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Crouzeix’s Conjecture and Perturbed Jordan Blocks
Anne Greenbaum Applied Math Dept., University of Washington
42nd Annual Iranian Mathematics Conference Vali-e-Asr University Rafsanjan, Iran September 5-8, 2011 Notation
• A an n × n matrix (bounded linear operator on a Hilbert space)
• p a polynomial (analytic function)
• kp(A)k≡ max kp(A)vk ≡ largest singular value of p(A). kvk2=1 2
• For S ⊂ C, kpkS ≡ supz∈S |p(z)|. • σ(A) ≡ spectrum of A.
• W (A) ≡ {q∗Aq : kqk =1}⊂ C is the field of values (numerical range) of A.
• r(A) ≡ maxz∈W (A) |z| is the numerical radius of A. Can kp(A)k be related to kpkS for some set S ⊂ C?
• If A is diagonalizable, A = V ΛV −1, then p(A)= Vp(Λ)V −1 and −1 kpkσ(A) ≤ kp(A)k ≤ kV k · kV k · kpkσ(A).
• If κ(V ) ≡ kV k · kV −1k is of moderate size, then σ(A) determines kp(A)k. What if κ(V ) is huge? Applications
• ketAk determines stability of y′(t)= Ay(t), t > 0; k(I + ∆tA)kk determines stability of Euler’s method.
• Even if κ(V ) is huge, eigenvalues still determine asymptotic behavior: tA limt→∞ ke k = 0 iff eigenvalues lie in the left half-plane. k limk→∞ kA k = 0 iff eigenvalues are less than 1 in magnitude. • However, if you know only σ(A), cannot distinguish: 6 10 15 10
5 10 10 10
4 10
5 10 3 10
0 2 10 10
1 ||
k −5 10 10 || A || exp(tA) || || exp(tA) 0 10 −10 10
−1 10 −15 10 −2 10
−20 −3 10 10
−4 −25 10 10 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 t k The Field of Values or Numerical Range
W (A)= {q∗Aq : q ∈ Cn, kqk =1}
• W (αI + βA)= α + βW (A), since q∗(αI + βA)q = α + βq∗Aq.
• For A finite dimensional, W (A) is closed (continuous image of compact unit ball in Cn); not necessarily so in infinite dimensions.
• W (A) (W (A)) contains σ(A): Av = λv, kvk =1=⇒ v∗Av = λ. • Toeplitz-Hausdorff theorem (1919). W (A) is convex. Method of proof: Reduce to 2 × 2 case. • If A is normal (A = V ΛV −1 where V −1 = V ∗), then W (A) is just the convex hull of the eigenvalues. But if A is highly nonnormal (κ(V ) is huge), then W (A) may contain much more:
7 Eigenvalues x 10 Field of Values 400 1
200 0.5
0 0
−200 −0.5
−400 −1 −1500 −1000 −500 0 −1 −0.5 0 0.5 1 7 x 10 Crouzeix’s Conjecture (2004)
For any square matrix A and any polynomial p:
kp(A)k≤ 2kpkW (A).
• 2 is best possible constant: 0 1 A = , W (A) = disk of radius 1 about 0, 0 0 2 kAk =1=2 max |z|. z∈W (A)
• Crouzeix proved 11.08. Cases Where Result is Known to Hold
• Obvious: Matrices whose eigenvectors have condition number at most 2, since A = V ΛV −1 =⇒ p(A)= Vp(Λ)V −1 so −1 kp(A)k ≤ kV k kV k max |p(z)|≤ κ(V )kpkW (A). z∈σ(A)
• 2 × 2 matrices, 3 × 3 nilpotent matrices (Crouzeix).
• Matrices whose field of values is a disk (Okubo and Ando / Badea).
• Perturbed Jordan blocks (G. and Choi). λ 1 ...... ... 1 ν λ • If p(z)= zk, k =1, 2,..., result follows from the power inequality (Berger and Pearcy, 1966): r(Ak) ≤ [r(A)]k. 1 Since 2kAk≤ r(A) ≤ kAk, k k k kA k≤ 2r(A ) ≤ 2[r(A)] =2kpkW (A).
iθ j If z = r(A)e ∈ W (A) then statement holds for p(z) = j cjz if i(ϕ−jθ) coeffs cj have args ϕ − jθ: cj = |cj|e . Then P
kp(A)k≤ |c | kAjk≤ 2 |c | [r(A)]j =2 c zj ≤ kpk . j j j W (A) Xj Xj Xj
Related Results
• Von Neumann’s inequality (1951): If A is a contraction (i.e. kAk≤ 1), then
kp(A)k ≤ kpkD, D≡ unit disk.
Proof: A has a unitary dilation, e.g., A (I − AA∗)1/2 −(I − A∗A)1/2 A∗ Also, A (I − AA∗)1/2 0 ... 0 0 0 I U = . . ... 0 0 I −(I − A∗A)1/2 A∗ 0 ... 0 is a unitary dilation of A and p(U) is a dilation of p(A). Therefore
kp(A)k ≤ kp(U)k = max |p(z)| ≤ kpkD. z∈σ(U) Corollary. kp(A)k ≤ kpkD(kAk,0), where D(kAk, 0) is the disk of radius kAk about the origin.
Proof: Define q so that q(A/kAk)= p(A) and apply von Neumann to q.
Corollary. If A˜ = A − αI, then
kp(A)k ≤ kpkD(kA˜k,α).
Proof: Define q so that q(A˜)= p(A), and apply previous Corollary to q. Related Results (Continued)
Definition. Will say that A and B are 2-similar if A is similar to B via a similarity transformation with condition number at most 2; that is, A = XBX−1 where κ(X) ≡ kXk kX−1k≤ 2.
