Information Theory, Inference, and Learning ... - MAELabs UCSD

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C.3: Perturbation theory 611
where
wb = e (b) (a)
L (0)f R , (C.27)
so, comparing (C.25) and (C.27), we have:
f (a)
R
e
b=a
(b)
L (0)Ve(a) R (0)
λ (a) (0) − λ (b) (0) e(b)
R
=
(0). (C.28)
Equations (C.23) and (C.28) are the solution to the first-order perturbation
theory problem, giving respectively the first derivative of the eigenvalue and
the eigenvectors.
Second-order perturbation theory
If we expand the eigenvector equation (C.11) to second order in ɛ, and assume
that the equation
H(ɛ) = H(0) + ɛV (C.29)
is exact, that is, H is a purely linear function of ɛ, then we have:
(H(0) + ɛV)(e (a) (a) 1
R (0) + ɛf R +
2 ɛ2g (a)
R + · · ·)
= (λ (a) (0) + ɛµ (a) + 1
2ɛ2ν (a) + · · ·)(e (a) (a)
R (0) + ɛf R
1 + 2ɛ2g (a)
R + · · ·) (C.30)
where g (a)
R and ν(a) are the second derivatives of the eigenvector and eigenvalue.
Equating the second-order terms in ɛ in equation (C.30),
Vf (a)
R
1
+
2 H(0)g(a)
1
R =
2 λ(a) (0)g (a) 1
R +
2 ν(a) e (a)
Hitting this equation on the left with e (a)
L (0), we obtain:
e (a) (a) 1
L (0)Vf R +
2 λ(a) e (a)
L (0)g(a) R
= 1
2λ(a) (0)e (a)
L (0)g(a) 1
R + 2ν(a) e (a)
L (0)e(a)
R (0) + µ(a) f (a)
R . (C.31)
R (0) + µ(a) e (a)
L
(a)
(0)f R . (C.32)
The term e (a) (a)
L (0)f R is equal to zero because of our constraints (C.19), so
e (a) (a) 1
L (0)Vf R =
2 ν(a) , (C.33)
so the second derivative of the eigenvalue with respect to ɛ is given by
1
2 ν(a) = e (a)
L (0)V
=
b=a
e
b=a
(b)
L (0)Ve(a) R (0)
λ (a) (0) − λ (b) (0) e(b)
R
[e (b)
L (0)Ve(a) R (0)][e(a) L (0)Ve(b)
λ (a) (0) − λ (b) (0)
This is as far as we will take the perturbation expansion.
Summary
(0) (C.34)
R (0)]
. (C.35)
If we introduce the abbreviation Vba for e (b)
L (0)Ve(a) R (0), we can write the
eigenvectors of H(ɛ) = H(0) + ɛV to first order as
e (a)
R
(ɛ) = e(a)
R (0) + ɛ
b=a
and the eigenvalues to second order as
λ (a) (ɛ) = λ (a)
2
(0) + ɛVaa + ɛ
Vba
λ (a) (0) − λ (b) (0) e(b)
R
b=a
(0) + · · · (C.36)
VbaVab
λ (a) (0) − λ (b) + · · · . (C.37)
(0)