Eigenvalues of rank one perturbations of matrices

Transcription

1 Eigenvalues of rank one perturbations of matrices André Ran VU University Amsterdam Joint work in various combinations with: Christian Mehl, Volker Mehrmann, Leiba Rodman, Michał Wojtylak Jan Fourie, Gilbert Groenewald, Dawie Janse van Rensburg Rank one perturbations André Ran 1

2 and motivation Single input control system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) A is n n, B is n 1, C is m n, D is m 1. To find: a dynamic controller which stabilizes the system. Well known result: first construct a state observer, then a static feedback based on the observed state. Algorithm: find matrices R and F so that Focus on A + BF here. σ(a RC) C σ(a + BF) C. Rank one perturbations André Ran 2

3 Closed loop system The feedback is then u(t) = Fx(t). Closed loop feedback system: ẋ(t) = (A + BF)x(t) Eigenvalues of closed loop system matrix are σ(a + BF).. Failures in control system: changes in B (actuator failure). Rank one perturbations André Ran 3

4 European Robot Arm Artists impression of the attached to ISS. s elbow joint.

5 European Robot Arm A linearized version of a mathematical model for the movement of the elbow joint in the European Robotic Arm is given by ẋ(t) = x(t) + 4K 0 u(t) K y(t) = x(t) + 0 u(t) K is the motor torque constant of the motor operating the joint. Possible failure: Increase in K with about 1% per second for a short while, starting from K = 0.5. Rank one perturbations André Ran 4

6 Ten seconds failure Placing the eigenvalues of A + BF at 4.5, 5.5, 6.5, 7.5. Graph shows eigenvalues of closed loop system for values of K varying from 0.5 to Figure 3. Instability after 10 seconds. Rank one perturbations André Ran 5

7 Ten seconds failure II Placing the eigenvalues of A + BF at 20, 21, 22, 23, again for values of K varying from 0.5 to 0.55, we see that the system remains stable Figure 4. Stability. Rank one perturbations André Ran 6

8 statement Goal: to find explanation for the phenomenon observed here. Broader goal: A an n n real matrix, study the behaviour of eigenvalues of rank one perturbations A + tuv, where u and v are real n-vectors, and t R. Focus: u and v in R n generic. In other words: what is the generic behaviour of eigenvalues of rank one perturbations of A? Interest in several questions:, asymptotics for small t, asymptotics for large t, generic behaviour. Rank one perturbations André Ran 7

9 Let J n (λ) denote the standard n n upper triangular Jordan block with eigenvalues λ: J n (λ) = λ λ λ Rank one perturbations André Ran 8

10 Let J n (λ) denote the standard n n upper triangular Jordan block with eigenvalues λ: J n (λ) = λ λ λ General Jordan canonical form: Ordering of the sizes A = p j=1 κ j i=1 J n i,j (λ j ) n 1,j n 2,j n κj,j. Rank one perturbations André Ran 8

11 Example matrix with three eigenvalues: 0, 1, 2, with blocks of size 4, 4, 3, 1 corresponding to 0 5, 2 corresponding to 1 2 corresponding to 2 A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) Rank one perturbations André Ran 9

12 Example matrix with three eigenvalues: 0, 1, 2, with blocks of size 4, 4, 3, 1 corresponding to 0 5, 2 corresponding to 1 2 corresponding to 2 A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) What happens to the Jordan form when we apply a rank one perturbation? B(t) = A + tuv, where u, v R 21 and t R. Rank one perturbations André Ran 9

13 Example matrix with three eigenvalues: 0, 1, 2, with blocks of size 4, 4, 3, 1 corresponding to 0 5, 2 corresponding to 1 2 corresponding to 2 A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) What happens to the Jordan form when we apply a rank one perturbation? B(t) = A + tuv, where u, v R 21 and t R. What can we say about the eigenvalues as functions of t? Rank one perturbations André Ran 9

14 :. Beautiful result by L. Hörmander and A. Melin (1994) S.V. Savchenko (2003) F. Dopico and J. Moro (2003) Rank one perturbations André Ran 10

15 :. Beautiful result by L. Hörmander and A. Melin (1994) S.V. Savchenko (2003) F. Dopico and J. Moro (2003) Generically (in an algebraic sense) the of B(t) is as follows: Rank one perturbations André Ran 10

16 :. Beautiful result by L. Hörmander and A. Melin (1994) S.V. Savchenko (2003) F. Dopico and J. Moro (2003) Generically (in an algebraic sense) the of B(t) is as follows: for every eigenvalue of A only the largest Jordan block is lost, Rank one perturbations André Ran 10

