We should compute that an element subalgebra is associative even
if we circumvent the element method::
- sage: set_random_seed()
sage: J = random_eja(field=QQ,orthonormalize=False)
sage: x = J.random_element()
sage: A = x.subalgebra_generated_by(orthonormalize=False)
TESTS::
- sage: set_random_seed()
sage: J = random_eja()
sage: J(1)
Traceback (most recent call last):
TESTS::
- sage: set_random_seed()
sage: J = random_eja()
sage: n = J.dimension()
sage: bi = J.zero()
Our inner product is "associative," which means the following for
a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: (x*y).inner_product(z) == y.inner_product(x*z)
Ensure that this is the usual inner product for the algebras
over `R^n`::
- sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
one). This is in Faraut and Koranyi, and also my "On the
symmetry..." paper::
- sage: set_random_seed()
sage: J = BilinearFormEJA.random_instance()
sage: n = J.dimension()
sage: x = J.random_element()
The values we've presupplied to the constructors agree with
the computation::
- sage: set_random_seed()
sage: J = random_eja()
sage: J.is_associative() == J._jordan_product_is_associative()
True
Ensure that we can convert any element back and forth
faithfully between its matrix and algebra representations::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J(x.to_matrix()) == x
Our inner product is "associative," which means the following for
a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: (x*y).inner_product(z) == y.inner_product(x*z)
Ensure that this is the usual inner product for the algebras
over `R^n`::
- sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
one). This is in Faraut and Koranyi, and also my "On the
symmetry..." paper::
- sage: set_random_seed()
sage: J = BilinearFormEJA.random_instance()
sage: n = J.dimension()
sage: x = J.random_element()
The identity element acts like the identity, regardless of
whether or not we orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
regardless of the base field and whether or not we
orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
Ensure that the cached unit element (often precomputed by
hand) agrees with the computed one::
- sage: set_random_seed()
sage: J = random_eja()
sage: cached = J.one()
sage: J.one.clear_cache()
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: cached = J.one()
sage: J.one.clear_cache()
Every algebra decomposes trivially with respect to its identity
element::
- sage: set_random_seed()
sage: J = random_eja()
sage: J0,J5,J1 = J.peirce_decomposition(J.one())
sage: J0.dimension() == 0 and J5.dimension() == 0
elements in the two subalgebras are the projections onto their
respective subspaces of the superalgebra's identity element::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: if not J.is_trivial():
for idx in range(count) )
+ def operator_polynomial_matrix(self):
+ r"""
+ Return the matrix of polynomials (over this algebra's
+ :meth:`coordinate_polynomial_ring`) that, when evaluated at
+ the basis coordinates of an element `x`, produces the basis
+ representation of `L_{x}`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 0 0 0]
+ [ 0 X1 0 0]
+ [ 0 0 X2 0]
+ [ 0 0 0 X3]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 X1 X2 X3]
+ [X1 X0 0 0]
+ [X2 0 X0 0]
+ [X3 0 0 X0]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ """
+ R = self.coordinate_polynomial_ring()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( v*self.monomial(k).operator().matrix()[i,j]
+ for (k,v) in enumerate(R.gens()) )
+
+ n = self.dimension()
+ return matrix(R, n, n, L_x_i_j)
+
@cached_method
def _charpoly_coefficients(self):
r"""
The theory shows that these are all homogeneous polynomials of
a known degree::
- sage: set_random_seed()
sage: J = random_eja()
sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
True
"""
n = self.dimension()
R = self.coordinate_polynomial_ring()
- vars = R.gens()
F = R.fraction_field()
- def L_x_i_j(i,j):
- # From a result in my book, these are the entries of the
- # basis representation of L_x.
