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 More precisely, for an n × n matrix, the diameter of the interval containing similar eigenvalues could be as large as sum of the reciprocals of the first n integers times tol. Warning: Use caution when using the tol parameter to group eigenvalues. See the examples below to see how this can go wrong. EXAMPLES: sage: m = matrix(RDF, 2, 2, [1,2,3,4]) sage: ev = m.eigenvalues(); ev [-0.372281323..., 5.37228132...] sage: ev[0].parent() Complex Double Field sage: m = matrix(RDF, 2, 2, [0,1,-1,0]) sage: m.eigenvalues(algorithm=’default’) [1.0*I, -1.0*I] sage: m = matrix(CDF, 2, 2, [I,1,-I,0]) sage: m.eigenvalues() [-0.624810533... + 1.30024259...*I, 0.624810533... - 0.30024259...*I] The adjacency matrix of a graph will be symmetric, and the eigenvalues will be real. sage: A = graphs.PetersenGraph().adjacency_matrix() sage: A = A.change_ring(RDF) sage: ev = A.eigenvalues(algorithm=’symmetric’); ev [-2.0, -2.0, -2.0, -2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 3.0] sage: ev[0].parent() Real Double Field The matrix A is “random”, but the construction of B provides a positive-definite Hermitian matrix. Note that the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a positive-definite matrix will be positive. sage: A = matrix([[ 4*I + 5, 8*I + 1, 7*I + 5, 3*I + 5], ... [ 7*I - 2, -4*I + 7, -2*I + 4, 8*I + 8], ... [-2*I + 1, 6*I + 6, 5*I + 5, -I - 4], ... [ 5*I + 1, 6*I + 2, I - 4, -I + 3]]) sage: B = (A*A.conjugate_transpose()).change_ring(CDF) sage: ev = B.eigenvalues(algorithm=’hermitian’); ev [2.68144025..., 49.5167998..., 274.086188..., 390.71557...] sage: ev[0].parent() Real Double Field A tolerance can be given to aid in grouping eigenvalues that are similar numerically. However, if the parameter is too small it might split too finely. Too large, and it can go wrong very badly. Use with care. sage: G = graphs.PetersenGraph() sage: G.spectrum() [3, 1, 1, 1, 1, 1, -2, -2, -2, -2] sage: A = G.adjacency_matrix().change_ring(RDF) sage: A.eigenvalues(algorithm=’symmetric’, tol=1.0e-5) [(-2.0, 4), (1.0, 5), (3.0, 1)] sage: A.eigenvalues(algorithm=’symmetric’, tol=2.5) [(-2.0, 4), (1.33333333333, 6)] 364 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 An (extreme) example of properly grouping similar eigenvalues. sage: G = graphs.HigmanSimsGraph() sage: A = G.adjacency_matrix().change_ring(RDF) sage: A.eigenvalues(algorithm=’symmetric’, tol=1.0e-5) [(-8.0, 22), (2.0, 77), (22.0, 1)] TESTS: Testing bad input. sage: A = matrix(CDF, 2, range(4)) sage: A.eigenvalues(algorithm=’junk’) Traceback (most recent call last): ... ValueError: algorithm must be ’default’, ’symmetric’, or ’hermitian’, not junk sage: A = matrix(CDF, 2, 3, range(6)) sage: A.eigenvalues() Traceback (most recent call last): ... ValueError: matrix must be square, not 2 x 3 sage: A = matrix(CDF, 2, [1, 2, 3, 4*I]) sage: A.eigenvalues(algorithm=’symmetric’) Traceback (most recent call last): ... TypeError: cannot apply symmetric algorithm to matrix with complex entries sage: A = matrix(CDF, 2, 2, range(4)) sage: A.eigenvalues(tol=’junk’) Traceback (most recent call last): ... TypeError: tolerance parameter must be a real number, not junk sage: A = matrix(CDF, 2, 2, range(4)) sage: A.eigenvalues(tol=-0.01) Traceback (most recent call last): ... ValueError: tolerance parameter must be positive, not -0.01 A very small matrix. sage: matrix(CDF,0,0).eigenvalues() [] eigenvectors_left() Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF). OUTPUT: Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an associated left eigenvector. If the matrix is of size n, then there are n triples. Values are computed with the SciPy library. The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine. 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 forcing their 365
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