def test_diagonal_symmetrical():
m = Matrix(2,2,[0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = Matrix(2,2,[1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3,3,[1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])
assert not m.is_symmetric()
assert m.is_diagonal()
m = Matrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = Matrix(((5, 0, 0), (0, 6, 0)))
assert m.is_diagonal()
x, y = symbols('x y')
m = Matrix(3,3,[1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])
assert m.is_symmetric()
assert not m.is_symmetric(simplify=False)
assert m.expand().is_symmetric(simplify=False)
def test_diagonal_symmetrical():
m = Matrix(2,2,[0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = Matrix(2,2,[1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3,3,[1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])
assert not m.is_symmetric()
assert m.is_diagonal()
m = Matrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = Matrix(((5, 0, 0), (0, 6, 0)))
assert m.is_diagonal()
x, y = symbols('x y')
m = Matrix(3,3,[1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])
assert m.is_symmetric()
assert not m.is_symmetric(simplify=False)
assert m.expand().is_symmetric(simplify=False)
def test_get_diag_blocks2():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b).get_diag_blocks() == [a, b, b]
assert diag(a, b, c).get_diag_blocks() == [a, b, c]
assert diag(a, c, b).get_diag_blocks() == [a, c, b]
assert diag(c, c, b).get_diag_blocks() == [c, c, b]
def test_get_diag_blocks2():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b).get_diag_blocks() == [a, b, b]
assert diag(a, b, c).get_diag_blocks() == [a, b, c]
assert diag(a, c, b).get_diag_blocks() == [a, c, b]
assert diag(c, c, b).get_diag_blocks() == [c, c, b]
def test_diag():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b) == Matrix([
[1, 2, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0],
[0, 0, y, 3, 0, 0],
[0, 0, 0, 0, 3, x],
[0, 0, 0, 0, y, 3],
])
assert diag(a, b, c) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0, 0],
[0, 0, y, 3, 0, 0, 0],
[0, 0, 0, 0, 3, x, 3],
[0, 0, 0, 0, y, 3, z],
[0, 0, 0, 0, x, y, z],
])
assert diag(a, c, b) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 3, 0, 0],
[0, 0, y, 3, z, 0, 0],
[0, 0, x, y, z, 0, 0],
[0, 0, 0, 0, 0, 3, x],
[0, 0, 0, 0, 0, y, 3],
])
a = Matrix([x, y, z])
b = Matrix([[1, 2], [3, 4]])
c = Matrix([[5, 6]])
assert diag(a, 7, b, c) == Matrix([
[x, 0, 0, 0, 0, 0],
[y, 0, 0, 0, 0, 0],
[z, 0, 0, 0, 0, 0],
[0, 7, 0, 0, 0, 0],
[0, 0, 1, 2, 0, 0],
[0, 0, 3, 4, 0, 0],
[0, 0, 0, 0, 5, 6],
])
def test_diag():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b) == Matrix([
[1, 2, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0],
[0, 0, y, 3, 0, 0],
[0, 0, 0, 0, 3, x],
[0, 0, 0, 0, y, 3],
])
assert diag(a, b, c) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0, 0],
[0, 0, y, 3, 0, 0, 0],
[0, 0, 0, 0, 3, x, 3],
[0, 0, 0, 0, y, 3, z],
[0, 0, 0, 0, x, y, z],
])
assert diag(a, c, b) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 3, 0, 0],
[0, 0, y, 3, z, 0, 0],
[0, 0, x, y, z, 0, 0],
[0, 0, 0, 0, 0, 3, x],
[0, 0, 0, 0, 0, y, 3],
])
a = Matrix([x, y, z])
b = Matrix([[1, 2], [3, 4]])
c = Matrix([[5, 6]])
assert diag(a, 7, b, c) == Matrix([
[x, 0, 0, 0, 0, 0],
[y, 0, 0, 0, 0, 0],
[z, 0, 0, 0, 0, 0],
[0, 7, 0, 0, 0, 0],
[0, 0, 1, 2, 0, 0],
[0, 0, 3, 4, 0, 0],
[0, 0, 0, 0, 5, 6],
])
def test_inv_block():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A = diag(a, b, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), b.inv())
A = diag(a, b, c)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), c.inv())
A = diag(a, c, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), c.inv(), b.inv())
A = diag(a, a, b, a, c, a)
assert A.inv(try_block_diag=True) == diag(
a.inv(), a.inv(), b.inv(), a.inv(), c.inv(), a.inv())
assert A.inv(try_block_diag=True, method="ADJ") == diag(
a.inv(method="ADJ"), a.inv(method="ADJ"), b.inv(method="ADJ"),
a.inv(method="ADJ"), c.inv(method="ADJ"), a.inv(method="ADJ"))
def test_inv_block():
x, y, z = symbols("x y z")
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A = diag(a, b, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), b.inv())
