); only relevant for the padic algorithm. •algorithm: “default” – use PARI for up to 7 rows, then use integer “pari” – use PARI “integer” – clear denominators and call det on integer matrix Note: It would be VERY VERY hard for det to fail even with proof=False. ALGORITHM: Clear denominators and call the integer determinant function. EXAMPLES: sage: m = matrix(QQ,3,[1,2/3,4/5, 2,2,2, 5,3,2/5]) sage: m.determinant() -34/15 sage: m.charpoly() x^3 - 17/5*x^2 - 122/15*x + 34/15 echelon_form(algorithm=’default’, height_guess=None, proof=None, **kwds) INPUT: •algorithm –‘default’ (default): use heuristic choice –‘padic’: an algorithm based on the IML p-adic solver. –‘multimodular’: uses a multimodular algorithm the uses linbox modulo many primes. –‘classical’: just clear each column using Gauss elimination •height_guess, **kwds - all passed to the multimodular algorithm; ignored by the p-adic algorithm. •proof - bool or None (default: None, see proof.linear_algebra or sage.structure.proof). Passed to the multimodular algorithm. Note that the Sage global default is proof=True. OUTPUT: the reduced row echelon form of self. EXAMPLES: sage: a = matrix(QQ, 4, range(16)); a[0,0] = 1/19; a[0,1] = 1/5; a [1/19 1/5 2 3] [ 4 5 6 7] [ 8 9 10 11] [ 12 13 14 15] sage: a.echelon_form() 343
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