Uniform individual asymptotics for the eigenvalues and eigenvectors of large Toeplitz matrices

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1 Uniform individual asymptotics for the eigenvalues and eigenvectors of large Toeplitz matrices Sergei Grudsky CINVESTAV, Mexico City, Mexico The International Workshop WIENER-HOPF METHOD, TOEPLITZ OPERATORS, AND THEIR APPLICATIONS Boca del Rio, Veracruz, Mexico November 03-07, 2015 Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

2 Abstract The asymptotic behavior of the spectrum of large Toeplitz matrices has been studied for almost one century now. Among this huge work, we can find the Szegö theorems on the eigenvalue distribution and the asymptotics for the determinants, as well as other theorems about the individual asymptotics for the smallest and largest eigenvalues. Results about uniform individual asymptotics for all the eigenvalues and eigenvectors appeared only five years ago. The goal of the present lecture is to review this area, to talk about the obtained results, and to discuss some open problems. This review is based on joint works with Manuel Bogoya, Albrecht Böttcher, and Egor Maximenko. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

3 Main object. Spectral properties of larger finite Toeplitz matrices a 0 a 1 a 2... a (n 1) a 1 a 0 a 1... a (n 2) A n = (a j k ) n 1 j,k=0 = a 2 a 1 a 0... a (n 3) a n 1 a n 2 a n 3... a 0 a(t) = j= a j t j, t T -symbol of {A n } n=1 Eigenvalues, eigenvectors singular values, condition numbers, invertibility and norms of inverses, e.t.c. n 1000 is a business of numerical linear algebra. Statistical physics - n = is a business of asymptotic theory. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

4 I. Two parameters: n- dimensions of matrices; j- number of eigenvalue Asymptotics by n uniformly in j. 1 j n II. Distance between λ j and λ j+1 is small: ( ) 1 λ j λ j+1 = O normal case n ( ) 1 λ j λ j+1 = O n γ special case λ j = λ j+1 exceptional case Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

5 Publications about asymptotics of individual eigenvalues and eigenvectors. 1. Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maksimenko. Inside the Eigenvalues of Certian of Hermitian Toeplitz Band Matrices. Computational and Applied Mathematics, 233 (2010), pp. 2. Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maksimenko. On the Structure of the Eigenvectors of Large Hermitian Toeplitz Band Matrices. Operator Theory: Advances and Applications, 210 (2010), pp. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

6 3. Deift P, Its A, and Krasovsky I. Eigenvalues of Toeplitz matrices in the bulk of the spectrum. Bulletin of the Institute of Mathematics Academia Sinica (New Series) 7 (2012), pp. 4. J. M. Bogoya, Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maksimenko. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. Journal of Mathematical Analysis and Applications Volume 422, Issue 2, 15 February 2015, pp. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

7 5. H. Dai, Z. Geary, L.P. Kadanoff. Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. Journal of Statistical Mechanics: Theory and Experiment, May, 2009, PO J. M. Bogoya, Albrecht Böttcher and Sergei M. Grudsky. Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices. Operator Theory: Advances and Applications, 220 (2012), pp. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

8 7. J. M. Bogoya, Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maksimenko. Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai Geary and Kadanoff. Linea Algebra and its Applications, 436 (2012), pp. 8. Albrecht Böttcher, Sergei Grudsky and Arieh Iserles. Spectral theory of large Wiener-Hopf operators with complex-symmetric kernels and rational symbols. Mathematical Proceedings of Cambridge Philosophical Society, 151 (2011), pp. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

9 Main results-simple loop case For α 0, we denote by W α the weighted Wiener algebra of all functions a : T C whose Fourier coefficients satisfy a α := j= a j ( j + 1) α <. Let m be the entire part of α. It is readily seen that if a W α then the function g defined by g(σ):= a(e iσ ) is a 2π-periodic C m function on R. In what follows we consider real-valued simple-loop functions in W α. To be more precise, for α 2, we let SL α denote the set of all a W α such that g has the following properties: the range of g is a segment [0, M] with M > 0, g(0) = g(2π) = 0, g (0) = g (2π) > 0, and there is a ϕ 0 (0, 2π) such that g(ϕ 0 ) = M, g (σ) > 0 for σ (0, ϕ 0 ), g (σ) < 0 for σ (ϕ 0, 2π), and g (ϕ 0 ) < 0. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

