Inverse Problems for Selfadjoint Matrix Polynomials

Transcription

1 Peter Lancaster, University of Calgary, Canada Castro Urdiales, SPAIN, Castro Urdiales, SPAIN, /

2 Preliminaries Given A 0, A 1,, A l C n n (or possibly in R n n ), L(λ) := l A j λ j, λ C, j=0 det A l 0. Spectrum of L : σ(l) := {λ C : detl(λ) = 0} (the eigenvalues). Eigenvectors: If λ 0 σ(l) vectors x C n such that x 0 and L(λ 0 )x = 0 are eigenvectors. Direct problems: Given L(λ) find σ(l), eigenvectors, etc.. Inverse problems: Given a set of candidates for eigenvalues and eigenvectors (possibly incomplete) find system(s) L(λ) consistent with this spectral data. Castro Urdiales, SPAIN, /

3 The ultimate inverse problem: One can argue that the ultimate inverse problem is the expression of the coefficients A j in terms of spectral data. i.e. when the spectral data is properly (completely) defined it should determine the coefficients of the system. Hence our interest in canonical forms. Castro Urdiales, SPAIN, /

4 Important special cases 1. l = l = 2: the quadratic eigenvalue problem. 3. A j R n n for all j. 4. A j = A j C n n for all j, A l > 0? (et al?) An equivalent first-degree problem is selfadjoint in a natural inner-product on C ln ln. We say that L(λ) is selfadjoint. 5. A T j = A j R n n for all j, A l > 0? (et al?) Selfadjoint...on R ln ln. Note that the role of eigenvalues of L(λ) as the singularitites of L(λ) 1 is of great importance. Castro Urdiales, SPAIN, /

5 Sign characteristics (from GLR, 2005). Theorem Let L(λ) be an n n self-adjoint matrix polynomial with non-singular leading coefficient and let µ 1 (λ),..., µ n (λ) be the real analytic functions of real λ such that det{µ j (λ)i n L(λ)} = 0 for j = 1,..., n. Let λ 1 < < λ r be the different real eigenvalues of L(λ) (zeros of detl(λ))). For j = 1, 2,..., n and every i = 1,..., r, write µ j (λ) = (λ λ i ) m ij ν ij (λ) where ν ij (λ i ) 0 is real. Castro Urdiales, SPAIN, /

6 Sign characteristics (from GLR, 2005). Theorem Let L(λ) be an n n self-adjoint matrix polynomial with non-singular leading coefficient and let µ 1 (λ),..., µ n (λ) be the real analytic functions of real λ such that det{µ j (λ)i n L(λ)} = 0 for j = 1,..., n. Let λ 1 < < λ r be the different real eigenvalues of L(λ) (zeros of detl(λ))). For j = 1, 2,..., n and every i = 1,..., r, write µ j (λ) = (λ λ i ) m ij ν ij (λ) where ν ij (λ i ) 0 is real. Then the non-zero numbers among m i1,..., m in are the partial multiplicities of L(λ) associated with λ i, and the sign of ν ij (λ i ) (for m ij 0) is the sign characteristic associated with the elementary divisors (λ λ i ) m ij of L(λ). Castro Urdiales, SPAIN, /

7 Sign characteristics and inverse problems Spectral properties for selfadjoint matrix functions include sign characteristics - as above. So they have a role to play in the formulation of inverse problems. For example, suppose that the leading coefficient A l is NOT positive definite, and consider the behaviour of the eigenfunctions µ j (λ) as λ. Theorem Let (π, n π, 0) be the inertia of L l and assume that there exists at least one real eigenvalue of L(λ). Let λ max be the largest real eigenvalue of L(λ). Then there are π indices {i 1,..., i π } in {1, 2,..., n} such that, for all λ > λ max, µ j (λ) > 0 if j {i 1,..., i π } and µ j (λ) < 0 if j / {i 1,..., i π }. Castro Urdiales, SPAIN, /

