return self.from_vector(coords)
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
"""
Return an integer "size" that is an upper bound on the size of
this algebra when it is used in a random test
interpreted to be far less than the dimension) should override
with a smaller number.
"""
- return 5
+ raise NotImplementedError
def _repr_(self):
"""
# appeal to the "long vectors" isometry.
oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
- # Now we use basis linear algebra to find the coefficients,
+ # Now we use basic linear algebra to find the coefficients,
# of the matrices-as-vectors-linear-combination, which should
# work for the original algebra basis too.
- A = matrix.column(self.base_ring(), oper_vecs)
+ A = matrix(self.base_ring(), oper_vecs)
# We used the isometry on the left-hand side already, but we
# still need to do it for the right-hand side. Recall that we
# wanted something that summed to the identity matrix.
b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )
- # Now if there's an identity element in the algebra, this should work.
- coeffs = A.solve_right(b)
- return self.linear_combination(zip(self.gens(), coeffs))
+ # Now if there's an identity element in the algebra, this
+ # should work. We solve on the left to avoid having to
+ # transpose the matrix "A".
+ return self.from_vector(A.solve_left(b))
def peirce_decomposition(self, c):
Beware, this will crash for "most instances" because the
constructor below looks wrong.
"""
- if cls is TrivialEJA:
- # The TrivialEJA class doesn't take an "n" argument because
- # there's only one.
- return cls(field)
-
- n = ZZ.random_element(cls._max_test_case_size() + 1)
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, field, **kwargs)
@cached_method
class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
# Play it safe, since this will be squared and the underlying
# field can have dimension 4 (quaternions) too.
return 2
The dimension of this algebra is `(n^2 + n) / 2`::
sage: set_random_seed()
- sage: n_max = RealSymmetricEJA._max_test_case_size()
+ sage: n_max = RealSymmetricEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = RealSymmetricEJA(n)
sage: J.dimension() == (n^2 + n)/2
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
return 4 # Dimension 10
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
+ sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_random_instance_size()
sage: n = ZZ.random_element(n_max)
sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
The dimension of this algebra is `n^2`::
sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_test_case_size()
+ sage: n_max = ComplexHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = ComplexHermitianEJA(n)
sage: J.dimension() == n^2
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
+ sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_random_instance_size()
sage: n = ZZ.random_element(n_max)
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
The dimension of this algebra is `2*n^2 - n`::
sage: set_random_seed()
- sage: n_max = QuaternionHermitianEJA._max_test_case_size()
+ sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
**kwargs)
self.rank.set_cache(n)
+ @staticmethod
+ def _max_random_instance_size():
+ return 5
+
def inner_product(self, x, y):
"""
Faster to reimplement than to use natural representations.
r"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the half-trace inner product and jordan product ``x*y =
- (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
- symmetric positive-definite "bilinear form" matrix. It has
- dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
- when ``B`` is the identity matrix of order ``n-1``.
+ (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
+ a symmetric positive-definite "bilinear form" matrix. Its
+ dimension is the size of `B`, and it has rank two in dimensions
+ larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
+ the identity matrix of order ``n``.
+
+ We insist that the one-by-one upper-left identity block of `B` be
+ passed in as well so that we can be passed a matrix of size zero
+ to construct a trivial algebra.
