def test_eigen():
R = Rational
M = Matrix.eye(3)
assert M.eigenvals(multiple=False) == {S.One: 3}
assert M.eigenvals(multiple=True) == [1, 1, 1]
assert M.eigenvects() == ([
(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])
])
assert M.left_eigenvects() == ([
(1, 3, [Matrix([[1, 0, 0]]),
Matrix([[0, 1, 0]]),
Matrix([[0, 0, 1]])])
])
M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]])
assert M.eigenvals() == {2 * S.One: 1, -S.One: 1, S.Zero: 1}
assert M.eigenvects() == ([(-1, 1, [Matrix([-1, 1, 0])]),
(0, 1, [Matrix([0, -1, 1])]),
(2, 1, [Matrix([R(2, 3), R(1, 3), 1])])])
assert M.left_eigenvects() == ([(-1, 1, [Matrix([[-2, 1, 1]])]),
(0, 1, [Matrix([[-1, -1, 1]])]),
(2, 1, [Matrix([[1, 1, 1]])])])
a = Symbol('a')
M = Matrix([[a, 0], [0, 1]])
assert M.eigenvals() == {a: 1, S.One: 1}
M = Matrix([[1, -1], [1, 3]])
assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3 * 33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3 * b / 8)
e = (R(33, 8) - 3 * b / 8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b / 2, 1, [
Matrix([
(12 + 24 / (c - b / 2)) / ((c - b / 2) * e) + 3 / (c - b / 2),
(6 + 12 / (c - b / 2)) / e, 1
])
]),
(0, 1, [Matrix([1, -2, 1])]),
(a + b / 2, 1, [
Matrix([
(12 + 24 / (c + b / 2)) / ((c + b / 2) * d) + 3 / (c + b / 2),
(6 + 12 / (c + b / 2)) / d, 1
])
]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I * eps], [-I * eps, abs(eps)]])
assert M.eigenvects() == ([
(0, 1, [Matrix([[-I * eps / abs(eps)], [1]])]),
(2 * abs(eps), 1, [Matrix([[I * eps / abs(eps)], [1]])]),
])
assert M.left_eigenvects() == ([
(0, 1, [Matrix([[I * eps / Abs(eps), 1]])]),
(2 * Abs(eps), 1, [Matrix([[-I * eps / Abs(eps), 1]])])
])
M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
M._eigenvects = M.eigenvects(simplify=False)
assert max(i.q for i in M._eigenvects[0][2][0]) > 1
M._eigenvects = M.eigenvects(simplify=True)
assert max(i.q for i in M._eigenvects[0][2][0]) == 1
M = Matrix([[Rational(1, 4), 1], [1, 1]])
assert M.eigenvects(simplify=True) == [
(Rational(5, 8) - sqrt(73) / 8, 1,
[Matrix([[-sqrt(73) / 8 - Rational(3, 8)], [1]])]),
(Rational(5, 8) + sqrt(73) / 8, 1,
[Matrix([[Rational(-3, 8) + sqrt(73) / 8], [1]])])
]
with dotprodsimp(True):
assert M.eigenvects(simplify=False) == [
(Rational(5, 8) - sqrt(73) / 8, 1,
[Matrix([[-1 / (-Rational(3, 8) + sqrt(73) / 8)], [1]])]),
(Rational(5, 8) + sqrt(73) / 8, 1,
[Matrix([[8 / (3 + sqrt(73))], [1]])])
]
# issue 10719
assert Matrix([]).eigenvals() == {}
assert Matrix([]).eigenvals(multiple=True) == []
assert Matrix([]).eigenvects() == []
# issue 15119
raises(NonSquareMatrixError,
lambda: Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
raises(
NonSquareMatrixError, lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(
error_when_incomplete=False))
raises(
NonSquareMatrixError, lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(
error_when_incomplete=False))
m = Matrix([[1, 2], [3, 4]])
assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
def test_eigen():
R = Rational
assert eye(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye(3).charpoly(y) == Poly((y - 1)**3, y)
M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
assert M.eigenvals(multiple=False) == {S.One: 3}
assert M.eigenvals(multiple=True) == [1, 1, 1]
assert M.eigenvects() == ([
(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])
])
assert M.left_eigenvects() == ([
