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from sympy.core.numbers import I
from sympy.core.symbol import symbols
from sympy.testing.pytest import raises
from sympy.matrices import Matrix, zeros, eye
from sympy.core.symbol import Symbol
from sympy.core.numbers import Rational
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.simplify.simplify import simplify
from sympy.abc import x
# Matrix tests
def test_row_op():
e = eye(3)
raises(ValueError, lambda: e.elementary_row_op("abc"))
raises(ValueError, lambda: e.elementary_row_op())
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=-5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=-5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=-5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=1, k=5))
# test various ways to set arguments
assert e.elementary_row_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row1=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", 0, 5, 1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row1=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_col_op():
e = eye(3)
raises(ValueError, lambda: e.elementary_col_op("abc"))
raises(ValueError, lambda: e.elementary_col_op())
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=-5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=-5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=-5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=1, k=5))
# test various ways to set arguments
assert e.elementary_col_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col1=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", 0, 5, 1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col1=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_is_echelon():
zro = zeros(3)
ident = eye(3)
assert zro.is_echelon
assert ident.is_echelon
a = Matrix(0, 0, [])
assert a.is_echelon
a = Matrix(2, 3, [3, 2, 1, 0, 0, 6])
assert a.is_echelon
a = Matrix(2, 3, [0, 0, 6, 3, 2, 1])
assert not a.is_echelon
x = Symbol('x')
a = Matrix(3, 1, [x, 0, 0])
assert a.is_echelon
a = Matrix(3, 1, [x, x, 0])
assert not a.is_echelon
a = Matrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
assert not a.is_echelon
def test_echelon_form():
# echelon form is not unique, but the result
# must be row-equivalent to the original matrix
# and it must be in echelon form.
a = zeros(3)
e = eye(3)
# we can assume the zero matrix and the identity matrix shouldn't change
assert a.echelon_form() == a
assert e.echelon_form() == e
a = Matrix(0, 0, [])
assert a.echelon_form() == a
a = Matrix(1, 1, [5])
assert a.echelon_form() == a
# now we get to the real tests
def verify_row_null_space(mat, rows, nulls):
for v in nulls:
assert all(t.is_zero for t in a_echelon*v)
for v in rows:
if not all(t.is_zero for t in v):
assert not all(t.is_zero for t in a_echelon*v.transpose())
a = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
nulls = [Matrix([
[ 1],
[-2],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
nulls = []
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = Matrix(3, 3, [2, 1, 3, 0, 0, 0, 2, 1, 3])
nulls = [Matrix([
[Rational(-1, 2)],
[ 1],
[ 0]]),
Matrix([
[Rational(-3, 2)],
[ 0],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
# this one requires a row swap
a = Matrix(3, 3, [2, 1, 3, 0, 0, 0, 1, 1, 3])
nulls = [Matrix([
[ 0],
[ -3],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = Matrix(3, 3, [0, 3, 3, 0, 2, 2, 0, 1, 1])
nulls = [Matrix([
[1],
[0],
[0]]),
Matrix([
[ 0],
[-1],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = Matrix(2, 3, [2, 2, 3, 3, 3, 0])
nulls = [Matrix([
[-1],
[1],
[0]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
def test_rref():
e = Matrix(0, 0, [])
assert e.rref(pivots=False) == e
e = Matrix(1, 1, [1])
a = Matrix(1, 1, [5])
assert e.rref(pivots=False) == a.rref(pivots=False) == e