SETUP::
When no bilinear form is specified, the identity matrix is used,
and the resulting algebra is the Jordan spin algebra::
- sage: J0 = BilinearFormEJA(3)
+ sage: B = matrix.identity(AA,3)
+ sage: J0 = BilinearFormEJA(B)
sage: J1 = JordanSpinEJA(3)
sage: J0.multiplication_table() == J0.multiplication_table()
True
We can create a zero-dimensional algebra::
- sage: J = BilinearFormEJA(0)
+ sage: B = matrix.identity(AA,0)
+ sage: J = BilinearFormEJA(B)
sage: J.basis()
Finite family {}
sage: set_random_seed()
sage: n = ZZ.random_element(5)
sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
- sage: B = M.transpose()*M
- sage: J = BilinearFormEJA(n, B=B)
+ sage: B11 = matrix.identity(QQ,1)
+ sage: B22 = M.transpose()*M
+ sage: B = block_matrix(2,2,[ [B11,0 ],
+ ....: [0, B22 ] ])
+ sage: J = BilinearFormEJA(B)
sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
sage: V = J.vector_space()
sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
sage: actual == expected
True
"""
- def __init__(self, n, field=AA, B=None, **kwargs):
- if B is None:
- self._B = matrix.identity(field, max(0,n-1))
- else:
- self._B = B
+ def __init__(self, B, field=AA, **kwargs):
+ self._B = B
+ n = B.nrows()
+
+ if not B.is_positive_definite():
+ raise TypeError("matrix B is not positive-definite")
V = VectorSpace(field, n)
mult_table = [[V.zero() for j in range(n)] for i in range(n)]
xbar = x[1:]
y0 = y[0]
ybar = y[1:]
- z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
+ z0 = (B*x).inner_product(y)
zbar = y0*xbar + x0*ybar
z = V([z0] + zbar.list())
mult_table[i][j] = z
**kwargs)
self.rank.set_cache(min(n,2))
+ @staticmethod
+ def _max_random_instance_size():
+ return 5
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if n == 0:
+ # Special case needed since we use (n-1) below.
+ B = matrix.identity(field, 0)
+ return cls(B, field, **kwargs)
+
+ B11 = matrix.identity(field,1)
+ M = matrix.random(field, n-1)
+ I = matrix.identity(field, n-1)
+ alpha = field.zero()
+ while alpha.is_zero():
+ alpha = field.random_element().abs()
+ B22 = M.transpose()*M + alpha*I
+
+ from sage.matrix.special import block_matrix
+ B = block_matrix(2,2, [ [B11, ZZ(0) ],
+ [ZZ(0), B22 ] ])
+
+ return cls(B, field, **kwargs)
+
def inner_product(self, x, y):
r"""
Half of the trace inner product.
paper::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,5)
- sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
- sage: B = M.transpose()*M
- sage: J = BilinearFormEJA(n, B=B)
+ sage: J = BilinearFormEJA.random_instance()
+ sage: n = J.dimension()
sage: x = J.random_element()
sage: y = J.random_element()
- sage: x.inner_product(y) == (x*y).trace()/2
+ sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
True
"""
- xvec = x.to_vector()
- xbar = xvec[1:]
- yvec = y.to_vector()
- ybar = yvec[1:]
- return x[0]*y[0] + (self._B*xbar).inner_product(ybar)
+ return (self._B*x.to_vector()).inner_product(y.to_vector())
class JordanSpinEJA(BilinearFormEJA):
def __init__(self, n, field=AA, **kwargs):
# This is a special case of the BilinearFormEJA with the identity
# matrix as its bilinear form.
- super(JordanSpinEJA, self).__init__(n, field, **kwargs)
+ B = matrix.identity(field, n)
+ super(JordanSpinEJA, self).__init__(B, field, **kwargs)
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+
+ Needed here to override the implementation for ``BilinearFormEJA``.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
# largest subalgebra generated by any element.
self.rank.set_cache(0)
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ # We don't take a "size" argument so the superclass method is
+ # inappropriate for us.
+ return cls(field, **kwargs)
class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
r"""
iota_left = lambda x: self.from_vector(I1*x.to_vector())
iota_right = lambda x: self.from_vector(I2*+x.to_vector())
return (iota_left, iota_right)
+
+ def inner_product(self, x, y):
+ r"""
+ The standard Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ SETUP::
+
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLE::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: x1 = J1.one()
+ sage: x2 = x1
+ sage: y1 = J2.one()
+ sage: y2 = y1
+ sage: x1.inner_product(x2)
+ 3
+ sage: y1.inner_product(y2)
+ 2
+ sage: J.one().inner_product(J.one())
+ 5
+
+ """
+ (pi_left, pi_right) = self.projections()
+ x1 = pi_left(x)
+ x2 = pi_right(x)
+ y1 = pi_left(y)
+ y2 = pi_right(y)
+
+ return (x1.inner_product(y1) + x2.inner_product(y2))