(1, 3, [Matrix([[1, 0, 0]]),
Matrix([[0, 1, 0]]),
Matrix([[0, 0, 1]])])
])
M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]])
assert M.eigenvals() == {2 * S.One: 1, -S.One: 1, S.Zero: 1}
assert M.eigenvects() == ([(-1, 1, [Matrix([-1, 1, 0])]),
(0, 1, [Matrix([0, -1, 1])]),
(2, 1, [Matrix([R(2, 3), R(1, 3), 1])])])
assert M.left_eigenvects() == ([(-1, 1, [Matrix([[-2, 1, 1]])]),
(0, 1, [Matrix([[-1, -1, 1]])]),
(2, 1, [Matrix([[1, 1, 1]])])])
a = Symbol('a')
M = Matrix([[a, 0], [0, 1]])
assert M.eigenvals() == {a: 1, S.One: 1}
M = Matrix([[1, -1], [1, 3]])
assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3 * 33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3 * b / 8)
e = (R(33, 8) - 3 * b / 8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b / 2, 1, [
Matrix([
(12 + 24 / (c - b / 2)) / ((c - b / 2) * e) + 3 / (c - b / 2),
(6 + 12 / (c - b / 2)) / e, 1
])
]),
(0, 1, [Matrix([1, -2, 1])]),
(a + b / 2, 1, [
Matrix([
(12 + 24 / (c + b / 2)) / ((c + b / 2) * d) + 3 / (c + b / 2),
(6 + 12 / (c + b / 2)) / d, 1
])
]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I * eps], [-I * eps, abs(eps)]])
assert M.eigenvects() == ([
(0, 1, [Matrix([[-I * eps / abs(eps)], [1]])]),
(2 * abs(eps), 1, [Matrix([[I * eps / abs(eps)], [1]])]),
])
assert M.left_eigenvects() == ([
(0, 1, [Matrix([[I * eps / Abs(eps), 1]])]),
(2 * Abs(eps), 1, [Matrix([[-I * eps / Abs(eps), 1]])])
])
M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
M._eigenvects = M.eigenvects(simplify=False)
assert max(i.q for i in M._eigenvects[0][2][0]) > 1
M._eigenvects = M.eigenvects(simplify=True)
assert max(i.q for i in M._eigenvects[0][2][0]) == 1
M = Matrix([[Rational(1, 4), 1], [1, 1]])
assert M.eigenvects(simplify=True) == [
(Rational(5, 8) - sqrt(73) / 8, 1,
[Matrix([[-sqrt(73) / 8 - Rational(3, 8)], [1]])]),
(Rational(5, 8) + sqrt(73) / 8, 1,
[Matrix([[Rational(-3, 8) + sqrt(73) / 8], [1]])])
]
assert M.eigenvects(simplify=False) == [
(Rational(5, 8) - sqrt(73) / 8, 1,
[Matrix([[-1 / (-Rational(3, 8) + sqrt(73) / 8)], [1]])]),
(Rational(5, 8) + sqrt(73) / 8, 1,
[Matrix([[8 / (3 + sqrt(73))], [1]])])
]
m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
evals = {
Rational(5, 4) - sqrt(385) / 20: 1,
sqrt(385) / 20 + Rational(5, 4): 1,
S.Zero: 1
}
assert m.eigenvals() == evals
nevals = list(sorted(m.eigenvals(rational=False).keys()))
sevals = list(sorted(evals.keys()))
assert all(abs(nevals[i] - sevals[i]) < 1e-9 for i in range(len(nevals)))
# issue 10719
assert Matrix([]).eigenvals() == {}
assert Matrix([]).eigenvects() == []
# issue 15119
raises(NonSquareMatrixError,
lambda: Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
raises(NonSquareMatrixError,
lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
raises(
NonSquareMatrixError, lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(
error_when_incomplete=False))
raises(
NonSquareMatrixError, lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(
error_when_incomplete=False))
# issue 15125
from sympy.core.function import count_ops
q = Symbol("q", positive=True)
m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
assert count_ops(m.eigenvals(simplify=False)) > count_ops(
m.eigenvals(simplify=True))
assert count_ops(m.eigenvals(simplify=lambda x: x)) > count_ops(
m.eigenvals(simplify=True))
assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