8 Quadratics using divisors (with Tisseur and then Chorianopoulos) Consider quadratic functions with Hermitian coeffts.: L(λ) = L 2 λ 2 + L 1 λ + L 0, det L 2 0. We would like to express L(λ) in factored form, L(λ) = (I λ S)L 2 (I λ A). More precisely, given nonsingular Hermitian L 2 and the right divisor I λ A, describe the class of matrices S for which L(λ) is selfadjoint. Castro Urdiales, SPAIN, /

9 Quadratics using divisors (with Tisseur and then Chorianopoulos) Consider quadratic functions with Hermitian coeffts.: L(λ) = L 2 λ 2 + L 1 λ + L 0, det L 2 0. We would like to express L(λ) in factored form, L(λ) = (I λ S)L 2 (I λ A). More precisely, given nonsingular Hermitian L 2 and the right divisor I λ A, describe the class of matrices S for which L(λ) is selfadjoint. First define T H = {A L 2 + Z 1 : A Z 1 Z 1 A = 0 and Z 1 = Z 1}. Theorem Given L 2 nonsingular and a right divisor I λ A, the function L(λ) is Hermitian when λ R if and only if S = TL 1 2 and T T H. Castro Urdiales, SPAIN, /

10 Example Nonsingular leading coefficient: L 2 = (Diagonal) right divisor I λ A where A = diag [ 1, 3, 4, 2 + i, 5 3i ]. Calculated left divisor S has spectrum , ± i, 2 i, 5 + 3i. Castro Urdiales, SPAIN, /

11 Updating problems: Given L(λ) and a complete description of σ(l), adjust the coefficients of L(λ) to produce desired changes in σ(l). xxxxxxxxxxxxxxxxxxxxxxxx Standard pair: X F n ln, T F ln ln, (F = R or C) for which X XT det. 0. XT l 1 Standard pair for L(λ) is a standard pair (X, T ) for which L(X, T ) := L l XT l + + L 1 XT + L 0 X = 0. With T in Jordan form, columns of X determine eigenvectors (and generalized eigenvectors) of L(λ). Castro Urdiales, SPAIN, /

12 From pairs to triples If the leading coefft. L l is invertible, then a triple (X, T, Y ) is a standard triple for L(λ) if (X, T ) is a standard pair for L(λ) and Y = X XT. XT l L l 1. Similarity of standard triples: {Standard Triples} = {XS, S 1 TS, S 1 Y } : (X, T, Y ) is a st. triple.} JORDAN forms JORDAN pairs JORDAN triples. Castro Urdiales, SPAIN, /

13 Hermitian/real-symmetric systems Symmetries in coefficients Symmetries in canonical structures Prime example of a standard triple: X = [ I 0 0 ], T = C R, Y = where C R is the (right) companion matrix of L(λ). Any triple similar to the above is also a standard triple L 1 l, Castro Urdiales, SPAIN, /

14 Symmetries in terms of standard triples Definition: (a) A real standard triple (X, T, Y ) is real self-adjoint if there is a real nonsingular H such that Y T = XH 1, T T = HTH 1, X T = HY. (b) A complex standard triple (X, T, Y ) is self-adjoint if there is a nonsingular Hermitian H such that Theorem Y = XH 1, T = HTH 1, X = HY. Let L(λ) have real coefficients with A l nonsingular. Then: (a) If L(λ) admits a real selfadjoint standard triple, then L(λ) is real and symmetric. (b) If L(λ) is real and symmetric then all its real standard triples are selfadjoint. Castro Urdiales, SPAIN, /

15 APPLICATION - AN UPDATING PROBLEM M, D, K R n n all real-symmetric, M > 0. L(λ) = Mλ 2 + Dλ + K. Castro Urdiales, SPAIN, /

16 APPLICATION - AN UPDATING PROBLEM M, D, K R n n all real-symmetric, M > 0. All e.v. distinct. Jordan pair: J = L(λ) = Mλ 2 + Dλ + K. U 1 + iw U U U 1 iw X = [ X c X R1 X R2 X c ]., Castro Urdiales, SPAIN, /

17 APPLICATION - AN UPDATING PROBLEM M, D, K R n n all real-symmetric, M > 0. All e.v. distinct. Jordan pair: J = L(λ) = Mλ 2 + Dλ + K. U 1 + iw U U U 1 iw X = [ X c X R1 X R2 X c ]., To complete the selfadjoint Jordan triple (X, J, PX ) we need: I n r P = 0 I r I r 0. I n r Castro Urdiales, SPAIN, /

