Lecture 15: Quadratic approximation and the second variation formula

Transcription

1 Lecture 15: Quadratic approximation and the second variation formula Symmetric ilinear forms and inner products (15.1) The associated self-adjoint operator. Let V e a finite dimensional real vector space and B : V ˆ V Ñ R a symmetric ilinear form. In (14.35) we proved that there is a asis e 1,..., e n of V in which B is diagonal, in the sense that Bpe i, e j q if i j. By scaling the asis elements we can arrange that the diagonal entries Bpe i, e i q are either, `1, or 1; see (14.37). Now suppose V also has an inner product x, y. Define a linear operator S B : V Ñ V y (15.) Bpξ 1, ξ q xξ 1, S B pξ qy, ξ 1, ξ P V. The symmetry of B implies that S B is self-adjoint in the sense that (15.3) xξ 1, S B pξ qy xs B pξ 1 q, ξ y, ξ 1, ξ P V. (15.4) Diagonalization. The operator S B is diagonalizale, as we now prove. Theorem Let V e a finite dimensional real inner product space and S : V Ñ V a self-adjoint operator. Then S has a (nonzero) eigenvector. Of course, not every linear operator has an eigenvector: a nontrivial rotation in the plane does not fix any line. The following proof is essentially a reprise of the second proof of Proposition 9.5. Proof. Consider the functions f, g : V Ñ R defined y (15.6) fpξq 1 xξ, Spξqy, gpξq 1 xξ, ξy. Let SpV q g 1 p1q Ă V e the unit sphere. Since SpV q is compact, f has a maximum on SpV q, say at e 1 P SpV q. The Lagrange multiplier criterion implies that there exists λ 1 P R such that df e1 λ 1 dg e1, in other words Spe 1 q λ 1 e 1. Corollary In the situation of Theorem 15.5 the operator S is diagonalizale. Proof. Let V 1 e the orthogonal complement to the eigenvector e 1. The self-adjointness (15.3) implies that if ξ P V 1, then Spξ q P V 1 and the restriction of S to V 1 is self-adjoint. Theorem 15.5 produces an eigenvector e of this restriction with eigenvalue λ ď λ 1. Repeat the argument a total of dim V times to produce an orthonormal asis e 1,..., e n of eigenvectors with eigenvalues λ 1 ě λ ě ě λ n. It follows immediately from (15.) that B is diagonalized as well: #, i j; (15.8) Bpe i, e j q λ i, i j, The inner product gives meaning to the diagonal entries; compare (14.37). Multivariale Analysis (M375T, Spring 19), Dan Freed, March 17, 19 1

2 D. S. Freed The second derivative test and quadratic approximation (15.9) Introduction. Resume our standard setup (5.3) with B R. Let p P U e a critical point of f, i.e., df p. Then we would like to say that the function (15.1) p ` ξ ÞÝÑ fpp q ` 1 d f p pξ, ξq is a good approximation to f near p. But this is not necessarily true. Example Take U A R and consider the five functions (15.1) f 1 pxq `x f pxq x f 3 pxq x 3 f 4 pxq `x 4 f 5 pxq x 4 Each function has a critical point at x. The quadratic approximation (15.1) works in fact is exact for f 1, f. For f 3, f 4, f 5 the quadratic approximation is a constant function and that does not predict the local ehavior: x is an inflection point of f 3, a local minimum of f 4, and a local maximum of f 5. (15.13) Second derivative test for local extrema. The quadratic approximation is guaranteed to e good if d f p is nondegenerate and if A is finite dimensional. We first prove a special case, when d f p is positive definite. The analogous theorem for d f p negative definite follows y applying the following to f. Theorem Suppose A is finite dimensional, f is a C 1 function, p P U is a critical point, f is twice differentiale at p, and d f p is positive definite. Then f has a strict local minimum at p. The strictness means that there exists a neighorhood U 1 Ă U of p such that fppq ą fpp q for all p P U 1 ztp u. Remark The following proof works for A infinite dimensional if the given norm on V is equivalent to the norm (15.16) defined y the second differential. In finite dimensions all norms are equivalent, and so we can and do use (15.16) as the norm in the definition of differentiaility. Proof. Since d f is positive definite, (15.16) }ξ} d f p pξ, ξq, ξ P V, is a norm on V. The twice differentiaility of f at p is the assertion: given ɛ ą there exists δ ą such that if ξ P V satisfies }ξ} ă δ, then p ` ξ P U and (15.17) ˇ df p`ξ df p d f p pξq ˇˇ ď ɛ}ξ}.

