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: es = A.eigenspaces_right(format=’galois’); es [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5]) ] sage: es = A.eigenspaces_right(format=’galois’, algebraic_multiplicity=True); es [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1], 1), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5], 1) ] sage: e, v, n = es[0]; v = v.basis()[0] sage: delta = v*e - A*v sage: abs(abs(delta)) < 1e-10 True The same computation, but with implicit base change to a field: sage: A = matrix(ZZ, 3, range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: A.eigenspaces_right(format=’galois’) [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5]) ] 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: B = matrix(RR, 3, 3, range(9)) sage: B.eigenspaces_right() Traceback (most recent call last): ... NotImplementedError: eigenspaces cannot be computed reliably for inexact rings such as Real consult numerical or symbolic matrix classes for other options sage: em = B.change_ring(RDF).eigenmatrix_right() sage: eigenvalues = em[0]; eigenvalues.dense_matrix().zero_at(1e-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 [ 0.164763817... 0.799699663... 0.408248290...] [ 0.505774475... 0.104205787... -0.816496580...] 234 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 0.846785134... -0.591288087... 0.408248290...] sage: x, y = var(’x y’) sage: S = matrix([[x, y], [y, 3*x^2]]) sage: em = S.eigenmatrix_right() 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 + x^2 + 4*y^2))/y 1/2*(3*x^2 - x + sqrt(9*x^4 - 6*x^3 + TESTS: sage: B = matrix(QQ, 2, 3, range(6)) sage: B.eigenspaces_right() 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_right(format=’junk’) Traceback (most recent call last): ... ValueError: format keyword must be None, ’all’ or ’galois’, not junk sage: B.eigenspaces_right(algebraic_multiplicity=’garbage’) Traceback (most recent call last): ... ValueError: algebraic_multiplicity keyword must be True or False right_eigenvectors(extend=True) Compute the right eigenvectors of a matrix. For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding right eigenspace, and n is the algebraic multiplicity of the eigenvalue. If extend = True (the default), this will return eigenspaces over the algebraic closure of the base field where this is implemented; otherwise it will restrict to eigenvalues in the base field. EXAMPLES: We compute the right eigenvectors of a 3 × 3 rational matrix. sage: A = matrix(QQ,3,3,range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: es = A.eigenvectors_right(); es [(0, [ (1, -2, 1) ], 1), (-1.348469228349535, [(1, 0.1303061543300932, -0.7393876913398137)], 1), (13.34846922834954, [(1, 3.069693845669907, 5.139387691339814)], 1)] sage: A.eigenvectors_right(extend=False) [(0, [ (1, -2, 1) ], 1)] sage: eval, [evec], mult = es[0] sage: delta = eval*evec - A*evec sage: abs(abs(delta)) < 1e-10 235
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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