Numerical Ranges of Weighted Shift Matrices

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1 Numerical Ranges of Weighted Shift Matrices Hwa-Long Gau {º( ³. Department of Mathematics, National Central University, Chung-Li 32, Taiwan (jointly with Ming-Cheng Tsaia, Han-Chun Wang) July 18, 211 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 1/17

2 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

3 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

4 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

5 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

6 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

7 Numerical range A M n Definition (numerical range of A) W (A) = { Ax, x : x C n, x = 1} Note : (1) W (A) C is always compact and convex. (2) W (U AU) = W (A), where U is unitary. (3) σ(a) W (A) (4) N M n is normal W (N) = conv (σ(n)) a 1 (5) A = [ a b c ] W(A) = b c a W(N) = (6) A = A 1 A k W (A) = conv (W (A 1) W (A k )) a 2 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 2/17

8 Weighted Shift Matrices A = a an 1 a n A is called a weighted shift matrix with weights a 1,..., a n. Note : 1 1 a 2. (1) A =..... an 1 1 a 1 a 1 a 1. (2).... =.... an 1... an 1 where a n a n e iθ θ = n j=1 arg(a j) Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 3/17

9 n-symmetry of W (A) Hence, we have A = ω na, ω n = e 2πi/n. That is, W (A) = ω nw (A) W (A).5 W (A) n = 6 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 4/17

10 Special Cases If a 1 = a 2 = = a n r. Then In this case, A is normal. 1 A = re iθ W (A) n = 7 Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 5/17

11 Special Cases If a j =, for some j. Then In this case, a j+1. A =..... aj 1. W (A) = {z C : z r} where r is the maximal root of the equation n 2 S l ( a j+1 2,..., a j 1 2, )( 1 4 )l z n 2l =. l= 1. In particular, J n = W (Jn) = {z C : z cos π n+1 }. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 6/17

12 (n 1) (n 1) principal submatrices a 2. A[1] =..... an 1 a j+1,..., A[j] = aj 2,... Now, if W (A) has a line segment on the line x = r, r >. Then r is the maximal eigenvalue of Re A with multiplicity at least two. By the interlacing property, r is also the maximal eigenvalue of Re A[j] for all j. That is, W (A[j]) = {z C : z r} for all j W (A[j]) Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 7/17

13 Results of Tsai and Wu, 211 Theorem (Tsai & Wu, 211) a 1. A =..... an 1, a j for all j. a n Then W (A) has a line segment if and only if W (A[1]) = = W (A[n]) W (A[j]) Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 8/17

14 Results of Tsai and Wu, 211 Lemma 1 If w(a[j 1]) = w(a[j]) = w(a[j + 1]) r, then r is either the largest or the second largest eigenvalue of Re (e iθ A) for some θ. Proof. May assume j = n, a n < and a j > for j = 1,..., n 1. Compute the determinant of (ri n Re A) and obtain det(ri n Re A) =. Lemma 2 If w(a[1]) = = w(a[n]) r, then r is a multiple eigenvalue of Re (e iθ A) for some θ. Proof. Let p(z) = det(zin Re A), then p (r) = n n=1 det(ri n 1 Re A[j]) =. Hence r is a multiple eigenvalue of Re A. Lemma 3 If the maximal eigenvalue r of Re A is multiple, then W (A) has a line segment on the line x = r Proof. Take a vector x ker(ri n Re A) such that Im Ax, x. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 9/17

15 Refinements Lemma 1 If w(a[j]) = w(a[k]) = w(a[l]) r for some 1 j < k < l n, then r is either the largest or the second largest eigenvalue of Re (e iθ A) for some θ. Proof. May assume a n < and a j > for j = 1,..., n 1. Compute the determinant of (ri n Re A) and obtain det(ri n Re A) =. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 1/17

16 Refinements Lemma 2 If w(a[j]) = w(a[k]) = w(a[l]) r for some 1 j < k < l n, then r is a multiple eigenvalue of Re (e iθ A) for some θ. Proof. Assume j = 1, l = n and Re A has eigenvalues λ 1 λ n. We want to show that r = λ 1 = λ 2. If r = λ 1. Take x, y [ C n 1 ] such that (Re [ A[1])x ] = rx and (Re A[n])y = ry. y Let x = C n and y = C n. x Then x, y ker(λ 1I n Re A). Since every entry of x and y is nonzero, hence dim ker(λ 1I n Re A) 2. Hence r is a multiple eigenvalue of Re A. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 11/17

17 Refinements Next, if r = λ 2 λ 1. Let u and v are unit eigenvectors of Re A w.r.t. λ 1 and λ 2, resp., N = [ span ] {u, v}. We have x x (( C n 1 {} ) N ) [ ] and y (( {} C n 1) N ). y Say x = au + bv and y = cu + dv with a 2 + b 2 = 1 = c 2 + d 2. Then r = w(a[n]) (Re (A[n])) x, x = (Re A) x, x = λ 1 a 2 +λ 2 b 2 r a 2 +r b 2 = r, the inequality here are actually equalities. λ 1 = λ 2 = r. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 12/17

18 Refinements Theorem A = a an 1 a n Then the following are equivalent: 1 W (A) has a line segment; 2 W (A[1]) = = W (A[n]);, a j for all j. 3 W (A[j]) = W (A[k]) = W (A[l]) for some 1 j < k < l n W (A[j]) Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 13/17

19 Results of Tsai, 211 Theorem 1 (Tsai, 211) If A is an n n weighted shift matrix with nonzero periodic weights a 1,..., a k, a 1,..., a k,..., a 1,..., a k, n = km. Then W (A) has a line segment. Moreover, C C... ω nc W (A) = W ( where C = a 1... C C ak 1 a k ) = W (.... ωn m 1 C ), Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 14/17

20 Questions Theorem 2 (Tsai, 211) W (A) contains a noncircular elliptic arc if and only if the aj s are nonzero, n is even, a 1 = a 3 = = a n 1, a 2 = a4 = = a n and a 1 = a 2. Question 1 If A is an n n weighted shift matrix and W (A) has a line segment. Are the weights periodic? Ans. No! We can find a 5 5 weighted shift matrix A such that W (A) has a line segment. Question 2 Let A and B are n n weighted shift matrices. If the weights of A are periodic and W (A) = W (B). Are the weights of B periodic? Ans. I don t know! Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 15/17

21 Recent result Theorem Let A and B are n n (n 3) weighted shift matrices with nonzero weights a 1,..., a n and b 1,..., b n. The following are equivalent : 1 W (A) = W (B); 2 p A (x, y, z) = p B (x, y, z); 3 S l ( a 1 2,..., a n 2 ) = S l ( b 1 2,..., b n 2 ), for all 1 l n 2 and n i=1 a i = n i=1 b i. Note. 1 p A (x, y, z) = det(xre A + yim A + zi n). 2 Our formulae involve the circularly symmetric function S r (a 1,... a n), where n and r are nonnegative integers. S is define to be 1, while for r 1, S r (a 1,... a n) = { r k=1 a π(k) π : (1,..., r) (1,..., n), where π(k) + 1 < π(k + 1) for 1 k < r, and if π(1) = 1 then π(r) n}. Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 16/17

22 Thank you for your attention! Hwa-Long Gau Numerical Ranges of Weighted Shift Matrices 17/17

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