Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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
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