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: A.norm(p=’bogus’) Traceback (most recent call last): ... ValueError: matrix norm ’p’ must be +/- infinity, ’frob’ or an integer, not bogus sage: A.norm(p=632) Traceback (most recent call last): ... ValueError: matrix norm integer values of ’p’ must be -2, -1, 1 or 2, not 632 numpy(dtype=None) This method returns a copy of the matrix as a numpy array. It uses the numpy C/api so is very fast. INPUT: •dtype - The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence. EXAMPLES: sage: m = matrix(RDF,[[1,2],[3,4]]) sage: n = m.numpy() sage: import numpy sage: numpy.linalg.eig(n) (array([-0.37228132, 5.37228132]), array([[-0.82456484, -0.41597356], [ 0.56576746, -0.90937671]])) sage: m = matrix(RDF, 2, range(6)); m [0.0 1.0 2.0] [3.0 4.0 5.0] sage: m.numpy() array([[ 0., 1., 2.], [ 3., 4., 5.]]) Alternatively, numpy automatically calls this function (via the magic __array__() method) to convert Sage matrices to numpy arrays: sage: import numpy sage: m = matrix(RDF, 2, range(6)); m [0.0 1.0 2.0] [3.0 4.0 5.0] sage: numpy.array(m) array([[ 0., 1., 2.], [ 3., 4., 5.]]) sage: numpy.array(m).dtype dtype(’float64’) sage: m = matrix(CDF, 2, range(6)); m [0.0 1.0 2.0] [3.0 4.0 5.0] sage: numpy.array(m) array([[ 0.+0.j, 1.+0.j, 2.+0.j], [ 3.+0.j, 4.+0.j, 5.+0.j]]) sage: numpy.array(m).dtype dtype(’complex128’) TESTS: sage: m = matrix(RDF,0,5,[]); m [] sage: m.numpy() array([], shape=(0, 5), dtype=float64) sage: m = matrix(RDF,5,0,[]); m 380 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [] sage: m.numpy() array([], shape=(5, 0), dtype=float64) right_eigenspaces(var=’a’, algebraic_multiplicity=False) Computes the right eigenspaces of a matrix of double precision real or complex numbers (i.e. RDF or CDF). Warning: This method returns eigenspaces that are all of dimension one, since it is impossible to ascertain if the numerical results belong to the same eigenspace. So this is deprecated in favor of the eigenmatrix routines, such as sage.matrix.matrix2.Matrix.eigenmatrix_right(). INPUT: •var - ignored for numerical matrices •algebraic_multiplicity - must be set to False for numerical matrices, and will raise an error otherwise. OUTPUT: Return a list of pairs (e, V) where e is a (complex) eigenvalue and V is the associated right eigenspace as a vector space. No attempt is made to determine if an eigenvalue has multiplicity greater than one, so all the eigenspaces returned have dimension one. The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by creating an eigenspace with a canonical basis. EXAMPLES: This first test simply raises the deprecation warning. sage: A = identity_matrix(RDF, 2) sage: es = A.eigenspaces_right() doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are deprecated as of Sage version 5.0, please use "eigenmatrix_right" instead See http://trac.sagemath.org/11603 for details. sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]]) sage: m [ -9.0 -14.0 19.0 -74.0] [ -1.0 2.0 4.0 -11.0] [ -4.0 -12.0 6.0 -32.0] [ 0.0 -2.0 -1.0 1.0] sage: spectrum = m.eigenspaces_right() sage: spectrum[0][0] 2.0 sage: (RDF^4).subspace(spectrum[0][1].basis()) Vector space of degree 4 and dimension 1 over Real Double Field Basis matrix: [ 1.0 -2.0 3.0 1.0] sage: e, V = spectrum[2] sage: v = V.basis()[0] sage: diff = (m*v).change_ring(CDF) - e*v 381
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