MET Workshop: Exercises

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1 MET Workshop: Exercises Alex Blumenthal and Anthony Quas May 7, 206 Notation. R d is endowed with the standard inner product (, ) and Euclidean norm. M d d (R) denotes the space of n n real matrices. When R d = E F is a splitting of R d into complementary subspaces, we let π E//F denote the projection onto E, parallel to F ; that is, the range of π E//F is E, π E//F π E//F = π E//F, and ker π E//F = F. For a subspace E R d, let π E denote the orthogonal projection onto E. For a subspace E R d, let E denote the orthogonal complement to E. For k d and a linear operator A M d d (R), let σ k (A) denote the k-th singular value of A. Here, w,, w k denotes the subspace spanned by the vectors w,, w k R d. Preinaries from linear algebra. The Singular Value Decomposition (SVD) Definition. Let A M d d (R). The singular values σ (A),, σ d (A) of A may be defined by σ i (A) = λ i (A A), where λ i (A A) denotes the i-th highest eigenvalue of A A. Note that A A is positive semidefinite, and so λ (A A) λ d (A A) 0. Problem 2. Let S M d d (R) be a symmetric matrix (i.e. A T = A, where T denotes the transpose). Show that there is an orthogonal basis {v i } i d of R d consisting of eigenvectors for S. Problem 3 (Proof of the SVD). Assume that A M d d (R) is invertible. (a) Show that λ k (A A) = λ k (AA ) for each i d. (b) Let {v i } i d be an orthonormal eigenbasis for A A as in Problem 2, ordered so that A Av i = λ i (A A)v i for each i d. Let {w i } i d be an analogous orthonormal eigenbasis for AA, ordered the same way. Prove that Av i = ±σ i (A)w i for each i d. Problem 4. Show that if the invertible matrix A has singular values σ >... > σ d, then A has singular values σ d >... > σ.

2 Problem 5. Let A, B M d d (R) and assume both A, B are invertible. Prove that there exists a basis {v,, v d } of R d for which {Av i } i d is orthogonal and {Bv i } i d is orthogonal. Problem 6. Let A M d d (R), and do not assume that A is invertible. (a) Prove that σ i (A) = min{ A F : F R d, codim F = i } for each i d. Recall that codim F = d dim F for a subspace F R d. (b) Prove that σ i (A) = max{m(a E ) : E R d, dim E = i}, where { } Av m(a E ) := min v : v E \ {0} = (A E ). Note that m(a E ) = 0 if A E is not injective. (c) Prove that for each k d. k σ i (A) = max{det(a W ) : W R d, dim W = k} i= Problem 7. Let A M d d. Let A have singular vectors v,..., v k. Prove that v k is a vector in v,..., v k that is maximally expanded by A, i.e.,.2 Angles Av k = A v,,v k. Problem 8. Let E R d be a proper subspace. For v R d, define the minimal distance dist(v, E) := min{ v e : e E}. Prove that when v 0. dist(v, E) = π E v v Definition 9. The angle (v, w) [0, π/2] between two vectors v, w R d is defined by cos (v, w) = (v, w) v w. For a subspace E R d and a nonzero vector v R d, we define the minimal angle (v, E) = min{ (v, e) : e E, e = }. Problem 0. Prove that where E R d is a subspace and v R d \ {0}. sin (v, E) = dist(v, E), 2