a = Matrix(3, 1, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1], [0], [0]])
a = Matrix(1, 3, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1, 2, 3]])
a = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert a.rref(pivots=False) == Matrix([
[1, 0, -1],
[0, 1, 2],
[0, 0, 0]])
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
b = Matrix(3, 3, [1, 2, 3, 0, 0, 0, 0, 0, 0])
c = Matrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
d = Matrix(3, 3, [0, 0, 0, 0, 0, 0, 1, 2, 3])
assert a.rref(pivots=False) == \
b.rref(pivots=False) == \
c.rref(pivots=False) == \
d.rref(pivots=False) == b
e = eye(3)
z = zeros(3)
assert e.rref(pivots=False) == e
assert z.rref(pivots=False) == z
a = Matrix([
[ 0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[ 0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
mat, pivot_offsets = a.rref()
assert mat == Matrix([
[1, -5, 0, 0, 1, 1, -1],
[0, 0, 1, 0, 0, -1, 1],
[0, 0, 0, 1, 1, -2, 1],
[0, 0, 0, 0, 0, 0, 0]])
assert pivot_offsets == (0, 2, 3)
a = Matrix([[Rational(1, 19), Rational(1, 5), 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[ 12, 13, 14, 15]])
assert a.rref(pivots=False) == Matrix([
[1, 0, 0, Rational(-76, 157)],
[0, 1, 0, Rational(-5, 157)],
[0, 0, 1, Rational(238, 157)],
[0, 0, 0, 0]])
x = Symbol('x')
a = Matrix(2, 3, [x, 1, 1, sqrt(x), x, 1])
for i, j in zip(a.rref(pivots=False),
[1, 0, sqrt(x)*(-x + 1)/(-x**Rational(5, 2) + x),
0, 1, 1/(sqrt(x) + x + 1)]):
assert simplify(i - j).is_zero
def test_rref_rhs():
a, b, c, d = symbols('a b c d')
A = Matrix([[0, 0], [0, 0], [1, 2], [3, 4]])
B = Matrix([a, b, c, d])
assert A.rref_rhs(B) == (Matrix([
[1, 0],
[0, 1],
[0, 0],
[0, 0]]), Matrix([
[ -2*c + d],
[3*c/2 - d/2],
[ a],
[ b]]))
def test_issue_17827():
C = Matrix([
[3, 4, -1, 1],
[9, 12, -3, 3],
[0, 2, 1, 3],
[2, 3, 0, -2],
[0, 3, 3, -5],
[8, 15, 0, 6]
])
# Tests for row/col within valid range
D = C.elementary_row_op('n<->m', row1=2, row2=5)
E = C.elementary_row_op('n->n+km', row1=5, row2=3, k=-4)
F = C.elementary_row_op('n->kn', row=5, k=2)
assert(D[5, :] == Matrix([[0, 2, 1, 3]]))
assert(E[5, :] == Matrix([[0, 3, 0, 14]]))
assert(F[5, :] == Matrix([[16, 30, 0, 12]]))
# Tests for row/col out of range
raises(ValueError, lambda: C.elementary_row_op('n<->m', row1=2, row2=6))
raises(ValueError, lambda: C.elementary_row_op('n->kn', row=7, k=2))
raises(ValueError, lambda: C.elementary_row_op('n->n+km', row1=-1, row2=5, k=2))
def test_rank():
m = Matrix([[1, 2], [x, 1 - 1/x]])
assert m.rank() == 2
n = Matrix(3, 3, range(1, 10))
assert n.rank() == 2
p = zeros(3)
assert p.rank() == 0
def test_issue_11434():
ax, ay, bx, by, cx, cy, dx, dy, ex, ey, t0, t1 = \
symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
M = Matrix([[ax, ay, ax*t0, ay*t0, 0],
[bx, by, bx*t0, by*t0, 0],
[cx, cy, cx*t0, cy*t0, 1],
[dx, dy, dx*t0, dy*t0, 1],
[ex, ey, 2*ex*t1 - ex*t0, 2*ey*t1 - ey*t0, 0]])
assert M.rank() == 4
def test_rank_regression_from_so():
# see:
# https://stackoverflow.com/questions/19072700/why-does-sympy-give-me-the-wrong-answer-when-i-row-reduce-a-symbolic-matrix
nu, lamb = symbols('nu, lambda')
A = Matrix([[-3*nu, 1, 0, 0],
[ 3*nu, -2*nu - 1, 2, 0],
[ 0, 2*nu, (-1*nu) - lamb - 2, 3],
[ 0, 0, nu + lamb, -3]])
expected_reduced = Matrix([[1, 0, 0, 1/(nu**2*(-lamb - nu))],
[0, 1, 0, 3/(nu*(-lamb - nu))],
[0, 0, 1, 3/(-lamb - nu)],
[0, 0, 0, 0]])
expected_pivots = (0, 1, 2)
reduced, pivots = A.rref()
assert simplify(expected_reduced - reduced) == zeros(*A.shape)
assert pivots == expected_pivots
def test_issue_15872():
A = Matrix([[1, 1, 1, 0], [-2, -1, 0, -1], [0, 0, -1, -1], [0, 0, 2, 1]])
B = A - Matrix.eye(4) * I
assert B.rank() == 3
assert (B**2).rank() == 2
assert (B**3).rank() == 2
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