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 Hilbert matrices are famously ill-conditioned, while an identity matrix can hit the minimum with the right norm. sage: A = matrix(RDF, 10, [1/(i+j+1) for i in range(10) for j in range(10)]) sage: A.condition() 1.633...e+13 sage: id = identity_matrix(CDF, 10) sage: id.condition(p=1) 1.0 Return values are in RDF . sage: A = matrix(CDF, 2, range(1,5)) sage: A.condition() in RDF True Rectangular and singular matrices raise errors if p is not ‘sv’. sage: A = matrix(RDF, 2, 3, range(6)) sage: A.condition() Traceback (most recent call last): ... TypeError: matrix must be square if p is not ’sv’, not 2 x 3 sage: A.condition(’sv’) 7.34... sage: A = matrix(QQ, 5, range(25)) sage: A.is_singular() True sage: B = A.change_ring(CDF) sage: B.condition() Traceback (most recent call last): ... LinAlgError: Singular matrix Improper values of p are caught. sage: A = matrix(CDF, 2, range(1,5)) sage: A.condition(p=’bogus’) Traceback (most recent call last): ... ValueError: condition number ’p’ must be +/- infinity, ’frob’, ’sv’ or an integer, not bogus sage: A.condition(p=632) Traceback (most recent call last): ... ValueError: condition number integer values of ’p’ must be -2, -1, 1 or 2, not 632 TESTS: Some condition numbers, first by the definition which also exercises norm(), then by this method. sage: A = matrix(CDF, [[1,2,4],[5,3,9],[7,8,6]]) sage: c = A.norm(2)*A.inverse().norm(2) sage: d = A.condition(2) sage: abs(c-d) < 1.0e-12 True sage: c = A.norm(1)*A.inverse().norm(1) sage: d = A.condition(1) sage: abs(c-d) < 1.0e-12 True 360 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 determinant() Return the determinant of self. ALGORITHM: Use numpy EXAMPLES: sage: m = matrix(RDF,2,range(4)); m.det() -2.0 sage: m = matrix(RDF,0,[]); m.det() 1.0 sage: m = matrix(RDF, 2, range(6)); m.det() Traceback (most recent call last): ... ValueError: self must be a square matrix eigenspaces_left(var=’a’, algebraic_multiplicity=False) Computes the left 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 left 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_left() doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are deprecated as of Sage version 5.0, please use "eigenmatrix_left" instead See http://trac.sagemath.org/11603 for details. sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]]) sage: spectrum = m.eigenspaces_left() 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: 361
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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