3 .3 Grassmanian on R d Definition. For k d, the Grassmanian Gr(d, k) is the set of all k-dimensional subspaces of R d. We endow Gr(d) := d k= Gr(d, k) with the following metric: for E, E 2 Gr(d) we define d H (E, E 2 ) := π E π E2. Problem 2. The H stands for Hausdorff: the metric d H is known as the Hausdorff distance, and is in broader generality a metric on the space of compact subsets of R d : for two compact subsets A, B R d, we define d Haus (A, B) = max{max a A dist(a, B), max dist(b, A)}. b B For E, E 2 Gr(d, k), define B Ei = {v E i : v }. Compare d H (E, E 2 ) to d Haus (B E, B E2 ). Problem 3. Prove that d H is a complete metric on Gr(d), and that each Gr(d, k), k d, is connected. Problem 4. Prove that Gr(d, 2) is sequentially compact. Problem 5. Let E, E Gr(d) be such that d H (E, E ) <. Show that E and E are complements. Problem 6 (Harder). Let R d = E F be a splitting of R d into complementary subspaces, and let π E//F denote the projection onto E parallel to F. Show that if E R d is a subspace sufficiently close to E in the d H metric, then then E, F are complementary in R d. Can you find an estimate in terms of π E//F? Is it optimal? Problem 7 (Harder - for those that like manifolds). For a fixed k-codimensional subspace, W, of R d, let U denote those elements of Gr(d, k) that have a trivial intersection with W. (a) Prove that U is an open subset of Gr(d, k). (b) Fix an element V 0 U, a basis e,..., e k for V 0 and a basis f,..., f d k for W. For an element V U, since V W = R d, each e i may be expressed uniquely as v i + w i with v i V and w i W. Form a matrix, Φ(V ) whose ith column is the coefficients of w i in the (f j ) basis. (c) Prove that Φ is a bijection from U to R (d k) k. (d) Prove that the collection of (U, Φ) (as W varies over Gr(d, d k), V 0 varies over U and the bases vary over bases for W and V 0 ) forms a smooth manifold structure on Gr(d, k)..4 Volumes and wedges Problem 8. Verify that: v v i v j v k = v v j v i v k ; and v (cv + c v ) v k = v v v k + c v v v k If e,..., e d is a basis for V, then {e i e ik : i <... < i k } spans k V. Problem 9. Let V be a k-dimensional subspace of R d. Let two bases for V be e,..., e k and f,..., f k. Prove that f f k = c e e k, where c is the determinant of the matrix of coefficients of the f vectors in terms of the e vectors. 3

4 Problem 20. Prove that there is a linear map k A from k V to itself such that k A(v v k ) = (Av ) (Av k ) for all v,..., v k V. Problem 2. Let u, v and w be three orthonormal vectors in R 3. Prove that u v, u w and v w are orthonormal. Problem 22. Let v,, v d R d be a basis. Write P = { d i= α iv i : α i [0, ], i d}. Show that d Leb(P) = v d dist(v i, v i+,, v d ). i= Problem 23. Let R d = E F be a splitting into complementary subspaces E, F, and let A M d d (R) be an invertible matrix. Using Problem 22, estimate det(a E ) det(a F ) in terms of det(a) and the quantities π E//F, π AE//AF. What happens if E, F are subspaces spanned by singular vectors for A? 2 Subadditivity and the Kingman Subadditive Ergodic Theorem 2. Subadditive sequences Problem 24. Let {a n } n be a subadditive sequence of reals, i.e., for any m, n, we have that a m+n a m + a n. Prove that n a n converges (perhaps to ), and prove that it the iting value coincides with inf n n a n. Problem 25. Let {a n } n, {b n } n be sequences of reals for which a m+n a m + a n + b n for each m, n. Formulate conditions on {b n } under which n n a n converges. 2.2 Kingman Subadditive Ergodic Theorem (KSET) For the ensuing exercises, let us assume the KSET in the following form. Theorem 26 (KSET). Let (Ω, F, µ) be a measure space and let T : Ω Ω be a µ-invariant measurable transformation. Let {f n : Ω R} be a subadditive sequence of measurable functions for the mpt (Ω, F, µ, T ), i.e., for each m, n and for µ-almost every x Ω, f m+n (x) f m (x) + f n (x). Assume that f + L (µ). Then, the its f (x) = n f n(x) [, ) exist for µ-almost every x Ω, and f fn (x) (x)dµ(x) = dµ(x) = inf n 4 fn (x) n dµ(x).

5 For the next two problems, assume the setting and conclusions of the KSET as posed above. Problem 27. Prove that f is µ-almost surely T -invariant. Problem 28. Let T be an invertible ergodic measure-preserving transformation. Let (f n ) be a sub-additive sequence of functions over T : f n+m (ω) f n (ω) + f m (T n ω). Let g n (ω) = f n (T n ω). Prove that (g n ) is sub-additive over σ : g n+m (ω) g n (ω) + g m (T n ω). Prove that n g n(ω) exists a.e. and has the same it as f a.e. What if T is not ergodic? 2.3 First passage percolation Let E denote the collection of edges in the Z 2 lattice. Let ν be a probability measure on (0, ) (with x dν(x) < ). Let Ω = (0, ) E be the collection of all weightings of E and equip X with the probability measure ν E (so that in a realization ω Ω, each edge is assigned a weight from the distribution ν independently of all other edges). Define a Z 2 action, τv on Ω that translates the pattern of edge weightings through v. Now for v Z 2, define F v (ω) to be the length of the shortest path from 0 to v (where the length of a path is the sum of the weights of the edges). Problem 29. In this problem, we develop the application of the Kingman subbadditive ergodic theorem to the problem of first passage percolation on Z 2. (a) For u, v Z 2, ω Ω, prove that F u+v (ω) F u (ω) + F v (τ u (ω)). In particular, if v is any non-zero integer vector, then defining σ = τ v and f n (ω) = F nv (ω), (f n ) is a sub-additive sequence for the ergodic dynamical system σ : Ω Ω. (b) Show that (f (ω)) + dν(ω) <. (c) Show that ν is an ergodic invariant measure for σ : Ω Ω. (d) Applying (b) and (c), verify that the Kingman subadditive ergodic theorem implies that the it f n (ω) n exists and converges ν-almost surely to the constant g(v) R { }, where g(v) := F nv (ω)dν(ω). n (e) Check that g(v) > 0 for any v Z 2 \ {0}. In the next problem, we prove some properties of the function g. Problem 30. Ω 5

