def pprint(expr):
print(
latex_map(
mat_latex(expr),
OrderedDict([('fYY', r'f_{YY}'), ('fY', 'f_Y'),
('al', r'\alpha')])))
def calc_psd_range():
# in this method, the ambient is still
# R(n times p) + Symmetric(p)
# Horizontal is still
# al1*t(Y)*omg_Y + bt*omg_P*inv(R*R) - bt*inv(R*R)*omg_P)
# manifold is on pair (Y, P)
# Use pair B, D. B size n*(n-p), D size p(p+1)/2
# Use the embedding
# omg_Y = bt(Y*(inv(P)*D - D*inv(P))
# omg_P = al1*D
# so need to set up a new variable Y_0 and relation YY_0=0
Y, Y0 = stiefels('Y Y0')
sm.g_cstiefels[Y] = Y0
sm.g_cstiefels[Y0] = Y
B, eta_Y, eta_P = matrices('B eta_Y eta_P')
P, D = sym_symb('P D')
al0, al1, bt = scalars('al0 al1 bt')
def g(Y, P, omg_Y, omg_P):
return al0 * omg_Y + (al1 - al0) * Y * t(Y) * omg_Y, bt * inv(
P) * omg_P * inv(P)
def ginv(Y, P, omg_Y, omg_P):
return 1 / al0 * omg_Y + (
1 / al1 - 1 / al0) * Y * t(Y) * omg_Y, 1 / bt * P * omg_P * P
# check that ginv \circ g is id
e1, e2 = ginv(Y, P, *(g(Y, P, eta_Y, eta_P)))
e1 = mat_spfy(e1)
e2 = mat_spfy(e2)
print(e1, e2)
def base_ambient_inner(omg_Y, omg_P, xi_Y, xi_P):
return mat_spfy(
trace(mat_spfy(omg_Y * t(xi_Y))) +
trace(mat_spfy(omg_P * t(xi_P))))
def ambient_inner(Y, P, omg_Y, omg_P, xi_Y, xi_P):
return mat_spfy(
trace(
mat_spfy((al0 * omg_Y +
(al1 - al0) * Y * t(Y) * omg_Y) * t(xi_Y))) +
trace(mat_spfy(bt * inv(P) * omg_P * inv(P) * t(xi_P))))
def EN_inner(Y, P, Ba, Da, Bb, Db):
return trace(mat_spfy(Da * Db) + mat_spfy(Ba * t(Bb)))
qat = asym_symb('qat')
ipr = ambient_inner(Y, P, eta_Y, eta_P, Y * qat, P * qat - qat * P)
dqat = mat_spfy(xtrace(ipr, qat))
print(dqat)
def N(Y, P, B, D):
N_Y = mat_spfy(bt * Y * (inv(P) * D - D * inv(P))) + Y0 * B
N_P = mat_spfy(al1 * D)
return N_Y, N_P
def NT(Y, P, omg_Y, omg_P):
nB, nD = matrices('nB nD')
ipt = mat_spfy(base_ambient_inner(*N(Y, P, nB, nD), omg_Y, omg_P))
ntB = mat_spfy(xtrace(ipt, nB))
ntD1 = mat_spfy(xtrace(ipt, nD))
ntD = mat_spfy(Integer(1) / Integer(2) * (ntD1 + t(ntD1)))
return ntB, ntD
# check that image of N is horizontal:
print(
mat_spfy(
xtrace(
mat_spfy(
ambient_inner(Y, P, *N(Y, P, B, D), Y * qat,
P * qat - qat * P)), qat)))
NTe_B, NTe_D = NT(Y, P, eta_Y, eta_P)
mdict = {'bt': r'\beta', 'al': r'\alpha'}
print(latex_map(mat_latex(NTe_B), mdict))
print(latex_map(mat_latex(NTe_D), mdict))
gN = g(Y, P, *N(Y, P, B, D))
gN_B = mat_spfy(gN[0])
gN_D = mat_spfy(gN[1])
print(latex_map(mat_latex(gN_B), mdict))
print(latex_map(mat_latex(gN_D), mdict))
NTgN_B, NTgN_D = NT(Y, P, *gN)
print(latex_map(mat_latex(NTgN_B), mdict))
print(latex_map(mat_latex(NTgN_D), mdict))
NTg_B, NTg_D = NT(Y, P, *g(Y, P, eta_Y, eta_P))
print(latex_map(mat_latex(NTg_B), mdict))
# print(latex_map(sp.latex(NTgN_P), mdict))
print(latex_map(mat_latex(NTg_D), mdict))
def sym(x):
return mat_spfy(Integer(1) / Integer(2) * (x + t(x)))
xi_Y, xi_P, phi_Y, phi_P = matrices('xi_Y xi_P phi_Y phi_P')
gYPeta = g(Y, P, eta_Y, eta_P)
Dgxieta_Y = DDR(gYPeta[0], Y, xi_Y)
Dgxieta_P = DDR(gYPeta[1], P, xi_P)
gYPxi = g(Y, P, xi_Y, xi_P)
Dgetaxi_Y = DDR(gYPxi[0], Y, eta_Y)
Dgetaxi_P = DDR(gYPxi[1], P, eta_P)
Dgxiphi_Y = DDR(gYPxi[0], Y, phi_Y)
Dgxiphi_P = DDR(gYPxi[1], P, phi_P)
tr3 = mat_spfy(base_ambient_inner(Y, P, Dgxiphi_Y, Dgxiphi_P, eta_Y,
eta_P))
xcross_Y = xtrace(tr3, phi_Y)
xcross_P = xtrace(tr3, phi_P)
K_Y = (Integer(1) / Integer(2)) * (Dgxieta_Y + Dgetaxi_Y - xcross_Y)
K_P = (Integer(1) / Integer(2)) * (Dgxieta_P + Dgetaxi_P - xcross_P)
pprint(K_Y)
pprint(K_P)