17 :. Beautiful result by L. Hörmander and A. Melin (1994) S.V. Savchenko (2003) F. Dopico and J. Moro (2003) Generically (in an algebraic sense) the of B(t) is as follows: for every eigenvalue of A only the largest Jordan block is lost, all other eigenvalues are simple. Rank one perturbations André Ran 10

18 :. Beautiful result by L. Hörmander and A. Melin (1994) S.V. Savchenko (2003) F. Dopico and J. Moro (2003) Generically (in an algebraic sense) the of B(t) is as follows: for every eigenvalue of A only the largest Jordan block is lost, all other eigenvalues are simple. Also results for more general low rank perturbations. Rank one perturbations André Ran 10

19 Example In the example: A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) Rank one perturbations André Ran 11

20 Example In the example: A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) B(t) = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) Rank one perturbations André Ran 11

21 Example In the example: A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) B(t) = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) J 1 (t) J 2 (t) J 3 (t) Rank one perturbations André Ran 11

22 Example In the example: A = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) B(t) = J 4 (0) J 4 (0) J 3 (0) J 1 (0) J 5 (1) J 2 (1) J 2 (2) J 1 (t) J 2 (t) J 3 (t) where J 1 (t), J 2 (t), J 3 (t) are 4 4, 5 5, 2 2 matrices, generically (with respect to u, v and t) diagonalizable with disjoint spectra. Rank one perturbations André Ran 11

23 Comments on a proof A new proof (Mehl, Mehrmann, R., Rodman) uses a concept of systems theory: the Brunovski form. Given A, u and v, there is an invertible S such that S 1 AS is in Jordan canonical form, and simultaneously, v is in a nice form. Example. A = J 4 (0) J 3 (0), v = [ ] B = A + uv = u u u u u u u u u u u u u u Rank one perturbations André Ran 12

24 Example continued Rank one perturbations André Ran 13 B = A + uv = u u u u u u u u u u u u u u Jordan chain of lenght three for eigenvalue zero: B B B

25 : small t asymptotics. B(t) = A + tuv Well-known results for small t asymptotics of eigenvalues V.B. Lidskii (1966), M.I. Vishik and L.A. Lyusternik (1960), see also book T. Kato (Chapter 2). In the example A = J 4 (0) J 3 (0) small t asymptotics, blue t<0, red t> Rank one perturbations André Ran

26 : new eigenvalues The eigenvalues of B(t) = A + tuv which are not eigenvalues of A are nonderogatory for all t 0, and all vectors u and v. That is, corresponding to eigenvalues in σ(b(t)) \ σ(a) there is only one Jordan block. λ 0 σ(b(t)) \ σ(a) for some t 0 and some u and v, n = rank(a λ 0 I) rank (A + tuv λ 0 I) + rank(tuv ) = = rank (B(t) λ 0 I) + 1, So rank(b(t) λ 0 I) n 1, hence dim Ker (B(t) λ 0 I) 1. This says that λ 0 is a nonderogatory eigenvalue of B(t). Rank one perturbations André Ran 15

27 : large t asymptotics Define the polynomial p(λ) by B(t) = A + tuv. p(λ) = v m A (λ)(λ A) 1 u, where m A (λ) denotes the minimal polynomial of A. Let l denote the degree of p. Rank one perturbations André Ran 16

28 : large t asymptotics Define the polynomial p(λ) by B(t) = A + tuv. p(λ) = v m A (λ)(λ A) 1 u, where m A (λ) denotes the minimal polynomial of A. Let l denote the degree of p. Theorem As t ± the eigenvalues of B(t) which are not eigenvalues of A behave as follows as : exactly l 1 of them approximate the roots of p, and one goes to infinity, asymptotically along the ray in the complex plane going from zero through the number v u. Rank one perturbations André Ran 16

29 Example A = J 4 (0) J 3 (0) u and v random vectors in R 7 asymptotics of the eigenvalues in red (t>0) and cyan (t<0), roots of p in blue * Rank one perturbations André Ran 17

30 Example complex A = J 4 (0) J 3 (0) u and v random vectors in C 7 asymptotics of the eigenvalues in red/cyan, roots of p in blue*, asymptotic ray in black Rank one perturbations André Ran 18

31 Useful computation p B(t) (λ) = det(λi (A + tuv )) = det = det(λ A) det(i (λ A) 1 tuv ) = det(λ A)(1 tv (λ A) 1 u) = p A (λ) tv p A (λ)(λ A) 1 u. ( ) (λ A)(I (λ A) 1 tuv ) Rank one perturbations André Ran 19

