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: abs(abs(diff)) < 3e-14 True TESTS: sage: m.eigenspaces_right(algebraic_multiplicity=True) Traceback (most recent call last): ... ValueError: algebraic_multiplicity must be set to False for double precision matrices right_eigenvectors() 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] (-1.0, [(1.0, -0.5, 2.0, 0.5)], 1) round(ndigits=0) Returns a copy of the matrix where all entries have been rounded to a given precision in decimal digits (default 0 digits). INPUT: •ndigits - The precision in number of decimal digits OUTPUT: A modified copy of the matrix 382 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 EXAMPLES: sage: M=matrix(CDF, [[10.234r + 34.2343jr, 34e10r]]) sage: M [10.234 + 34.2343*I 3.4e+11] sage: M.round(2) [10.23 + 34.23*I 3.4e+11] sage: M.round() [10.0 + 34.0*I 3.4e+11] schur(base_ring=None) Returns the Schur decomposition of the matrix. INPUT: •base_ring - optional, defaults to the base ring of self. Use this to request the base ring of the returned matrices, which will affect the format of the results. OUTPUT: A pair of immutable matrices. The first is a unitary matrix Q. The second, T , is upper-triangular when returned over the complex numbers, while it is almost upper-triangular over the reals. In the latter case, there can be some 2 × 2 blocks on the diagonal which represent a pair of conjugate complex eigenvalues of self. If self is the matrix A, then A = QT (Q) t where the latter matrix is the conjugate-transpose of Q, which is also the inverse of Q, since Q is unitary. Note that in the case of a normal matrix (Hermitian, symmetric, and others), the upper-triangular matrix is a diagonal matrix with eigenvalues of self on the diagonal, and the unitary matrix has columns that form an orthonormal basis composed of eigenvectors of self. This is known as “orthonormal diagonalization”. Warning: The Schur decomposition is not unique, as there may be numerous choices for the vectors of the orthonormal basis, and consequently different possibilities for the upper-triangular matrix. However, the diagonal of the upper-triangular matrix will always contain the eigenvalues of the matrix (in the complex version), or 2 × 2 block matrices in the real version representing pairs of conjugate complex eigenvalues. In particular, results may vary across systems and processors. EXAMPLES: First over the complexes. The similar matrix is always upper-triangular in this case. sage: A = matrix(CDF, 4, 4, range(16)) + matrix(CDF, 4, 4, [x^3*I for x in range(0, 16)]) sage: Q, T = A.schur() sage: (Q*Q.conjugate().transpose()).zero_at(1.0e-12) [1.0 0.0 0.0 0.0] [0.0 1.0 0.0 0.0] [0.0 0.0 1.0 0.0] [0.0 0.0 0.0 1.0] sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)]) True sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] 383
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