- return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
- for k in range(n) )
-
- L_x = matrix(F, n, n, L_x_i_j)
+ L_x = self.operator_polynomial_matrix()
r = None
if self.rank.is_in_cache():
positive integer rank, unless the algebra is trivial in
which case its rank will be zero::
- sage: set_random_seed()
sage: J = random_eja()
sage: r = J.rank()
sage: r in ZZ
Ensure that computing the rank actually works, since the ranks
of all simple algebras are known and will be cached by default::
- sage: set_random_seed() # long time
sage: J = random_eja() # long time
sage: cached = J.rank() # long time
sage: J.rank.clear_cache() # long time
Our basis is normalized with respect to the algebra's inner
product, unless we specify otherwise::
- sage: set_random_seed()
sage: J = ConcreteEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
natural->EJA basis representation is an isometry and within the
EJA the operator is self-adjoint by the Jordan axiom::
- sage: set_random_seed()
sage: J = ConcreteEJA.random_instance()
sage: x = J.random_element()
sage: x.operator().is_self_adjoint()
return eja_class.random_instance(max_dimension, *args, **kwargs)
-class MatrixEJA(FiniteDimensionalEJA):
+class HermitianMatrixEJA(FiniteDimensionalEJA):
@staticmethod
def _denormalized_basis(A):
"""
- Returns a basis for the space of complex Hermitian n-by-n matrices.
+ Returns a basis for the given Hermitian matrix space.
Why do we embed these? Basically, because all of numerical linear
algebra assumes that you're working with vectors consisting of `n`
sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
....: QuaternionMatrixAlgebra,
....: OctonionMatrixAlgebra)
- sage: from mjo.eja.eja_algebra import MatrixEJA
+ sage: from mjo.eja.eja_algebra import HermitianMatrixEJA
TESTS::
- sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: A = MatrixSpace(QQ, n)
- sage: B = MatrixEJA._denormalized_basis(A)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
sage: all( M.is_hermitian() for M in B)
True
::
- sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: A = ComplexMatrixAlgebra(n, scalars=QQ)
- sage: B = MatrixEJA._denormalized_basis(A)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
sage: all( M.is_hermitian() for M in B)
True
::
- sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: A = QuaternionMatrixAlgebra(n, scalars=QQ)
- sage: B = MatrixEJA._denormalized_basis(A)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
sage: all( M.is_hermitian() for M in B )
True
::
- sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: A = OctonionMatrixAlgebra(n, scalars=QQ)
- sage: B = MatrixEJA._denormalized_basis(A)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
sage: all( M.is_hermitian() for M in B )
True
self.rank.set_cache(matrix_space.nrows())
self.one.set_cache( self(matrix_space.one()) )
-class RealSymmetricEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
+class RealSymmetricEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
The dimension of this algebra is `(n^2 + n) / 2`::
- sage: set_random_seed()
sage: d = RealSymmetricEJA._max_random_instance_dimension()
sage: n = RealSymmetricEJA._max_random_instance_size(d)
sage: J = RealSymmetricEJA(n)
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = RealSymmetricEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
-class ComplexHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
+class ComplexHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
The dimension of this algebra is `n^2`::
- sage: set_random_seed()
sage: d = ComplexHermitianEJA._max_random_instance_dimension()
sage: n = ComplexHermitianEJA._max_random_instance_size(d)
sage: J = ComplexHermitianEJA(n)
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = ComplexHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
return cls(n, **kwargs)
-class QuaternionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
+class QuaternionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
The dimension of this algebra is `2*n^2 - n`::
- sage: set_random_seed()
sage: d = QuaternionHermitianEJA._max_random_instance_dimension()
sage: n = QuaternionHermitianEJA._max_random_instance_size(d)
sage: J = QuaternionHermitianEJA(n)
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = QuaternionHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
n = ZZ.random_element(max_size + 1)
return cls(n, **kwargs)
-class OctonionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
+class OctonionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
SETUP::
matrix. We opt not to orthonormalize the basis, because if we
did, we would have to normalize the `s_{i}` in a similar manner::
- sage: set_random_seed()
sage: n = ZZ.random_element(5)
sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
sage: B11 = matrix.identity(QQ,1)
Ensure that we have the usual inner product on `R^n`::
- sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
The Jordan product is inherited from our factors and implemented by
our CombinatorialFreeModule Cartesian product superclass::
- sage: set_random_seed()
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
sage: J = cartesian_product([J1,J2])
The cached unit element is the same one that would be computed::
- sage: set_random_seed() # long time
sage: J1 = random_eja() # long time
sage: J2 = random_eja() # long time
sage: J = cartesian_product([J1,J2]) # long time
The answer never changes::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
The answer never changes::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
produce the identity map, and mismatching them should produce
the zero map::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
TESTS::
- sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: J = random_eja(max_dimension=n, field=QQ, orthonormalize=False)
sage: J.dimension() <= n
# if the sub-call also Decides on a cartesian product.
J2 = random_eja(new_max_dimension, *args, **kwargs)
return cartesian_product([J1,J2])
+
+
+class ComplexSkewHermitianEJA(RationalBasisEJA):
+ r"""
+ The EJA described in Faraut and Koranyi's Exercise III.1.b.