10 Let a SL α. Then for each λ [0, M], there are exactly one ϕ 1 (λ) [0, ϕ 0 ] such that g(ϕ 1 (λ)) = λ and exactly one ϕ 2 (λ) [ϕ 0, 2π] satisfying g(ϕ 2 (λ)) = λ. For each λ [0, M], the function g takes values less than or equal to λ on the segments [0, ϕ 1 (λ)] and [ϕ 2 (λ), 2π]. Denote by ϕ(λ) the arithmetic mean of the lengths of these two segments, ϕ(λ):= 1 2 (ϕ 1(λ) ϕ 2 (λ)) + π = 1 2 µ{ σ [0, 2π] : g(σ) λ }, where µ is the Lebesgue measure on [0, 2π]. The function ϕ : [0, M] [0, π] is continuous and bijective. We let ψ : [0, π] [0, M] stand for the inverse function. Put σ 1 (s) = ϕ 1 (ψ(s)) and σ 2 (s) = ϕ 2 (ψ(s)). Then g(σ 1 (s)) = g(σ 2 (s)) = ψ(s). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

11 Let further (g(σ) ψ(s))e is β(σ, s):= (e iσ e iσ1(s) )(e iσ e iσ2(s) ) ψ(s) g(σ) = 4 sin σ σ 1(s) 2 sin σ σ. 2(s) 2 We will show that β is a continuous and positive function on [0, 2π] [0, π]. We define the function η : [0, π] R by η(s):= θ(ψ(s)) = 1 2π 4π 0 log β(σ, s) tan σ σ dσ 1 2π 2(s) 4π 0 2 the integrals taken in the principal-value sense. log β(σ, s) tan σ σ dσ, 1(s) 2 Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

12 Theorem Let a SL α with α 2 and let λ (n) 1... λ (n) n be the eigenvalues of T n (a). If n is sufficiently large, then (i) the eigenvalues of T n (a) are all distinct, i.e., λ (n) 1 < λ (n) 2 <... < λ (n) n, (ii) the numbers s (n) j := ψ(λ (n) j ) (j = 1,..., n) satisfy (n + 1)s (n) j + η(s (n) j ) = πj + (n) 1 (j) with (n) 1 (j) = o(1/nα 2 ) as n, uniformly with respect to j, (iii) this equation has exactly one solution s (n) j [0, π] for each j = 1,..., n. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

13 To write down the individual asymptotics of the eigenvalues, we introduce the parameter d:= πj n + 1. Note that the dependence of d on j and n is suppressed. Theorem Let a SL α with α 2 and let s (n) j be as in previous Theorem. Then [α] 1 s (n) p k (d) j = d + (n + 1) k + (n) 2 (j) k=1 where (n) 2 (j) = o ( 1/n α 1) as n uniformly in j, p 1 (d) = η(d), p 2 = η(d)η (d). [α] is integer part of α. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

14 That is for 2 α < 3 we have s (n) j For 3 α < 4 we have e.t.c. s (n) j = d η(a) n o(1/nα 1 ), d = πj n + 1. = d η(a) n η(d)η (d) (n + 1) 2 + o(1/nα 1 ). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

15 Theorem Let α 2 and a SL α. Then [α] 1 λ (n) j = ψ(d) + k=1 c k (d) (n + 1) k + (n) 3 (j) (1) where (n) 3 (j) = o ( d(π d)/n α 1) as n, uniformly in j = 1, 2,..., n, and c 1 (d) = ψ (d)η(d), c 2 (d) = ψ (d)η 2 (d)/2 + ψ (d)η(d)η (d). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

