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"""
Routines for computing eigenvectors with DomainMatrix.
"""
from sympy.core.symbol import Dummy
from ..agca.extensions import FiniteExtension
from ..factortools import dup_factor_list
from ..polyroots import roots
from ..polytools import Poly
from ..rootoftools import CRootOf
from .domainmatrix import DomainMatrix
def dom_eigenvects(A, l=Dummy('lambda')):
charpoly = A.charpoly()
rows, cols = A.shape
domain = A.domain
_, factors = dup_factor_list(charpoly, domain)
rational_eigenvects = []
algebraic_eigenvects = []
for base, exp in factors:
if len(base) == 2:
field = domain
eigenval = -base[1] / base[0]
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (A - EE).nullspace(divide_last=True)
rational_eigenvects.append((field, eigenval, exp, basis))
else:
minpoly = Poly.from_list(base, l, domain=domain)
field = FiniteExtension(minpoly)
eigenval = field(l)
AA_items = [
[Poly.from_list([item], l, domain=domain).rep for item in row]
for row in A.rep.to_ddm()]
AA_items = [[field(item) for item in row] for row in AA_items]
AA = DomainMatrix(AA_items, (rows, cols), field)
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (AA - EE).nullspace(divide_last=True)
algebraic_eigenvects.append((field, minpoly, exp, basis))
return rational_eigenvects, algebraic_eigenvects
def dom_eigenvects_to_sympy(
rational_eigenvects, algebraic_eigenvects,
Matrix, **kwargs
):
result = []
for field, eigenvalue, multiplicity, eigenvects in rational_eigenvects:
eigenvects = eigenvects.rep.to_ddm()
eigenvalue = field.to_sympy(eigenvalue)
new_eigenvects = [
Matrix([field.to_sympy(x) for x in vect])
for vect in eigenvects]
result.append((eigenvalue, multiplicity, new_eigenvects))
for field, minpoly, multiplicity, eigenvects in algebraic_eigenvects:
eigenvects = eigenvects.rep.to_ddm()
l = minpoly.gens[0]
eigenvects = [[field.to_sympy(x) for x in vect] for vect in eigenvects]
degree = minpoly.degree()
minpoly = minpoly.as_expr()
eigenvals = roots(minpoly, l, **kwargs)
if len(eigenvals) != degree:
eigenvals = [CRootOf(minpoly, l, idx) for idx in range(degree)]
for eigenvalue in eigenvals:
new_eigenvects = [
Matrix([x.subs(l, eigenvalue) for x in vect])
for vect in eigenvects]
result.append((eigenvalue, multiplicity, new_eigenvects))
return result
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