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 eigenvectors to have their first non-zero entry equal to one. EXAMPLES: sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]]) sage: m [ -5.0 3.0 2.0 8.0] [ 10.0 2.0 4.0 -2.0] [ -1.0 -10.0 -10.0 -17.0] [ -2.0 7.0 6.0 13.0] sage: spectrum = m.left_eigenvectors() sage: for i in range(len(spectrum)): ... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] (2.0, [(1.0, 1.0, 1.0, 1.0)], 1) sage: spectrum[1] (1.0, [(1.0, 0.8, 0.8, 0.6)], 1) sage: spectrum[2] (-2.0, [(1.0, 0.4, 0.6, 0.2)], 1) sage: spectrum[3] (-1.0, [(1.0, 1.0, 2.0, 2.0)], 1) eigenvectors_right() Compute the right 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 right 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 eigenvectors to have their first non-zero entry equal to one. EXAMPLES: 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.right_eigenvectors() sage: for i in range(len(spectrum)): ... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] (2.0, [(1.0, -2.0, 3.0, 1.0)], 1) sage: spectrum[1] (1.0, [(1.0, -0.666666666667, 1.33333333333, 0.333333333333)], 1) sage: spectrum[2] (-2.0, [(1.0, -0.2, 1.0, 0.2)], 1) sage: spectrum[3] 366 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 (-1.0, [(1.0, -0.5, 2.0, 0.5)], 1) exp(algorithm=’pade’, order=None) Calculate the exponential of this matrix X, which is the matrix INPUT: e X = ∞∑ k=0 X k k! . •algorithm – ‘pade’, ‘eig’, or ‘taylor’; the algorithm used to compute the exponential. •order – for the Taylor series algorithm, this specifies the order of the Taylor series used. This is ignored for the other algorithms. The current default (from scipy) is 20. EXAMPLES: sage: A=matrix(RDF, 2, [1,2,3,4]); A [1.0 2.0] [3.0 4.0] sage: A.exp() [51.9689561987 74.736564567] [112.104846851 164.073803049] sage: A.exp(algorithm=’eig’) [51.9689561987 74.736564567] [112.104846851 164.073803049] sage: A.exp(algorithm=’taylor’, order=5) [19.9583333333 28.0833333333] [ 42.125 62.0833333333] sage: A.exp(algorithm=’taylor’) [51.9689035511 74.7364878369] [112.104731755 164.073635306] sage: A=matrix(CDF, 2, [1,2+I,3*I,4]); A [ 1.0 2.0 + 1.0*I] [ 3.0*I 4.0] sage: A.exp() [-19.6146029538 + 12.5177438468*I 3.79496364496 + 28.8837993066*I] [-32.3835809809 + 21.8842359579*I 2.26963300409 + 44.9013248277*I] sage: A.exp(algorithm=’eig’) [-19.6146029538 + 12.5177438468*I 3.79496364496 + 28.8837993066*I] [-32.3835809809 + 21.8842359579*I 2.26963300409 + 44.9013248277*I] sage: A.exp(algorithm=’taylor’, order=5) [ -6.29166666667 + 14.25*I 14.0833333333 + 15.7916666667*I] [ -10.5 + 26.375*I 20.0833333333 + 24.75*I] sage: A.exp(algorithm=’taylor’) [-19.6146006163 + 12.5177432169*I 3.79496442472 + 28.8837964828*I] [-32.3835771246 + 21.8842351994*I 2.26963458304 + 44.9013203415*I] is_hermitian(tol=1e-12, algorithm=’orthonormal’) Returns True if the matrix is equal to its conjugate-transpose. INPUT: •tol - default: 1e-12 - the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal. •algorithm - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure. 367
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