Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: abs(abs(delta)) < 1e-10
True
Notice the difference between considering ring extensions or not.
sage: M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]])
sage: M.eigenvectors_left()
[(2, [
(0, 0, 1)
], 1), (-1*I, [(1, -1*I, 0)], 1), (1*I, [(1, 1*I, 0)], 1)]
sage: M.eigenvectors_left(extend=False)
[(2, [
(0, 0, 1)
], 1)]
eigenvectors_right(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
True
elementary_divisors()
If self is a matrix over a principal ideal domain R, return elements d i for 1 ≤ i ≤ k = min(r, s) where r
and s are the number of rows and columns of self, such that the cokernel of self is isomorphic to
R/(d 1 ) ⊕ R/(d 2 ) ⊕ R/(d k )
with d i | d i+1 for all i. These are the diagonal entries of the Smith form of self (see smith_form()).
EXAMPLES:
sage: OE = EquationOrder(x^2 - x + 2, ’w’)
sage: w = OE.ring_generators()[0]
sage: m = Matrix([ [1, w],[w,7]])
sage: m.elementary_divisors()
[1, -w + 9]
166 Chapter 7. Base class for matrices, part 2