A = diag(a, b, c)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), c.inv())
A = diag(a, c, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), c.inv(), b.inv())
A = diag(a, a, b, a, c, a)
assert A.inv(try_block_diag=True) == diag(
a.inv(), a.inv(), b.inv(), a.inv(), c.inv(), a.inv())
assert A.inv(try_block_diag=True, method="ADJ") == diag(
a.inv(method="ADJ"), a.inv(method="ADJ"), b.inv(method="ADJ"),
a.inv(method="ADJ"), c.inv(method="ADJ"), a.inv(method="ADJ"))
def test_diagonal_symmetrical():
m = Matrix(2,2,[0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
m = Matrix(2,2,[1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3,3,[1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2,3,[0, 0, 0, 0, 0, 0])
assert not m.is_symmetric()
x, y = symbols('x','y')
m = Matrix(3,3,[1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])
assert m.is_symmetric()
def test_diagonal_symmetrical():
m = Matrix(2, 2, [0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
m = Matrix(2, 2, [1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])
assert not m.is_symmetric()
x, y = symbols('x', 'y')
m = Matrix(3, 3, [1, x**2 + 2 * x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
assert m.is_symmetric()
def test_diagonalization():
x, y, z = symbols('x y z')
m = Matrix(3,2,[-3, 1, -3, 20, 3, 10])
assert not m.is_diagonalizable()
assert not m.is_symmetric()
raises(NonSquareMatrixError, 'm.diagonalize()')
# diagonalizable
m = diag(1, 2, 3)
(P, D) = m.diagonalize()
assert P == eye(3)
assert D == m
m = Matrix(2,2,[0, 1, 1, 0])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, 3])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
assert D == m
m = Matrix(2,2,[1, 1, 0, 0])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, 0])
assert m.is_diagonal()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
# diagonalizable, complex only
m = Matrix(2,2,[0, 1, -1, 0])
assert not m.is_diagonalizable(True)
raises(MatrixError, '(D, P) = m.diagonalize(True)')
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, I])
raises(NotImplementedError, 'm.is_diagonalizable(True)')
# !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
# see issue 2193
# assert not m.is_diagonalizable(True)
# raises(MatrixError, '(P, D) = m.diagonalize(True)')
# (P, D) = m.diagonalize(True)
# not diagonalizable
m = Matrix(2,2,[0, 1, 0, 0])
assert not m.is_diagonalizable()
raises(MatrixError, '(D, P) = m.diagonalize()')
m = Matrix(3,3,[-3, 1, -3, 20, 3, 10, 2, -2, 4])
assert not m.is_diagonalizable()
raises(MatrixError, '(D, P) = m.diagonalize()')
# symbolic
a, b, c, d = symbols('a b c d')
m = Matrix(2,2,[a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
def test_diagonalization():
x, y, z = symbols('x y z')
m = Matrix(3,2,[-3, 1, -3, 20, 3, 10])
assert not m.is_diagonalizable()
assert not m.is_symmetric()
raises(NonSquareMatrixError, 'm.diagonalize()')
# diagonalizable
m = diag(1, 2, 3)
(P, D) = m.diagonalize()
assert P == eye(3)
assert D == m
m = Matrix(2,2,[0, 1, 1, 0])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, 3])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
assert D == m
m = Matrix(2,2,[1, 1, 0, 0])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, 0])
assert m.is_diagonal()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
# diagonalizable, complex only
m = Matrix(2,2,[0, 1, -1, 0])
assert not m.is_diagonalizable(True)
raises(MatrixError, '(D, P) = m.diagonalize(True)')
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2,2,[1, 0, 0, I])
raises(NotImplementedError, 'm.is_diagonalizable(True)')
# !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
# see issue 2193
# assert not m.is_diagonalizable(True)
# raises(MatrixError, '(P, D) = m.diagonalize(True)')
# (P, D) = m.diagonalize(True)
# not diagonalizable
m = Matrix(2,2,[0, 1, 0, 0])
assert not m.is_diagonalizable()
raises(MatrixError, '(D, P) = m.diagonalize()')
m = Matrix(3,3,[-3, 1, -3, 20, 3, 10, 2, -2, 4])
assert not m.is_diagonalizable()
raises(MatrixError, '(D, P) = m.diagonalize()')
# symbolic
a, b, c, d = symbols('a b c d')
m = Matrix(2,2,[a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()