3 Multivariale Analysis (Lecture 15) 3 Fix ă ɛ ă 1 and choose δ ą so that (15.17) holds if }ξ} ă δ. Now fix ξ P B δ pq and set gptq fpp ` tξ q, t P r, 1s. Then g 1 ptq df p`tξ pξ q. Evaluate the linear functionals in (15.17) on ξ and use the fact that df p to conclude (15.18) p1 ɛqt}ξ } ď g 1 ptq ď p1 ` ɛqt}ξ }. This is an inequality of real-valued functions of t. Integrating we conclude that (15.19) 1 ɛ }ξ } ď gp1q gpq ď 1 ` ɛ }ξ }, which is (15.) fpp q ` 1 ɛ }ξ } ď fpp ` ξ q ď fpp q ` 1 ` ɛ }ξ }. In particular, fpp ` ξ q ě fpp q and fpp ` ξ q ą fpp q if ξ. Therefore, f has a strict local minimum at p. (15.1) More general quadratic approximations. The inequalities (15.) sandwich the function f etween two quadratic functions, an approximation valid in a neighorhood of the critical point p with positive definite second differential. More generally, suppose p is a critical point of f with nondegenerate second differential. Choose a decomposition V P N such that d f p is positive definite on P and negative definite on N. Define the norm (15.) }ξ 1 ` ξ } on V. d f p pξ 1, ξ 1 q ` d f p pξ, ξ q, ξ 1 ` ξ P P N, Theorem Suppose A is finite dimensional, f is a C 1 function, p P U is a critical point, f is twice differentiale at p, and d f p is nondegenerate. Then in a neighorhood U 1 Ă U of p we have (15.4) fpp q ` 1 ɛ }ξ1 } 1 ` ɛ }ξ } ď fpp ` ξ 1 ` ξ q ď fpp q ` 1 ` ɛ }ξ1 } 1 ɛ }ξ }, in terms of the norm (15.), for ξ 1 ` ξ P V P N. Proof. We only need a small modification of the proof of Theorem Namely, (15.17) implies that for ξ ξ 1 ` ξ of norm less than δ and ď t ď 1 we have (15.5) ˇ df p`tξ 1`tξ pξ1 ` ξ q t}ξ 1 } ` t}ξ } ˇˇ ď ɛt`}ξ 1 } ` }ξ }, and so writing ξ ξ 1 ` ξ we replace (15.18) with the inequalities (15.6) p1 ɛqt}ξ 1 } p1 ` ɛqt}ξ } ď g 1 ptq ď p1 ` ɛqt}ξ 1 } p1 ɛqt}ξ }. The inequalities (15.4) follow y integrating (15.6).

4 4 D. S. Freed Second variation formula (15.7) Recalling the setup. In Lecture 1 we computed the first variation formula, that is, the differential of the length function. The setup is that V is a finite dimensional real inner product space, E an affine space over V, and p, q P E fixed points. Parametrized paths from p to q form an affine space (15.8) A γ : r, 1s ÝÑ E such that γpq p, γp1q q, γ P C pr, 1s, Eq ( whose tangent space is the vector space (15.9) X ξ : r, 1s ÝÑ V such that ξpq ξp1q, ξ P C pr, 1s, V q ( with norm (15.3) }ξ} max spr,1s } 9 ξpsq} V, where the dot denotes d{ds. Set (15.31) U tγ P A such that 9γpsq for all s P r, 1su. and define the length function (15.3) f : U ÝÑ R γ ÞÝÑ ds } 9γpsq} In Theorem 1.8 we proved that f is differentiale with differential (15.33) df γ pξq ξfpγq ds x 9γ, 9 ξy x 9γ, 9γy 1{. Furthermore, γ is a critical point if it is a constant velocity motion, or a reparametrization of a constant velocity motion. (15.34) The second directional derivative. Assume γ is a unit speed, so x 9γ, 9γy 1 and :γ. We do not prove that f is twice differentiale at γ, ut content ourselves with computing the iterated second directional derivative. Fix ξ 1, ξ P X. Then commuting differentiation and integration, as

5 in (1.1) and with the same justification, we have ξ 1 ξ fpγq d ˇ ξ fpγ ` tξ 1 q dt ˇt Multivariale Analysis (Lecture 15) 5 (15.35) d ˇ dt ˇt x 9γ ` tξ ds 9 1, ξ 9 y x 9γ ` tξ 9 1, 9γ ` tξ 9 1 y 1{ " ds xξ 9 1, ξ 9 y 1 * x 9γ, 9γy 3{ x 9γ, ξ 9 1 yx 9γ, ξ 9 y! xξ 9 1, ξ 9 y x 9γ, ξ 9 1 yx 9γ, ξ 9 ) y. This is a symmetric ilinear form in the variales ξ 1, ξ, as it should e. It has an infinite dimensional kernel due to reparametrization invariance. Namely, if ρ 1 : r, 1s Ñ R is a C function with ρ 1 pq ρ 1 p1q, then for ξ 1 psq ρ 1 psq 9γ we have 9 ξ 1 9ρ 1 9γ and (15.35) vanishes for all ξ P X. We claim that (15.35) is positive semidefinite. Namely, in general we write ξ P X as (15.36) ξpsq ρpsq 9γ ` η, xηpsq, 9γy, for η : r, 1s Ñ V with ηpq ηp1q. Differentiating the constraint we find x 9η, 9γy. Setting ξ 1 ξ ρ 9γ ` η we compute (15.37) ξξf pγq which is nonnegative. Remark Theorem does not apply since the second differential is only semidefinite, not definite, and the domain is infinite dimensional. We can work modulo the kernel to otain a positive definite form, ut it is not equivalent to (15.3) (on the quotient), so we would need further argument to prove that spoiler alert! the shortest distance etween two points in Euclidean space is a straight line segment. } 9η}