16 Here is the result for the extreme eigenvalues. Corollary Let a SL α with some α 3. (i) If j/(n + 1) 0 then λ (n) j = c 5j 2 (n + 1) 2 + c 6j 2 (n + 1) 3 + (n) 5 (j), where c 5 = π 2 g (0)/2, c 6 = π 2 g (0)η (0), and (n) 5 (j) = o(j/n3 ) as n. (ii) If j/(n + 1) 1 then λ (n) j = M + c 7(n + 1 j) 2 (n + 1) 2 + c 8(n + 1 j) 2 (n + 1) 3 + (n) 6 (j), where c 7 = π 2 g (ϕ 0 )/2, c 8 = π 2 g (ϕ 0 )η (π), and (n) 6 (j) = o(n + 1 j/n3 ) as n. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

17 This theorem is close to a result by Widom 1958, who considered the case where g is an even function and j is fixed. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

18 Local nature of the asymptotics Symmetric symbol: ψ(ϕ) = g(ϕ) = (a(e iϕ )) 1. Normal case: g (ϕ) 0 (ε < πj n+1 < π ε). Inner eigenvalues ) λ (n) j = g ( ( ) πj c πj 1 n+1 + n + 1 n + 1 Distance between next eigenvalues is ( ) 1 O n + O ( n 2). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

19 2. Exceptional case: λ (n) j ( ) πj n+1 ε. Extreme eigenvalue = g ( ) ( ) πj 1 + O n + 1 n 3, Distance between next eigenvalues is ( ) 1 O n 2 πj n + 1 ε. Asymptotics: is define by behavior of function g(ϕ) in neighborhood of point ϕ 0, where eigenvalues are located. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

20 More general restriction 1. Normal case: g(ϕ) g(ϕ 0 ) = (ϕ ϕ 0 ) g(ϕ), g(ϕ) W (= W 0 ). That is g(ϕ) W 1 2. Exceptional case: g(ϕ) g(ϕ 0 ) = (ϕ ϕ 0 ) 2 g(ϕ), g(ϕ) W (= W 0 ). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

21 Main ideas of the Proof Lemma Let a SL α, α 2 and n 1. A number λ = ψ(s) is an eigenvalue of T n (a) if and only if e i(n+1)σ 2(s) Θ n+2 (e iσ 1(s), s) Θ n+2 (e iσ 2(s), s) e i(n+1)σ 1(s) Θ n+2 (e iσ 2(s), s) Θ n+2 (e iσ 1(s), s) = 0, where, for every k 1, the functions Θ k and Θ k are defined by Θ k (t, s):= [T 1 k (b(, s))χ 0 ](t), Θk (t, s):= [T 1 k ( b(, s))χ 0 ](t 1 ), and b(t, s):= b(1/t, s), χ l (t) = t l, χ 0 (t) = 1. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

22 Proof. We are searching for all values of λ belonging to [0, M] such that the equation T n (a)x = λx has non-zero solutions X in L (n) 2. Using the change of variable λ = ψ(s) we can rewrite the latter equation as Equation (2) is equivalent to T n (a ψ(s))x = 0. (2) P n b(, s)p(, s)x = 0, (3) where p(t, s):= e is (t e iσ 1(s) )(t 1 e iσ 2(s) ). Multiply (3) by the function χ 1 to get (P n+1 P 1 )b(, s)χ 1 p(, s)x = 0. (4) Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

23 Here P n+1 P 1 is just one way to write the orthogonal projection of the space L 2 (T) onto the span of χ 1,..., χ n. Note that χ 1 p(, s)x L (n+2) 2 and put Y := T n+2 (a ψ(s))χ 1 X = P n+2 b(, s)χ 1 p(, s)x = T n+2 (b(, s))χ 1 p(, s)x. Then (4) can be written as (P n+1 P 1 )Y = 0. This means that Y has the form Y = y 0 χ 0 + y n+1 χ n+1. Since T n+2 (b(, s)) is invertible, it follows that T 1 n+2 (b(, s))y = χ 1p(, s)x, that is, y 0 [T 1 n+2 (b(, s))χ 0](t) + y n+1 [T 1 n+2 (b(, s))χ n+1](t) = tp(t, s)x (t). (5) Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

