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 [1.0 1.0 1.0 1.0] sage: e, V = spectrum[2] sage: v = V.basis()[0] sage: diff = (v*m).change_ring(CDF) - e*v sage: abs(abs(diff)) < 1e-14 True TESTS: sage: m.eigenspaces_left(algebraic_multiplicity=True) Traceback (most recent call last): ... ValueError: algebraic_multiplicity must be set to False for double precision matrices eigenspaces_right(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() 362 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 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 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 eigenvalues(algorithm=’default’, tol=None) Returns a list of eigenvalues. INPUT: •self - a square matrix •algorithm - default: ’default’ –’default’ - applicable to any matrix with double-precision floating point entries. Uses the eigvals() method from SciPy. –’symmetric’ - converts the matrix into a real matrix (i.e. with entries from RDF), then applies the algorithm for Hermitian matrices. This algorithm can be significantly faster than the ’default’ algorithm. –’hermitian’ - uses the eigh() method from SciPy, which applies only to real symmetric or complex Hermitian matrices. Since Hermitian is defined as a matrix equaling its conjugatetranspose, for a matrix with real entries this property is equivalent to being symmetric. This algorithm can be significantly faster than the ’default’ algorithm. •’tol’ - default: None - if set to a value other than None this is interpreted as a small real number used to aid in grouping eigenvalues that are numerically similar. See the output description for more information. Warning: When using the ’symmetric’ or ’hermitian’ algorithms, no check is made on the input matrix, and only the entries below, and on, the main diagonal are employed in the computation. Methods such as is_symmetric() and is_hermitian() could be used to verify this beforehand. OUTPUT: Default output for a square matrix of size n is a list of n eigenvalues from the complex double field, CDF. If the ’symmetric’ or ’hermitian’ algorithms are chosen, the returned eigenvalues are from the real double field, RDF. If a tolerance is specified, an attempt is made to group eigenvalues that are numerically similar. The return is then a list of pairs, where each pair is an eigenvalue followed by its multiplicity. The eigenvalue reported is the mean of the eigenvalues computed, and these eigenvalues are contained in an interval (or disk) whose radius is less than 5*tol for n < 10, 000 in the worst case. 363
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