Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.C.3: Perturbation theory 611wherew b = e (b) (a)L(0)fR , (C.27)so, comparing (C.25) and (C.27), we have:f (a)R= ∑ b≠ae (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 perturbationtheory problem, giving respectively the first derivative of the eigenvalue andthe eigenvectors.Second-order perturbation theoryIf we expand the eigenvector equation (C.11) to second order in ɛ, and assumethat the equationH(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)R(0) + ɛf(a)R+ 1 2 ɛ2 g (a)R + · · ·)= (λ (a) (0) + ɛµ (a) + 1 2 ɛ2 ν (a) + · · ·)(e (a)R(0) + ɛf (a)R+ 1 2 ɛ2 g (a)R+ · · ·) (C.30)where g (a)Rand 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) R= 1 2 λ(a) (0)g (a)RHitting this equation on the left with e (a)L(0), we obtain:e (a)LThe term e (a)L(0)Vf (a)R= 1 2 λ(a) (0)e (a)L(0)g(a) R+ 12 λ(a) e (a)L(0)g(a) R+ 12 ν(a) e (a)L+ 1 2 ν(a) e (a)R(0) + µ(a) f (a)R . (C.31)(0)e(a) R(0) + µ(a) e (a)L(a)(0)fR . (C.32)(0)f (a) is equal to zero because of our constraints (C.19), soRe (a)L(a)(0)VfR = 1 2 ν(a) , (C.33)so the second derivative of the eigenvalue with respect to ɛ is given by12 ν(a) = e (a)L(0)V ∑ e (b)L(0)Ve(a) R (0)λ (a) (0) − λ (b) (0) e(b) R (0)b≠a= ∑ b≠a[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(C.34)R (0)] . (C.35)If we introduce the abbreviation V ba for e (b)L(0)Ve(a) R(0), we can write theeigenvectors of H(ɛ) = H(0) + ɛV to first order ase (a)R(ɛ) = e(a) R(0) + ɛ ∑ b≠aV baλ (a) (0) − λ (b) (0) e(b) R (0) + · · ·(C.36)and the eigenvalues to second order asλ (a) (ɛ) = λ (a) (0) + ɛV aa + ɛ 2 ∑ b≠aV ba V abλ (a) (0) − λ (b) (0) + · · · .(C.37)