32 Useful computation p B(t) (λ) = det(λi (A + tuv )) = det = det(λ A) det(i (λ A) 1 tuv ) = det(λ A)(1 tv (λ A) 1 u) = p A (λ) tv p A (λ)(λ A) 1 u. p B(t) and p A have a common divisor ( ) (λ A)(I (λ A) 1 tuv ) q(λ) = Π p i=1 Πk i j=2 (λ λ i) n i,j, we divide through by this expression. This yields p B(t) (λ) q(λ) = m A (λ) tv m A (λ)(λ A) 1 u = m A (λ) tp(λ). Rank one perturbations André Ran 19

33 No intersections Generically the eigenvalue curves do not intersect. Rank one perturbations André Ran 20

34 No intersections Generically the eigenvalue curves do not intersect. What means generically here???? Rank one perturbations André Ran 20

35 No intersections Generically the eigenvalue curves do not intersect. What means generically here???? Generically (in an algebraic sense) there is no triple eigenvalue: λ 0 is a triple eigenvalue if and only if m A (λ 0 ) tp(λ 0 ) = 0 m A(λ 0 ) tp (λ 0 ) = 0 m A(λ 0 ) tp (λ 0 ) = 0 Rank one perturbations André Ran 20

36 No intersections Generically the eigenvalue curves do not intersect. What means generically here???? Generically (in an algebraic sense) there is no triple eigenvalue: λ 0 is a triple eigenvalue if and only if m A (λ 0 ) tp(λ 0 ) = 0 m A(λ 0 ) tp (λ 0 ) = 0 m A(λ 0 ) tp (λ 0 ) = 0 Eliminate t from first equation, insert in second and third: λ 0 is a triple eigenvalue if and only if λ 0 is a common root of two polynomials. Use the resultant to see that generically this does not happen. Rank one perturbations André Ran 20

37 Example of triple crossing A = J 4 (0) J 1 (2) and tuning of u and v produces asymptotics of the eigenvalues in red/cyan, roots of p in blue,bifurcation points in blue Rank one perturbations André Ran 21

38 Double eigenvalues Complex case. Proposition Given A C n n and given u C n the set {v C n t R, z C \ σ(a) : z is a double eigenvalue of A + tuv } has empty interior. Rank one perturbations André Ran 22

39 Double eigenvalues Complex case. Proposition Given A C n n and given u C n the set {v C n t R, z C \ σ(a) : z is a double eigenvalue of A + tuv } has empty interior. So, generically (in a topological sense) there are no double eigenvalues. Rank one perturbations André Ran 22

40 Double eigenvalues Complex case. Proposition Given A C n n and given u C n the set {v C n t R, z C \ σ(a) : z is a double eigenvalue of A + tuv } has empty interior. So, generically (in a topological sense) there are no double eigenvalues. 2 asymptotics of the eigenvalues in red/cyan, roots of p in blue*, asymptotic ray in black Rank one perturbations André Ran 22

41 Double eigenvalues Real case. Proposition Given A R n n and given u R n the set { v R n t R, z C \ (σ(a) R) : z is a double eigenvalue of A + tuv } has empty interior. Rank one perturbations André Ran 23

42 Double eigenvalues Real case. Proposition Given A R n n and given u R n the set { v R n t R, z C \ (σ(a) R) : z is a double eigenvalue of A + tuv } has empty interior. So, generically (in a topological sense) there are no non-real double eigenvalues. Rank one perturbations André Ran 23

43 Double eigenvalues Real case. Proposition Given A R n n and given u R n the set { v R n t R, z C \ (σ(a) R) : z is a double eigenvalue of A + tuv } has empty interior. So, generically (in a topological sense) there are no non-real double eigenvalues. asymptotics of the eigenvalues in red/cyan, roots of p in blue*,bifurcation points in blue o Rank one perturbations André Ran 23

44 The story becomes more interesting when structure comes into play. Rank one perturbations André Ran 24

45 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Rank one perturbations André Ran 24

46 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? Rank one perturbations André Ran 24

47 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. Rank one perturbations André Ran 24

48 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. Rank one perturbations André Ran 24

49 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. In R n let H = H be invertible, let A be real and H-Hamiltonian, that is HA = A H. (MMRR) Rank one perturbations André Ran 24

50 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. In R n let H = H be invertible, let A be real and H-Hamiltonian, that is HA = A H. (MMRR) Surprising extra results! Rank one perturbations André Ran 24

51 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. In R n let H = H be invertible, let A be real and H-Hamiltonian, that is HA = A H. (MMRR) Surprising extra results! Curious? Rank one perturbations André Ran 24

