def run_eighe(A, verbose=False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eighe(A)
B = Q * mp.diag(D) * Q.transpose_conj()
C = A - B
E = Q * Q.transpose_conj() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp(0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def run_eighe(A, verbose = False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eighe(A)
B = Q * mp.diag(D) * Q.transpose_conj()
C = A - B
E = Q * Q.transpose_conj() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp( 0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def conjugator_into_SL2R(SL2C_matrices):
"""
Returns a matrix C in SL(2, C) so that C^-1 * M * C is
(essentially) in SL(2, R) for all the input matrices M.
"""
ans, sig, form = preserves_hermitian_form(SL2C_matrices)
if ans is None:
raise ValueError('No invariant hermitian form found')
if sig == 'definite':
raise ValueError('Conjugate into SU(2), not SL(2, R)')
if sig == 'both':
raise ValueError('This degnerate case not implemented')
assert sig == 'indefinite'
J = sage_matrix_to_mpmath(form)
eigs, U = mp.eighe(J)
C = U * mp.diag([1 / mp.sqrt(abs(e)) for e in eigs])
sq_two = mp.sqrt(2)
sq_two_i = mp.mpc(imag=sq_two)
S = mp.matrix([[1 / sq_two_i, 1 / sq_two_i], [-1 / sq_two, 1 / sq_two]])
C = C * S.H
C = (1 / mp.sqrt(mp.det(C))) * C
return mpmath_matrix_to_sage(C)