6 (a) For k Z, v Z 2, show that g(kv) = k g(v). (b) Extend g to Q 2 by defining g(v) = n g(nv) when nv Z 2. Conclude from part (a) that this extension of g is well-defined, and that g(cv) = c g(v) for c Q, v Q 2. (c) Prove that g(v + w) g(v) + g(w). (d) Conclude that g is continuous on Q 2, hence admits a unique continuous continuation to all of R 2. Check that g is positive-valued on all of R Lyapunov exponents Problem 3. Check using Problem 4 and the Kingman sub-additive ergodic theorem that the Lyapunov exponents for the inverse cocycle B(ω) = A(σ ω) over σ are λ k,..., λ, with multiplicities m k,..., m. Problem 32. Check from the previous exercise and the Kingman sub-additive ergodic theorem that the Lyapunov exponents for the dual cocycle C (n) (ω) over σ, where C(ω) = A(σ ω) T over σ are the same as those for the cocycle A (n) ω over σ. 3 Guided proof of MET for a single 2 2 matrix. Problem 33 (A warmup). Why is the unstable manifold defined as W u (p) = {x: d(t n ( x, p) ) as n }? (that is: why are inverse powers of T used?) (Think about the map T (x) = x 0 2 for a concrete example). Problem 34. In this exercise, we will prove, in a roundabout way, the one-sided Multiplicative Ergodic Theorem for a matrix A M 2 2 (R). (a) Show that the its L k = n n log σ k(a n ) () exist for each k 2. Hint: use Problem 6, part (c) and Fekete s Lemma. (b) Assume that L > L 2. For each n, let F n denote the singular subspace corresponding to σ 2 (A n ), i.e., F n is the (unique) -dimensional subspace of R 2 for which A n Fn = σ 2 (A n ). Show that the sequence of subspaces {F n } is Cauchy by obtaining a bound on (Id π Fn+ ) Fn. (c) Let F denote the iting subspace as in (b). Show the following: For any v F \ {0}, for any v R 2 \ F, In particular, observe that F is invariant, i.e., AF F. n log An v = L 2, and n log An v = L. 6

7 (d) Identify the collection of its L, L 2 in terms of the eigenvalues of A. Identify F in terms of the eigenvectors of A. At this point, we have proved the following. Proposition 35 (One-sided MET for a single 2 2 matrix). Let A M 2 2 (R), and let L, L 2 be as in Problem 34. If L > L 2, then there exists a subspace F R 2 for which (i) AF F, (ii) for any v F \ {0}, we have n log An v = L 2, and for any v R 2 \ F, we have n log An v = L. Problem 36. Let A M 2 2 (R) (i.e., A is a two-by-two matrix with real entries), and assume that A has two distinct eigenvalues λ, λ 2 for which λ > λ 2 > 0. (a) Determine n log An v for each v R 2. Does your result change if is replaced with any norm on R n? (b) Let E, E 2 denote the eigenspaces corresponding to λ, λ 2, respectively, and assume that v R 2 \ E 2. Show that (An v, E ) = 0. (c) Determine the value of n log (An v, E ). Problem 37. In this exercise, we prove, again in a roundabout way, the two-sided MET for a matrix A M 2 2 (R). Let us assume that L > L 2, where L, L 2 are as in Problem 34. For now, let us assume that A is invertible. (a) Let ˆL, ˆL 2 denote the its as in () with A replacing A, and show that ˆL = L 2 and ˆL 2 = L. (b) Apply Problem 34, item (b) with A replacing A, and let E denote the iting subspace. Show that For any v E \ {0}, for any v R 2 \ E, In particular, observe that E is invariant, i.e., AE = E. (c) Identify the subspace E in terms of the eigenvectors of A. n log A n v = L, and n log A n v = L 2. Problem 38. In this exercise, we prove in an alternative way the two-sided MET for a matrix A M 2 2 (R), this time without explicitly using the fact that A is invertible. (a) Let L, L 2 denote the (possibly ) its as in () with A T, the transpose of A, replacing A. Show that L = L, L 2 = L 2. (b) Apply Problem 34, item (b) with A T replacing A, and let F denote the iting subspace. Note that automatically, F is invariant under A. Show that F complements F (as in Problem 34). 7

