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: random_matrix(ZZ, 5, x=-10, y=10, density=0.75) [-6 1 0 0 0] [ 9 0 0 4 1] [-6 0 0 -8 0] [ 0 4 0 6 0] [ 1 -9 0 0 -8] sage: random_matrix(ZZ, 5, x=20, y=30, density=0.75) [ 0 28 0 27 0] [25 28 20 0 0] [ 0 21 0 21 0] [ 0 28 22 0 0] [ 0 0 0 26 24] It is possible to construct sparse matrices, where it may now be advisable (but not required) to control the density of nonzero entries. sage: A=random_matrix(ZZ, 5, 5) sage: A.is_sparse() False sage: A=random_matrix(ZZ, 5, 5, sparse=True) sage: A.is_sparse() True sage: random_matrix(ZZ, 5, 5, density=0.3, sparse=True) [ 4 0 0 0 -1] [ 0 0 0 0 -7] [ 0 0 2 0 0] [ 0 0 1 0 -4] [ 0 0 0 0 0] For algorithm testing you might want to control the number of bits, say 10,000 entries, each limited to 16 bits. sage: A = random_matrix(ZZ, 100, 100, x=2^16); A 100 x 100 dense matrix over Integer Ring (type ’print A.str()’ to see all of the entries) Random rational matrices. Now num_bound and den_bound control the generation of random elements, by specifying limits on the absolute value of numerators and denominators (respectively). Entries will be positive and negative (map the absolute value function through the entries to get all positive values), and zeros are avoided unless the density is set. If either the numerator or denominator bound (or both) is not used, then the values default to the distribution for ZZ described above that is most frequently positive or negative one. sage: random_matrix(QQ, 2, 8, num_bound=20, den_bound=4) [ -1/2 6 13 -12 -2/3 -1/4 5 5] [ -9/2 5/3 19 15/2 19/2 20/3 -13/4 0] sage: random_matrix(QQ, 4, density = 0.5, sparse=True) [ 0 71 0 -1/2] [ 0 0 0 0] [31/85 0 -31/2 0] [ 1 -1/4 0 0] sage: A = random_matrix(QQ, 3, 10, num_bound = 99, den_bound = 99) sage: positives = map(abs, A.list()) sage: matrix(QQ, 3, 10, positives) [61/18 47/41 1/22 1/2 75/68 6/7 1 1/2 72/41 7/3] [33/13 9/2 40/21 45/46 17/22 1 70/79 97/71 7/24 12/5] [ 13/8 8/25 1/3 61/14 92/45 4/85 3/38 95/16 82/71 1/5] 48 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: random_matrix(QQ, 4, 10, den_bound = 10) [ -1 0 1/8 1/6 2/9 -1/6 1/5 -1/8 1/5 -1/5] [ 1/9 1/5 -1 2/9 1/4 -1/7 1/8 -1/9 0 2] [ 2/3 2 1/8 -2 0 0 -2 2 0 -1/2] [ 0 2 1 -2/3 0 0 1/6 0 -1/3 -2/9] Random matrices over other rings. Several classes of matrices have specialized randomize() methods. You can locate these with the Sage command: search_def(’randomize’) The default implementation of randomize() relies on the random_element() method for the base ring. The density and sparse keywords behave as described above. sage: K.=FiniteField(3^2) sage: random_matrix(K, 2, 5) [ 1 a 1 2*a + 1 2] [ 2*a a + 2 0 2 1] sage: random_matrix(RR, 3, 4, density=0.66) [ 0.000000000000000 -0.806696574554030 -0.693915509972359 0.000000000000000] [ 0.629781664418083 0.000000000000000 -0.833709843116637 0.000000000000000] [ 0.922346867410064 0.000000000000000 0.000000000000000 -0.940316454178921] sage: A = random_matrix(ComplexField(32), 3, density=0.8, sparse=True); A [ 0.000000000 0.399739209 + 0.909948633*I 0.000000000] [-0.361911424 - 0.455087671*I -0.687810605 + 0.460619713*I 0.625520058 - 0.360952012*I] [ 0.000000000 0.000000000 -0.162196416 - 0.193242896*I] sage: A.is_sparse() True Random matrices in echelon form. The algorithm=’echelon_form’ keyword, along with a requested number of non-zero rows (num_pivots) will return a random matrix in echelon form. When the base ring is QQ the result has integer entries. Other exact rings may be also specified. sage: A=random_matrix(QQ, 4, 8, algorithm=’echelon_form’, num_pivots=3); A # random [ 1 -5 0 -2 0 1 1 -2] [ 0 0 1 -5 0 -3 -1 0] [ 0 0 0 0 1 2 -2 1] [ 0 0 0 0 0 0 0 0] sage: A.base_ring() Rational Field sage: (A.nrows(), A.ncols()) (4, 8) sage: A in sage.matrix.matrix_space.MatrixSpace(ZZ, 4, 8) True sage: A.rank() 3 sage: A==A.rref() True For more, see the documentation of the random_rref_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available: from sage.matrix.constructor import random_rref_matrix Random matrices with predictable echelon forms. The algorithm=’echelonizable’ keyword, along with a requested rank (rank) and optional size control (upper_bound) will return a random matrix in echelon form. When the base ring is ZZ or QQ the result has integer entries, whose magnitudes can be limited by the 49
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