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 (a2, Vector space of degree 7 and dimension 2 over Number Field in a2 with defining polynomi [ 0 1 0 1 -1 1 -1] [ 0 0 1 0 -1 2 -1], 2), (-1.414213562373095, Vector space of degree 7 and dimension 2 over Algebraic Field User basis matrix: [ 0 1 0 -1 0.414213562 [ 0 0 1 0 (1.414213562373095, Vector space of degree 7 and dimension 2 over Algebraic Field User basis matrix: [ 0 1 0 -1 -2.414 [ 0 0 1 0 ] sage: A.eigenspaces_left(format=’galois’, algebraic_multiplicity=True) [ (3, Vector space of degree 7 and dimension 1 over Rational Field User basis matrix: [ 1 0 1/7 0 -1/7 0 -2/7], 1), (-2, Vector space of degree 7 and dimension 2 over Rational Field User basis matrix: [ 0 1 0 1 -1 1 -1] [ 0 0 1 0 -1 2 -1], 2), User basis matrix: [ 0 1 0 -1 -a2 - 1 1 -1] [ 0 0 1 0 -1 0 -a2 + 1], 2) ] Next we compute the left eigenspaces over the finite field of order 11. sage: A = ModularSymbols(43, base_ring=GF(11), sign=1).T(2).matrix(); A [ 3 9 0 0] [ 0 9 0 1] [ 0 10 9 2] [ 0 9 0 2] sage: A.base_ring() Finite Field of size 11 sage: A.charpoly() x^4 + 10*x^3 + 3*x^2 + 2*x + 1 sage: A.eigenspaces_left(format=’galois’, var = ’beta’) [ (9, Vector space of degree 4 and dimension 1 over Finite Field of size 11 User basis matrix: [0 0 1 5]), (3, Vector space of degree 4 and dimension 1 over Finite Field of size 11 User basis matrix: [1 6 0 6]), (beta2, Vector space of degree 4 and dimension 1 over Univariate Quotient Polynomial Ring in User basis matrix: [ 0 1 0 5*beta2 + 10]) ] This method is only applicable to exact matrices. The “eigenmatrix” routines for matrices with doubleprecision floating-point entries (RDF, CDF) are the best alternative. (Since some platforms return eigenvectors that are the negatives of those given here, this one example is not tested here.) There are also “eigenmatrix” routines for matrices with symbolic entries. sage: A = matrix(QQ, 3, 3, range(9)) sage: A.change_ring(RR).eigenspaces_left() Traceback (most recent call last): ... 160 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 NotImplementedError: eigenspaces cannot be computed reliably for inexact rings such as Real consult numerical or symbolic matrix classes for other options sage: em = A.change_ring(RDF).eigenmatrix_left() sage: eigenvalues = em[0]; eigenvalues.dense_matrix().zero_at(2e-15) [ 13.3484692283 0.0 0.0] [ 0.0 -1.34846922835 0.0] [ 0.0 0.0 0.0] sage: eigenvectors = em[1]; eigenvectors # not tested [ 0.440242867... 0.567868371... 0.695493875...] [ 0.897878732... 0.278434036... -0.341010658...] [ 0.408248290... -0.816496580... 0.408248290...] sage: x, y = var(’x y’) sage: S = matrix([[x, y], [y, 3*x^2]]) sage: em = S.eigenmatrix_left() sage: eigenvalues = em[0]; eigenvalues [3/2*x^2 + 1/2*x - 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) [ 0 3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - sage: eigenvectors = em[1]; eigenvectors [ 1 1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + [ 1 1/2*(3*x^2 - x + sqrt(9*x^4 - 6*x^3 + A request for ’all’ the eigenvalues, when it is not possible, will raise an error. Using the ’galois’ format option is more likely to be successful. sage: F. = FiniteField(11^2) sage: A = matrix(F, [[b + 1, b + 1], [10*b + 4, 5*b + 4]]) sage: A.eigenspaces_left(format=’all’) Traceback (most recent call last): ... NotImplementedError: unable to construct eigenspaces for eigenvalues outside the base field, try the keyword option: format=’galois’ sage: A.eigenspaces_left(format=’galois’) [ (a0, Vector space of degree 2 and dimension 1 over Univariate Quotient Polynomial Ring in a0 User basis matrix: [ 1 6*b*a0 + 3*b + 1]) ] TESTS: We make sure that trac ticket #13308 is fixed. sage: M = ModularSymbols(Gamma1(23), sign=1) sage: m = M.cuspidal_subspace().hecke_matrix(2) sage: [j*m==i[0]*j for i in m.eigenspaces_left(format=’all’) for j in i[1].basis()] # long t [True, True, True, True, True, True, True, True, True, True, True, True] sage: B = matrix(QQ, 2, 3, range(6)) sage: B.eigenspaces_left() Traceback (most recent call last): ... TypeError: matrix must be square, not 2 x 3 sage: B = matrix(QQ, 4, 4, range(16)) sage: B.eigenspaces_left(format=’junk’) Traceback (most recent call last): 161
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