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: CC(a) 0.309016994374947 + 0.951056516295154*I sage: M = matrix(C, 1, 2, [a^2, a+a^3]) sage: M.conjugate_transpose() [ zeta5^3] [-zeta5^3 - zeta5 - 1] Conjugation does not make sense over rings not containing complex numbers. sage: N = matrix(GF(5), 2, [0,1,2,3]) sage: N.conjugate_transpose() Traceback (most recent call last): ... AttributeError: ’sage.rings.finite_rings.integer_mod.IntegerMod_int’ object has no attribute AUTHOR: Rob Beezer (2010-12-13) cyclic_subspace(v, var=None, basis=’echelon’) Create a cyclic subspace for a vector, and optionally, a minimal polynomial for the iterated powers. These subspaces are also known as Krylov subspaces. They are spanned by the vectors INPUT: {v, Av, A 2 v, A 3 v, . . . } •self - a square matrix with entries from a field. •v - a vector with a degree equal to the size of the matrix and entries compatible with the entries of the matrix. •var - default: None - if specified as a string or a generator of a polynomial ring, then this will be used to construct a polynomial reflecting a relation of linear dependence on the powers A i v and this will cause the polynomial to be returned along with the subspace. A generator must create polynomials with coefficients from the same field as the matrix entries. •basis - default: echelon - the basis for the subspace is “echelonized” by default, but the keyword ‘iterates’ will return a subspace with a user basis equal to the largest linearly independent set {v, Av, A 2 v, A 3 v, . . . , A k−1 v}. OUTPUT: Suppose k is the smallest power such that {v, Av, A 2 v, A 3 v, . . . , A k v} is linearly dependent. Then the subspace returned will have dimension k and be spanned by the powers 0 through k − 1. If a polynomial is requested through the use of the var keyword, then a pair is returned, with the polynomial first and the subspace second. The polynomial is the unique monic polynomial whose coefficients provide a relation of linear dependence on the first k powers. For less convenient, but more flexible output, see the helper method “_cyclic_subspace” in this module. EXAMPLES: sage: A = matrix(QQ, [[5,4,2,1],[0,1,-1,-1],[-1,-1,3,0],[1,1,-1,2]]) sage: v = vector(QQ, [0,1,0,0]) sage: E = A.cyclic_subspace(v); E Vector space of degree 4 and dimension 3 over Rational Field Basis matrix: [ 1 0 0 0] [ 0 1 0 0] 144 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 0 0 1 -1] sage: F = A.cyclic_subspace(v, basis=’iterates’); F Vector space of degree 4 and dimension 3 over Rational Field User basis matrix: [ 0 1 0 0] [ 4 1 -1 1] [23 1 -8 8] sage: E == F True sage: p, S = A.cyclic_subspace(v, var=’T’); p T^3 - 9*T^2 + 24*T - 16 sage: gen = polygen(QQ, ’z’) sage: p, S = A.cyclic_subspace(v, var=gen); p z^3 - 9*z^2 + 24*z - 16 sage: p.degree() == E.dimension() True The polynomial has coefficients that yield a non-trivial relation of linear dependence on the iterates. Or, equivalently, evaluating the polynomial with the matrix will create a matrix that annihilates the vector. sage: A = matrix(QQ, [[15, 37/3, -16, -104/3, -29, -7/3, 35, 2/3, -29/3, -1/3], ... [ 2, 9, -1, -6, -6, 0, 7, 0, -2, 0], ... [24, 74/3, -29, -208/3, -58, -14/3, 70, 4/3, -58/3, -2/3], ... [-6, -19, 3, 21, 19, 0, -21, 0, 6, 0], ... [2, 6, -1, -6, -3, 0, 7, 0, -2, 0], ... [-96, -296/3, 128, 832/3, 232, 65/3, -279, -16/3, 232/3, 8/3], ... [0, 0, 0, 0, 0, 0, 3, 0, 0, 0], ... [20, 26/3, -30, -199/3, -42, -14/3, 70, 13/3, -55/3, -2/3], ... [18, 57, -9, -54, -57, 0, 63, 0, -15, 0], ... [0, 0, 0, 0, 0, 0, 0, 0, 0, 3]]) sage: u = zero_vector(QQ, 10); u[0] = 1 sage: p, S = A.cyclic_subspace(u, var=’t’, basis=’iterates’) sage: S Vector space of degree 10 and dimension 3 over Rational Field User basis matrix: [ 1 0 0 0 0 0 0 0 0 0] [ 15 2 24 -6 2 -96 0 20 18 0] [ 79 12 140 -36 12 -560 0 116 108 0] sage: p t^3 - 9*t^2 + 27*t - 27 sage: k = p.degree() sage: coeffs = p.list() sage: iterates = S.basis() + [A^k*u] sage: sum(coeffs[i]*iterates[i] for i in range(k+1)) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) sage: u in p(A).right_kernel() True TESTS: A small case. sage: A = matrix(QQ, 5, range(25)) sage: u = zero_vector(QQ, 5) sage: A.cyclic_subspace(u) Vector space of degree 5 and dimension 0 over Rational Field Basis matrix: [] Various problem inputs. Notice the vector must have entries that coerce into the base ring of the matrix, 145
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