8 (c) In the case when A is invertible, show that E = F with E as in Problem 37. Problem 39. In this exercise, we show yet another way of obtaining the two-sided MET for a matrix A M 2 2 (R), again not explicitly using the fact that A is invertible. (a) Let L > L 2 be as in Problem 34, and for each n let E n denote the one-dimensional subspace corresponding to σ (A n ) i.e., E n is the unique one-dimenisonal subspace with the property that m(a n En ) = min{ A n e : e E n, e = } = σ (A n ). Letting E n = A n E n, show that {E n} is a Cauchy sequence. (b) Let Ē denote the iting subspace from part (a). Show that Ē, F are complements, and that AĒ = Ē. (c) Show that Ē coincides with the subspace E as in Problem 37 when A is invertible. 4 The projective Markov chain and Lyapunov exponents Throughout these exercises we refer to the following construction. Definition 40. Let T be an almost-surely invertible random variable on M 2 2 (R) and consider the IID matrix product {T n } distributed like T. Let {ψ n } n denote the Markov chain on the projective space P = [0, π) of R 2 for which u ψn+ = T n u ψn ; Here, we write u ψ = (cos ψ, sin ψ), and for v R 2 \ {0} we write v P for the equivalence class of v. The Markov chain {ψ n } n is referred to as the projective Markov chain for the matrix product {T n }. The transition probabilities P (ψ, ) for ψ P = [0, π) for the projective Markov chain are given by P (ψ, A) = P(T u ψ A). for Borel A P. Recall that a probability measure ν on P is stationary when ν(a) = dν(ψ)p (ψ, A) P for any Borel A P. Problem 4. Let {ψ n } be the projective Markov chain for an IID matrix product {T n }. Prove the following Lemma. Lemma 42. (a) Let ν be any stationary measure for the Markov chain {ψ n } on P, which we regard as a measure on [0, π). Then, λ log T u ψ dν(ψ)dp(t ). (b) If ν is absolutely continuous with respect to Lebesgue measure on P, then λ = log T u ψ dν(ψ)dp(t ). Hint: for (a), use the Birkhoff ergodic theorem. For (b), use the Multiplicative Ergodic Theorem. 8

9 5 Computing LE of simple matrix cocycles Below, for L > 0, we set and for θ [0, 2π), H L = ( L 0 0 L ) ( cos θ sin θ R θ = sin θ cos θ ) Problem 43. Define the random matrix T by setting T = R π/2 with probability p (0, ], and T = H 2 with probability p. Let {T n } be an IID sequence with the same law as T. Show that the Lyapunov exponents of this random matrix product are both zero. Problem 44. Define the random matrix T by setting T = R 2π/3 with probability p (0, ], and T = H 4 with probability p. Let {T n } be an IID sequence with the same law as T. Show that the top Lyapunov exponent of this random matrix product is positive when p is sufficiently small. Hint: look for an invariant subset of projective space under the actions of both R 2π/3 and H 4. Problem 45. Define the random matrix T by setting T = R θ with probability p (0, ], where θ is distributed uniformly in [0, 2π), and T = H 2 with probability p. Let {T n } be an IID sequence with the same law as T. Show that for p sufficiently small, the top Lyapunov exponent of this random matrix product is positive. Hint: consider the orientation of the vector T n (, 0) after each time a rotation is sampled. Problem 46. Consider the random matrix product of the form T = R θ H L, where L > 0 is a large fixed constant and θ is distributed according to an absolutely continuous law on [0, 2π) with a bounded density ρ. (a) Let ν be any stationary measure for the associated projective Markov chain. Show that ν is absolutely continuous, and bound the density of ν in terms of the bound on the density of ρ. (b) Apply Lemma 42 to obtain a lower bound for λ in terms of L. Conclude that λ > 0 for L sufficiently large. Hint: P = { ψ π/2 > ɛ} { ψ π/2 ɛ}. Problem 47. Let f : S S denote rotation by an irrational angle α/2π; parametrizing S = [0, ), this means that f(x) = x + α(mod ). Define the cocycle A : S M 2 2 (R) by A x = R 2πx for x [0, ). Does there exist an equivariant subspace of R 2 for this cocycle? 9

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