• Okubo and Ando (1975): If r(A) ≤ 1, then A is 2-similar to a contraction.
• Corollary. kp(A)k≤ 2kpkD(r(A),0).
Proof: Define q so that q(A/r(A)) = p(A). By Okubo and Ando, kq(A/r(A))k ≤ 2kq(C)k. By von Neumann, kq(C)k ≤ kqkD = kpkD(r(A),0). • Corollary.
kp(A)k≤ 2kpksmallest disk containing W (A) Proof: Translate center of smallest enclosing disk to origin and apply previous Corol- lary.
• Corollary. If W (A) is a disk, then Crouzeix’s conjecture holds for A. A Stronger Conjecture
For any square matrix A (whose field of values has nonempty interior), there is a bijective holomorphic mapping g : W (A) → D such that g(A) is 2-similar to a contraction.
=⇒ Crouzeix’s conjecture: p(A) = (p ◦ g−1)(g(A)) = X(p ◦ g−1)(C)X−1, κ(X) ≤ 2, kCk≤ 1. Therefore kp(A)k≤ 2k(p ◦ g−1)(C)k and by von Neumann’s theorem −1 kp(A)k≤ 2k(p ◦ g )kD =2kpkW (A).
Note: Result would follow from Okubo and Ando if r(g(A)) ≤ 1, but usually this is not the case. Perturbed Jordan Blocks
λ 1 ...... J = ν ... 1 ν λ n×n • Eigenvalues are λ + ν1/n. • For 2−n/(n−1) ≤ |ν|≤ 2n/(n−1), eigenvectors have condition number less than or equal to 2. • Most interesting case is |ν| < 2−n/(n−1). • Assume wlog λ = 0, ν ≥ 0. Field of Values of Perturbed Jordan Block
0 1 ...... J = , ν ≥ 0. ν ... 1 ν 0 n×n • For ν = 0, W (Jν) is a disk about the origin of radius cos(π/(n + 1)).
• For ν = 1, W (Jν) is a regular n-gon (convex hull of the eigenvalues).
• For 0 < ν < 1, W (Jν) is like an n-gon with rounded corners. n = 3, nu = 0.3 n = 4, nu = 0.3
0.6
0.4 0.5
0.2
0 0
−0.2
−0.4 −0.5 −0.6
−0.5 0 0.5 1 −1 −0.5 0 0.5 1
n = 5, nu = 0.3
0.5
0
−0.5
−1 −0.5 0 0.5 1
• No simple formula for ∂W (Jν). ′ • No simple formula for g : W (Jν) →D, g(0) = 0, g (0) > 0.
• But, by reflection, each eigenvalue is multiplied by the same constant cν, so g(Jν)= cνJν. Theorem. Assume 0 < ν < 2−n/(n−1). Crouzeix’s conjecture (and 1/(n−1) the stronger conjecture) holds for Jν if and only if cν ≤ 2 .
1/(n−1) −1 n 2 n−1 Proof: If cν ≤ 2 , then g(Jν) = D Jcν νD, where D = diag(1,cν,cν,...,cν ) n has condition number at most 2. Since kJcν νk = 1, the result follows.
1/(n−1) Conversely, if cν > 2 , then (n−1) n−1 n−1 n−1 n−1 kg(Jν) k = cν kJν k = cν > 2=2 kg kW (Jν).
−1/(n−1) Theorem. If n ≥ 6, then W (Jν) contains a disk of radius 2 , 1/(n−1) so cν ≤ 2 . In fact, since W (Jν) contains a disk of radius cos(π/n), −(n−1) kp(Jν)k≤ cos(π/n) kpkW (Jν), −(n−1) and limn→∞ cos(π/n) = 1. • Conformal mapping from W (Jν) to D has the form n g(z)= cνz + (z − ν)zh(z), for some analytic function h. • Try a mapping of the form g˜(z)=21/(n−1)z + (zn − ν)zα,
and see if you can find α such that this maps W (Jν) to a region 1/(n−1) containing the unit disk. Then cν ≤ 2 . • For the remaining cases n = 3, 4, 5, using Mathematica for symbol manipulation, we were able to find such mappings. Final Theorem
Crouzeix’s conjecture holds for matrices of the form λ 1 ...... J = , ν ... 1 ν λ n×n for all λ, ν, and n. Moreover, if g is the bijective conformal mapping ′ from W (Jν) to D with g(λ) =0, g (λ) > 0, then g(Jν) is 2-similar to a contraction. Future Work
• Try to generalize to matrices A with the property that g(A) = cA. For example, 0 α1 ...... A = . ... α n−1 αn 0 In this case, g(A) is 2-similar to a contraction if and only if A is 2- similar to a matrix with norm at most 1/c. Do not know of any work on the smallest-norm matrix to which a given matrix is 2-similar.
• For numerical testing of Crouzeix’s conjecture, try the (stronger?) statement that g(A) is 2-similar to a contraction. For A = Jν, these two assertions are equivalent. Do not know if they are in general. • What about lower bounds? kp(A)k≥ mkpkS. Can’t do it if S contains more than σ(A) since if p is the minimal polynomial of A, then kp(A)k = 0 but kpkS > 0.
However, p(A) = fp(A) for infinitely many functions fp; let fp,S be the one with minimal infinity norm on S and look for m such that
kp(A)k≥ mkfp,SkS.
For perturbed Jordan blocks with |ν|≤ 1, have shown also that
kp(Jν)k≥ 1 · kfp,W (Jν)kW (Jν).