+ """
+ @staticmethod
+ def _denormalized_basis(A):
+ """
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexSkewHermitianEJA
+
+ TESTS::
+
+ sage: n = ZZ.random_element(1,2)
+ sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ)
+ sage: B = ComplexSkewHermitianEJA._denormalized_basis(A)
+ sage: all( M.is_skew_hermitian() for M in B)
+ True
+
+ """
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
+
+ basis = []
+
+ # The size of the blocks. We're going to treat these thing as
+ # 2x2 block matrices,
+ #
+ # [ x1 x2 ]
+ # [ -x2^* x1-conj ]
+ #
+ # where x1 is skew-Hermitian and x2 is symmetric.
+ #
+ m = A.nrows()/2
+
+ # We only loop through the top half of the matrix, because the
+ # bottom can be constructed from the top.
+ for i in range(m):
+
+ # First do the top-left block, which is skew-Hermitian.
+ # We can compute the bottom-right block in the process.
+ for j in range(i+1):
+ if i == j:
+ # Top-left block's entry.
+ E_ii = gen(A, (i,j,es[1]))
+
+ # Bottom-right block's entry.
+ E_ii += gen(A, (i+m,j+m,es[1])).conjugate()
+ basis.append(E_ii)
+ else:
+ for e in es:
+ # Top-left block's entry.
+ E_ij = gen(A, (i,j,e))
+ E_ij -= E_ij.conjugate_transpose()
+
+ # Bottom-right block's entry.
+ F_ij = gen(A, (i+m,j+m,e)).conjugate()
+ F_ij -= F_ij.conjugate_transpose()
+
+ basis.append(E_ij + F_ij)
+
+ # Now do the top-right block, which is symmetric, and compute
+ # the bottom-left block along the way.
+ for j in range(m,i+m+1):
+ if (i+m) == j:
+ # A symmetric (not Hermitian!) complex matrix can
+ # have both real and complex entries on its
+ # diagonal.
+ for e in es:
+ # Top-right block's entry.
+ E_ii = gen(A, (i,j,e))
+
+ # Bottom-left block's entry.
+ E_ii -= gen(A, (i-m,j-m,e)).conjugate()
+ basis.append(E_ii)
+ else:
+ for e in es:
+ # Top-right block's entry. BEWARE! We're not
+ # reflecting across the main diagonal as in
+ # (i,j)~(j,i). We're only reflecting across
+ # the diagonal for the top-right block.
+ E_ij = gen(A, (i,j,e))
+
+ # Shift it back to non-offset coords, transpose,
+ # and put it back:
+ #
+ # (i,j) -> (i,j-m) -> (j-m, i) -> (j-m, i+m)
+ E_ij += gen(A, (j-m,i+m,e))
+
+ # Bottom-left's block's below-diagonal entry.
+ # Just shift the top-right coords down m and
+ # left m.
+ F_ij = -gen(A, (i+m,j-m,e)).conjugate()
+ F_ij += -gen(A, (j,i,e)).conjugate()
+
+ basis.append(E_ij + F_ij)
+
+ return tuple( basis )
+
+
+ def __init__(self, n, field=AA, **kwargs):
+ # New code; always check the axioms.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(2*n, scalars=field)
+
+ I_n = matrix.identity(ZZ, n)
+ J = matrix.block(ZZ, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
+ J = A.from_list(J.rows())
+
+ def jordan_product(X,Y):
+ return (X*J*Y + Y*J*X)/2
+
+ def inner_product(X,Y):
+ return (X*Y.conjugate_transpose()).trace().real()
+
+ super().__init__(self._denormalized_basis(A),
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=A,
+ **kwargs)
projects / sage.d.git / blobdiff