24 Now recall notation (5). Taking into account the identity it is easy to verify that Therefore (5) can be written as W n+2 T n+2 (b)w n+2 = T n+2 ( b), [T 1 n+2 (b(, s))χ n+1](t) = t n+1 Θn+2 (t, s). y 0 Θ n+2 (t, s) + y n+1 t n+1 Θn+2 (t, s) = tp(t, s)x (t). (6) Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

25 Thanks to the factor p(t, s), the right-hand side vanishes at both t = e iσ 1(s) and t = e iσ 2(s). Consequently, y 0 and y n+1 must satisfy the homogeneous system of linear algebraic equations given by Θ n+2 (e iσ 1(s), s)y 0 + e i(n+1)σ 1(s) Θn+2 (e iσ 1(s), s)y n+1 = 0, Θ n+2 (e iσ 2(s), s)y 0 + e i(n+1)σ 2(s) Θn+2 (e iσ 2(s), s)y n+1 = 0. (7) If y 0 = y n+1 = 0, then, by (6), the function X is zero. Therefore the initial equation (2) has a non-trivial solution X if and only if the determinant of system (7) is zero. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

26 Recall that b ± (, s) are the Wiener-Hopf factors of b(, s): b + (t, s) = b(t, s) = b + (t, s) b (t, s) u j (s)t j and b (t, s) = j=0 T 1 (b(, s)) = b 1 + v j (s)t j j=0 (, s)pb 1(, s), [T 1 (b(, s))χ 0 ](t) = [b+ 1 (, s)pb 1 (, s)x 0](t) = b 1 (t, s). + Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

27 L n ( ( diag λ (n) j T n (a) x n = f n ) n j=1 λ (n) λ (n) λ (n) λ (n) n Singular Value Decomposition. ) L n X n = f n, Y 1 Y 2 Y 3. Y n Ȳ = L n X, F = L n f. L = L 1 n = F 1 F 2 F 3. F n Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

28 Example 1. Consider the non-rational symbol a(e iσ ) = g(σ) = g 2 σ 2 +g 3 σ 3 +g 4 σ 4+β +g 5 σ 5 +g 6 σ 6 +g 7 σ 7, where β [0, 1) and the coefficients g 2,..., g 7 are chosen in such a manner that σ [0, 2π], g(2π) = g (2π) = 0 and g (k) (2π) = g (k) (0) for k = 2, 3, 4. Elementary computations yield g 2 = (24 38β + 13β 2 + 2β 3 β 4 )/(2π) 2, g 3 = (24 50β + 35β 2 10β 3 + β 4 )/(2π) 3, g 4 = 240/(2π) 4+β, g 5 = ( β 201β β 3 + 3β 4 )/(2π) 5, g 6 = ( β + 209β 2 54β 3 5β 4 )/(2π) 6, g 7 = (48 20β 50β β 3 + 2β 4 )/(2π) 7. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

29 π 2π 0 π/4 π/2 3π/4 π Figure: Graph of g(σ) = a(e iσ ) (left), and η(s) (right) for β = 1/5. 0 π/2 π π/2 π Figure: The functions c 1 (d) (left) and c 2 (d) (right). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

30 n ε (n,1) (n + 1)ε (n,1) ε (n,2) (n + 1) 2 ε (n,2) ε (n,3) (n + 1) 3 ε (n,3) ˆε (n) (n + 1) 4.2 ˆε (n) Table: Maximum errors and normalized maximum errors for the eigenvalues of T n (a) obtained with our formula (1), ε (n,p) with p = 1, 2, 3, and by fixed-point iterations, ˆε (n), for different values of n. The data were obtained by comparison with the solutions given by Wolfram Mathematica. Note that Table 1 shows that ˆε (n) = O(1/(n + 1) 4.2 ) as n. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

31 Eigenvectors For γ R, we define z γ k as eiσ k(s (n) j )γ. Given a function f : T C, let f p be its pth Fourier coefficient, and for a vector X, let X p stand for its pth component. Let θ = (θ p ) n+1 p=0 be the vector in the first column of the matrix T 1 n+2 (b(, s(n) j )). For t C, we put θ(t) = θ 0 + θ 1 t + + θ n+1 t n+1. The following theorem describes the components of the eigenvectors of T n (a). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

