INPUT:
- - ``A`` - The matrix whose eigenvectors we want to compute.
+ - ``A`` -- The matrix whose unit eigenvectors we want to compute.
OUTPUT:
A list of (eigenvalue, eigenvector) pairs where each eigenvector is
- associated with its paired eigenvalue of ``A`` and has norm `1`.
+ associated with its paired eigenvalue of ``A`` and has norm `1`. If
+ the base ring of ``A`` is not algebraically closed, then returned
+ eigenvectors may (necessarily) be over its algebraic closure and not
+ the base ring of ``A`` itself.
SETUP::
EXAMPLES::
sage: A = matrix(QQ, [[0, 2, 3], [2, 0, 0], [3, 0, 0]])
- sage: unit_evs = unit_eigenvectors(A)
+ sage: unit_evs = list(unit_eigenvectors(A))
sage: bool(unit_evs[0][1].norm() == 1)
True
sage: bool(unit_evs[1][1].norm() == 1)
True
"""
- # This will give us a list of lists whose elements are the
- # eigenvectors we want.
- ev_lists = [ (val,vecs) for (val,vecs,multiplicity)
- in A.eigenvectors_right() ]
-
- # Pair each eigenvector with its eigenvalue and normalize it.
- evs = [ [(l, vec/vec.norm()) for vec in vecs] for (l,vecs) in ev_lists ]
-
- # Flatten the list, abusing the fact that "+" is overloaded on lists.
- return sum(evs, [])
-
+ return ( (val,vec.normalized())
+ for (val,vecs,multiplicity) in A.eigenvectors_right()
+ for vec in vecs )
all_evs = unit_eigenvectors(A)
evs = [ (val,vec) for (val,vec) in all_evs if not val == 0 ]
- d = [ sqrt(val) for (val,vec) in evs ]
+ d = ( val.sqrt() for (val,vec) in evs )
root_D = diagonal_matrix(d).change_ring(A.base_ring())
- Q = matrix(A.base_ring(), [ vec for (val,vec) in evs ]).transpose()
+ Q = matrix(A.base_ring(), ( vec for (val,vec) in evs )).transpose()
return Q*root_D*Q.transpose()