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 [ 0 0 12] sage: P.is_sparse() False symplectic_form() Find a symplectic basis for self if self is an anti-symmetric, alternating matrix. Returns a pair (F, C) such that the rows of C form a symplectic basis for self and F = C * self * C.transpose(). Raises a ValueError if self is not anti-symmetric, or self is not alternating. Anti-symmetric means that M = −M t . Alternating means that the diagonal of M is identically zero. A symplectic basis is a basis of the form e 1 , . . . , e j , f 1 , . . . f j , z 1 , . . . , z k such that •z i Mv t = 0 for all vectors v •e i Me j t = 0 for all i, j •f i Mf j t = 0 for all i, j •e i Mf i t = 1 for all i •e i Mf j t = 0 for all i not equal j. The ordering for the factors d i |d i+1 and for the placement of zeroes was chosen to agree with the output of smith_form. See the example for a pictorial description of such a basis. EXAMPLES: sage: E = matrix(ZZ, 5, 5, [0, 14, 0, -8, -2, -14, 0, -3, -11, 4, 0, 3, 0, 0, 0, 8, 11, 0, 0 [ 0 14 0 -8 -2] [-14 0 -3 -11 4] [ 0 3 0 0 0] [ 8 11 0 0 8] [ 2 -4 0 -8 0] sage: F, C = E.symplectic_form() sage: F [ 0 0 1 0 0] [ 0 0 0 2 0] [-1 0 0 0 0] [ 0 -2 0 0 0] [ 0 0 0 0 0] sage: F == C * E * C.transpose() True sage: E.smith_form()[0] [1 0 0 0 0] [0 1 0 0 0] [0 0 2 0 0] [0 0 0 2 0] [0 0 0 0 0] transpose() Returns the transpose of self, without changing self. EXAMPLES: We create a matrix, compute its transpose, and note that the original matrix is not changed. 336 Chapter 17. Dense matrices over the integer ring
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A = matrix(ZZ,2,3,xrange(6)) sage: type(A) sage: B = A.transpose() sage: print B [0 3] [1 4] [2 5] sage: print A [0 1 2] [3 4 5] .T is a convenient shortcut for the transpose: sage: A.T [0 3] [1 4] [2 5] sage: A.subdivide(None, 1); A [0|1 2] [3|4 5] sage: A.transpose() [0 3] [---] [1 4] [2 5] sage.matrix.matrix_integer_dense.tune_multiplication(k, nmin=10, nmax=200, bitmin=2, bitmax=64) Compare various multiplication algorithms. INPUT: •k - integer; affects numbers of trials •nmin - integer; smallest matrix to use •nmax - integer; largest matrix to use •bitmin - integer; smallest bitsize •bitmax - integer; largest bitsize OUTPUT: •prints what doing then who wins - multimodular or classical EXAMPLES: sage: from sage.matrix.matrix_integer_dense import tune_multiplication sage: tune_multiplication(2, nmin=10, nmax=60, bitmin=2,bitmax=8) 10 2 0.2 ... 337
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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