18 Updating - contd.. Define the moments of the system: Γ j = X (J j P)X C n n j = 1, 2, 3,..., then M = Γ 1 1, D = MΓ 2M, K = MΓ 3 M + DΓ 1 D. Castro Urdiales, SPAIN, /

19 Updating - contd.. Define the moments of the system: Γ j = X (J j P)X C n n j = 1, 2, 3,..., then M = Γ 1 1, D = MΓ 2M, K = MΓ 3 M + DΓ 1 D. This gives a formal recursive solution to the inverse problem: Given the spectral data in the form of the Jordan triple (X, J, PX ), these formulae generate the coefficients of the system. Castro Urdiales, SPAIN, /

20 Updating - STRATEGY: Given a system with real symmetric M, D, K and M > 0, compute a Jordan triple (X, J, PX ). Make the updates in X and J to produce ˆX, Ĵ so that: (a) the canonical matrix P is not disturbed, and (b) conditions XPX = 0 and X (JP)X > 0 are maintained. Compute the moments defined by ( ˆX, P, Ĵ) and hence new coefficients ˆM, ˆD, ˆK. Castro Urdiales, SPAIN, /

21 A CANONICAL selfadjoint triple for REAL, SEMISIMPLE systems with A l nonsingular. 1. Should display ALL eigenvalue/sign-characteristic/eigenvector information in convenient form. 2. Found by searching among the similarity-class of all selfadjoint triples. SPECTRAL DATA: Let δ be the signature of leading coefft. L l, and χ = 0 or 1 according as l is even or odd. Let r 1,..., r q+χδ be the real eigenvalues of positive type, r q+χδ+1,..., r 2q+χδ be the real e.v. of negative type and construct diagonal matrices of size q + χδ and q: R + = diag[r 1,..., r q+χδ ], R = diag[r q+χδ+1,..., r 2(q+χδ) ]. Castro Urdiales, SPAIN, /

22 A CANONICAL selfadjoint triple for REAL, SEMISIMPLE systems with A l nonsingular Write the 2s conjugate pairs of eigenvalues as follows: β j = µ j +iν j, β j+1 = β j = µ j iν j (ν j > 0), j = 1, 3,..., 2s 1, and set M = diag[µ 1, µ 3,..., µ 2s 1 ], N = diag[ν 1, ν 3,..., ν 2s 1 ]. Castro Urdiales, SPAIN, /

23 A CANONICAL TRIPLE (L., Prells and Zaballa, 2012) If non-real eigenvectors are u j ± iv j, for j = 1, 2,, s they can be normalized in such a way that, with V = [v 1 v s ], U = [u 1 u s ] R n s there is a REAL (CANONICAL) STANDARD TRIPLE: (X, J, PX T ) with [ J = Π T M N J R Π = Diag(R +, R, N M ] ) R nl nl, P = Π T P R Π = Diag (I q+χδ, I q, I s, I s ) R nl nl, X = X R Π = [ X + X V U ] R n nl. (Still in the semisimple case. Admits real and complex spectrum.) Castro Urdiales, SPAIN, /

24 FIRST PROPERTIES Moment conditions: XJ k PX T = 0, k = 0, 1,..., l 2, Resolvent form: XJ (l 1) PX T = L 1 l. { λ r L(λ) 1 XJ r (I λ J) 1 PX T, r = 0, 1,..., l 1, = XJ l (I λi J) 1 PX T + L 1 l, r = l, Castro Urdiales, SPAIN, /

25 CASE OF EVEN DEGREE: l = 2m: Moment conditions can be written: X XJ. XJ m P ˆ X T J T X T (J T ) m 1 X T = 0. Now separate real spectrum of +ve and -ve types - along with separation of conjugate pairs (ref canonical forms) and define: 2 A = 6 4 A 0 A 1. A m = 6 4 X + U 0 X +R + U 1.. X +R+ m 1 U m , B = 6 4 B 0 B 1. B m = 6 4 X V 0 X R V 1.. X R m 1 V m Castro Urdiales, SPAIN, /