52 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. In R n let H = H be invertible, let A be real and H-Hamiltonian, that is HA = A H. (MMRR) Surprising extra results! Curious? Go and listen to Christian Mehl! Rank one perturbations André Ran 24

53 The story becomes more interesting when structure comes into play. In C n let H = H be invertible, let A be H-selfadjoint. Rank-one perturbations studied by de Snoo, Winkler, Wojtylak in Π 1 case. Curious? See Michał Wojtylak s talk. General case studied by Mehl, Mehrmann, R., Rodman. Also the real case. In R n let H = H be invertible, let A be real and H-Hamiltonian, that is HA = A H. (MMRR) Surprising extra results! Curious? Go and listen to Christian Mehl! Here we discuss the H-positive real case. H = H real and invertible, A real and HA + A T H 0. Rank one perturbations André Ran 24

54 H-positive real matrices H = H, HA + A H 0. Consider rank one perturbations A + tuv, where uv is H-positive real. Joint work with Dawie Janse van Rensburg, Jan Fourie and Gilbert Groenewald. Rank one perturbations André Ran 25

55 H-positive real matrices H = H, HA + A H 0. Consider rank one perturbations A + tuv, where uv is H-positive real. Joint work with Dawie Janse van Rensburg, Jan Fourie and Gilbert Groenewald. Proposition A rank one matrix uv is H-positive real if and only if v = c u H for some c > 0. Rank one perturbations André Ran 25

56 H-positive real matrices H = H, HA + A H 0. Consider rank one perturbations A + tuv, where uv is H-positive real. Joint work with Dawie Janse van Rensburg, Jan Fourie and Gilbert Groenewald. Proposition A rank one matrix uv is H-positive real if and only if v = c u H for some c > 0. H-positive real rank one perturbations of H-positive real matrices. Rank one perturbations André Ran 25

57 H-positive real matrices H = H, HA + A H 0. Consider rank one perturbations A + tuv, where uv is H-positive real. Joint work with Dawie Janse van Rensburg, Jan Fourie and Gilbert Groenewald. Proposition A rank one matrix uv is H-positive real if and only if v = c u H for some c > 0. H-positive real rank one perturbations of H-positive real matrices. Theorem Put B(t) = A + tuu H, with t > 0. Then (σ(b(t)) \ σ(a)) ir =. Rank one perturbations André Ran 25

58 Example Theorem in words: the new eigenvalues of B(t) aren t on the pure imaginary axis. Example: J = γ I γ I 2, γ = 0 0 γ [ ] 0 2, H = I 2 0 I 2 0 I Rank one perturbations André Ran 26

59 Example Theorem in words: the new eigenvalues of B(t) aren t on the pure imaginary axis. Example: J = γ I γ I 2, γ = 0 0 γ 3 [ ] 0 2, H = 2 0 blue is unstructured rank one perturbation, red is positive real perturbation 0 0 I 2 0 I 2 0 I Rank one perturbations André Ran 26

60 Asymptotics But: Rank one perturbations André Ran 27

61 Asymptotics But: asymptotically the eigenvalues may be on the imaginary line. Rank one perturbations André Ran 27

62 Asymptotics But: asymptotically the eigenvalues may be on the imaginary line. asymptotics of the eigenvalues, two blocks size 3, eigenvalue Rank one perturbations André Ran 27

63 Chr. Mehl, V. Mehrmann, A.C.M. Ran, L. Rodman. Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations LAA 2010, to appear. Chr. Mehl, V. Mehrmann, A.C.M. Ran, L. Rodman. Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations LAA 2010, to appear. Works with M. Wojtylak and D.B. Janse van Rensburg, J.H. Fourie and G.J. Groenewald reported on here are in preparation. Rank one perturbations André Ran 28

64 Thompson, Robert C. Invariant factors under rank one perturbations. Canad. J. Math. 32 (1980), no. 1, L. Hörmander, and A. Melin. A remark on perturbations of compact operators. Math. Scand 75 (1994), F. Dopico and J. Moro. Low rank perturbation of, SIAM J. Matrix Anal. Appl., 25, (2003), Savchenko, S. V. On a generic change in the spectral properties under perturbation by an operator of rank one. (Russian) Mat. Zametki 74 (2003), no. 4, ; translation in Math. Notes 74 (2003), no. 3 4, Savchenko, S. V. On the change in the spectral properties of a matrix under a perturbation of a sufficiently low rank. (Russian) Funktsional. Anal. i Prilozhen. 38 (2004), no. 1, 85 88; translation in Funct. Anal. Appl. 38 (2004), no. 1, Rank one perturbations André Ran 29

65 Thank you for your attention! Rank one perturbations André Ran 30