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.constructor.random_unimodular_matrix(parent, upper_bound=None, max_tries=100) This function is available as random_unimodular_matrix(...) and matrix.random_unimodular(...). Generate a random unimodular (determinant 1) matrix of a desired size over a desired ring. INPUT: •parent - A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the result. The base ring must be exact. •upper_bound - For large matrices over QQ or ZZ, upper_bound is the largest value matrix entries can achieve. But see the warning below. •max_tries - If designated, number of tries used to generate each new random row; only matters when upper_bound!=None. Used to prevent endless looping. (default: 100) A matrix not in reduced row-echelon form with the desired dimensions and properties. OUTPUT: An invertible matrix with the desired properties and determinant 1. Warning: When upper_bound is set, it is possible for this constructor to fail with a ValueError. This may happen when the upper_bound, rank and/or matrix dimensions are all so small that it becomes infeasible or unlikely to create the requested matrix. If you must have this routine return successfully, do not set upper_bound. Note: It is easiest to use this function via a call to the random_matrix() function with the algorithm=’unimodular’ keyword. We provide one example accessing this function directly, while the remainder will use this more general function. EXAMPLES: A matrix size 5 over QQ. sage: from sage.matrix.constructor import random_unimodular_matrix sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5) sage: A=random_unimodular_matrix(matrix_space); A # random [ -8 31 85 148 -419] [ 2 -9 -25 -45 127] [ -1 10 30 65 -176] [ -3 12 33 58 -164] [ 5 -21 -59 -109 304] sage: det(A) 1 A matrix size 6 with entries no larger than 50. sage: B=random_matrix(ZZ, 7, algorithm=’unimodular’, upper_bound=50);B # random [ -1 0 3 1 -2 2 9] [ 1 2 -5 0 14 19 -49] [ -3 -2 12 5 -6 -4 24] [ 1 2 -9 -3 3 4 -7] [ -2 -1 7 2 -8 -5 31] [ 2 2 -6 -3 8 16 -32] [ 1 2 -9 -2 5 6 -12] sage: det(B) 1 56 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 A matrix over the number Field in y with defining polynomial y 2 − 2y − 2. sage: y = var(’y’) sage: K=NumberField(y^2-2*y-2,’y’) sage: C=random_matrix(K, 3, algorithm=’unimodular’);C # random [ 2*y - 33 681*y - 787 31*y - 37] [ y + 6 -155*y + 83 -7*y + 4] [ -y 24*y + 51 y + 3] sage: det(C) 1 TESTS: Unimodular matrices are square. sage: random_matrix(QQ, 5, 6, algorithm=’unimodular’) Traceback (most recent call last): ... TypeError: a unimodular matrix must be square. Only matrices over ZZ and QQ can have size control. sage: F.=GF(5^7) sage: random_matrix(F, 5, algorithm=’unimodular’, upper_bound=20) Traceback (most recent call last): ... TypeError: only matrices over ZZ or QQ can have size control. AUTHOR: Billy Wonderly (2010-07) sage.matrix.constructor.vector_on_axis_rotation_matrix(v, i, ring=None) This function is available as vector_on_axis_rotation_matrix(...) and matrix.vector_on_axis_rotation(...). Return a rotation matrix M such that det(M) = 1 sending the vector v on the i-th axis so that all other coordinates of Mv are zero. Note: Such a matrix is not uniquely determined. This function returns one such matrix. INPUT: •v‘ - vector •i - integer •ring - ring (optional, default: None) of the resulting matrix OUTPUT: A matrix EXAMPLES: sage: from sage.matrix.constructor import vector_on_axis_rotation_matrix sage: v = vector((1,2,3)) sage: vector_on_axis_rotation_matrix(v, 2) * v (0, 0, sqrt(14)) sage: vector_on_axis_rotation_matrix(v, 1) * v (0, sqrt(14), 0) sage: vector_on_axis_rotation_matrix(v, 0) * v (sqrt(14), 0, 0) 57
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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