26 CASE OF EVEN DEGREE (the punch-line): With A and B as above: Theorem For any real-symmetric matrix polynomial L(λ) of even degree, l = 2m, with invertible leading coefficient, there is a real orthogonal matrix Θ R mn mn such that B = AΘ. Conversely, let J, P be real canonical matrices as described, and X = [ X + X V U ] R n ln with B := AΘ for some real orthogonal Θ and (XJ l 1 PX T ) nonsingular then (X, J, PX T ) is a real selfadjoint triple. Implication for inverse problems? Castro Urdiales, SPAIN, /

27 The quadratic case In the eigenvector matrix: X = X R Π = [ X + X V U ] R n 2n. we have [ X V ] = [ X + U ] Θ (began with Lancaster-Prells, 2005). Castro Urdiales, SPAIN, /

28 CONSTRUCTIONS Use the 2s non-real e.vs. µ j ± iν j (j=1,3,...,2s-1) to define M := diag[µ 1,, µ 2s 1 ], N := diag[ν 1,, ν 2s 1 ], in R s s. and then define [ Mr N r N r M r ] [ M N := N M ] r [ Is 0 0 I s ]. Castro Urdiales, SPAIN, /

29 CONSTRUCTIONS Use the 2s non-real e.vs. µ j ± iν j (j=1,3,...,2s-1) to define M := diag[µ 1,, µ 2s 1 ], N := diag[ν 1,, ν 2s 1 ], in R s s. and then define [ Mr N r N r M r H k (Θ) := [ I nm Θ ] ] [ ] r [ M N Is 0 := N M 0 I s := [ I nm Θ ] G k [ Inm Θ T R+ k M k 0 N k 0 0 R k 0 0 N k 0 M k ]. [ Inm Θ T ] ; defines G k R ln ln. ]. Castro Urdiales, SPAIN, /

30 Cauchy to the rescue Keeping in mind H k (Θ) = [ I nm Θ ] G k [ Inm Θ T ] and using the Cauchy interlacing inequalities we get (for the ev of H k (Θ)): Theorem For k = 1, 2,... write the (known) eigenvalues of G k in the form λ 1 (G k ) λ 2nm (G k ). Then for any nm nm orthogonal matrix Θ we have 2λ i (G k ) λ i (H k (Θ)) 2λ i+nm (G k ), 1 i nm. Castro Urdiales, SPAIN, /

31 Semisimple, real, symmetric, QUADRATICS We form a real canonical triple (X, J, PX T ) for L(λ) = L 2 λ 2 + L 1 λ + L 0. Castro Urdiales, SPAIN, /

32 Semisimple, real, symmetric, QUADRATICS We form a real canonical triple (X, J, PX T ) for L(λ) = L 2 λ 2 + L 1 λ + L 0. R + = Diag[r 1,, r q ], R = Diag[r q+1,, r 2q ] β j = µ j + iν j, β j+1 = β j = µ j iν j, ν j > 0, j = 1, 3,..., 2s 1. M = Diag[µ 1, µ 3,, µ 2s 1 ], N = Diag[ν 1, ν 3,, ν 2s 1 ] [ ] M N J = Diag[R +, R, ] R 2n 2n, N M P = Π T P R Π = Diag[I q, I q, I s, I s ] R 2n 2n, X = [ X + X V U ] R n 2n. (X +, X are n q and (V, U) are n s.) Castro Urdiales, SPAIN, /

33 An inverse quadratic problem: Given semisimple canonical matrices J, P R 2n 2n as above, can we always find an X R n 2n to complete a canonical triple (X, J, PX T )? NO! Recall that there are 2q real eigenvalues (counting multiplicities), and s pairs of complex conjugate eigenvalues 0 q, s n. To characterize this problem we need another parameter: Let p be the maximal multiplicity of any real eigenvalue (so that 1 p 2n). Theorem (L.-Zaballa) With the hypotheses above, there exists a canonical triple (X, J, PX T ) if and only if q + s p. (Notice that s = 0 is possible if q p. Also, q + s = p when the system is semisimple.) Castro Urdiales, SPAIN, /