32 Theorem Let a SL α. The vector X (n,j) = M (n,j) + L (n,j) + R (n,j) (8) whose p-th component, p = 0, 1,..., n 1, is given by := z n 1 2 p 1 θ(z 1 ) + ( 1) n j z n 1 2 p 2 θ(z 2 ), p := z n θ(z 1 ) 2πi θ(z 1 ) M (n,j) p L (n,j) R p (n,j) := L (n,j) n p 1, T ( θ(t) θ(z1 ) t z 1 θ(t) θ(z 2) t z 2 ) dt t p+1, is an eigenvector of T n (a) corresponding to the eigenvalue λ (n) j. Moreover, M (n,j) is conjugate symmetric, i.e., M (n,j) p = M (n,j) n p 1. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

33 Theorem Let a SL α. For k = 1, 2, let ẑ k := e iσ k(ŝ (n) j ), and, for p = 0, 1,..., n 1, put ˆM (n,j) p ˆL (n,j) := ẑ n 1 2 p n b + (ẑ 1 ) + ẑ p ( 1)n j 2 b + (ẑ 2 ), p := ẑ n b + (ẑ 1 ) 2πi b + (ẑ 1 ) ˆR p (n,j) :=ˆL (n,j) n 1 p. T ( b 1 + (t) b 1 + (ẑ 1) t ẑ 1 b 1 + (t) b 1 t ẑ 2 + (ẑ ) 2) Then there is a vector Ω (n,j) 1 such that [Ω (n,j) 1 ] p = o(1/n α 3 ) as n, uniformly in j and p, and such that dt t p+1, X (n,j) = ˆM (n,j) + ˆL (n,j) + ˆR (n,j) + Ω (n,j) 1 (9) is an eigenvector of T n (a) corresponding to the eigenvalue λ (n) j. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

34 Symbols with Fisher Harturg singularity. a α,β (t) = (1 t) α ( t) γ, 0 < α < β < 1. Conjecture of H.Dai, Z.Geary and L.P.Kadanoff, 2009 ( where ω j = exp λ (n) j a α,β (ω j exp i 2πj n ). { (2α + 1) log n }), Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

35 Complex value case where 1. f (t) H C. a(t) = t 1 (1 t) α f (t), α R + \N 2. f can be analytically extended to a neighborhood of T\{1}. 3. The range of the symbol a R(a) is a closed Jordan curve without loops and winding number -1 around each interior point. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

36 Figure: The map a(t) over the unit circle. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

37 Lemma Let a(t) = t 1 h(t) be a symbol that satisfies the following conditions: 1. h H. 2. R(a) is a closed Jordan curve in C without loops. 3. wind λ (a) = 1, for each λ in the interior of sp T (a). Then, for each λ in the interior of sp T (a), we have the equality [ ] D n (a λ) = ( 1) n ho n+1 1 h(t) λt for every n N., n Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

38 Theorem We have the following asymptotic expression for λ j : ( ) λ j = a(ω j ) + (α + 1)ω j a (ω j ) log(n) + ω ja (ω j ) a 2 (ω j ) log n n c o a (ω j )ωj 2 ( ) log(n) 2 + O, n, n ( ) where ω j = exp. i 2πj n Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

39 n = 4096 n= Figure: The solid blue line is the range of a. The black dots are sp T n (a) calculated by Matlab. The red crosses and the green stars are the approximations, for 1 and 2 terms respectively. Here we took α = 3/4. Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

40 New problems 1. Symbols of the kind: a(t) = a 1 t + a 0 + a 1 t + a 2 t 2 a 0 (t) = 1 t + t2 2. Symbols of the kind: a(t) = a 2 t 2 + a 1 t + a 0 + a 1 t + a 2 t 2 a 0 (t) = ( 1 t 2 + t ) 2 Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41

41 3. No simpleloop case. 4. Fisher-Harturg general case a(t) = (t t 0 ) α t β, α, β R (α, β C). Sergei Grudsky (CINVESTAV) Simple-loop Veracruz, November, / 41