34 Another inverse quadratic problem Recall that, for a real canonical triple (X, J, PX T ), Γ 1 = XJPX = A 1 2. Theorem Given (semisimple) candidates J, P as components of a real canonical triple (X, J, PX T ), there is an associated semisimple, real, symmetric quadratic matrix polynomial with Γ 1 (= XJPX ) nonsingular (or Γ 1 > 0) if and only if there is an orthogonal matrix Θ such that H 1 (Θ) := [ I nm Θ ] is nonsingular (resp. positive definite). R M 0 N 0 0 R 0 0 N 0 M [ Inm Θ T ] Castro Urdiales, SPAIN, /

35 EPILOGUE: GENERAL CANONICAL FORMS (L/Z) FIRST - reduction over C. Primitive matrices F j and G j : F 1 = [1], G 1 = [0], F 4 = , G 4 = etc.. Castro Urdiales, SPAIN, /

36 EPILOGUE: GENERAL CANONICAL FORMS (L/Z) P = q s ε j F lj F 2mk, j=1 k=1 PJ = q ε j (α j F lj + G lj ) s [ j=1 k=1 0 β k F mk + G mk β k F mk + G mk 0 ]. Theorem If L(λ) is Hermitian and A l is nonsingular, then there exists a selfadjoint Jordan triple of the form (X, J, PX ). The set of numbers ε j (= ±1) is determined uniquely by L(λ) up to permutation of the signs in the blocks of P corresponding to the Jordan blocks, α j F lj + G lj, of J with the same real eigenvalue and size. Castro Urdiales, SPAIN, /

37 And the Jordan form, J itself: The numbers ε j are, of course, ±1 and are the sign characteristics of the real eigenvalues. J = q (α j I lj + F lj ) s [ βk I mk + F mk G mk 0 0 β k I mk + F mk G mk j=1 k=1 ], and satisfies J P = PJ, i.e. J is selfadjoint in the (indefinite) P inner-product. Castro Urdiales, SPAIN, /

38 Epilogue: THE REAL SYMMETRIC CASE (L/Z) Reduction over the reals is more complicated and requires another class of primitive matrices of even size, 2m. For example, when m = 3 E 6 = Castro Urdiales, SPAIN, /

39 Epilogue: THE REAL SYMMETRIC CASE (L/Z) PK = j=1 P = q s ε j F lj F 2mk, j=1 j=1 k=1 q ε j (α j F lj + G lj ) s ( [ F2mj 2 0 µ j F 2mj + ν j E 2mj (Extract K from this.) ]) Theorem If L(λ) is real and symmetric and A l is nonsingular, then there exists a real Jordan triple of the form (X ρ, K, PX T ρ ). The set of numbers ε j (= ±1) is determined uniquely by L(λ) up to permutation of the signs in the blocks of P corresponding to the Jordan blocks of K with the same real eigenvalue and size. Castro Urdiales, SPAIN, /

40 REFERENCES: C. Chorianopoulos and P. Lancaster Inverse problems for Hermitian quadratic matrix polynomials, Indag. Math., 23, 2012, I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, 1982, and SIAM, I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra and Applications, Birkhäuser, Basel, P. Lancaster, Model-updating for self-adjoint quadratic eigenvalue problems, Lin. Alg and its Applications, 428, 2008, P. Lancaster, U. Prells Inverse problems for damped vibrating systems, J. of Sound and Vibration, 283, 2005, Castro Urdiales, SPAIN, /

41 REFERENCES (continued) P. Lancaster, U. Prells, I. Zaballa, An orthogonality property for real symmetric matrix polynomials with application to the inverse problem Operators and Matrices, to appear. P. Lancaster and F. Tisseur, Hermitian quadratic matrix polynomials: solvents and inverse problems, Lin. Alg and its Applications, 436, 2012, P. Lancaster, I. Zaballa, A review of canonical forms for selfadjoint matrix polynomials. Operator Theory: Advances and Applications, 218, 2012, P. Lancaster, I. Zaballa, On the inverse symmetric quadratic eigenvalue problem. Submitted. Castro Urdiales, SPAIN, /

42 Time to go! Castro Urdiales, SPAIN, /