Hyperbolic Matrix Polynomials and Definite Linearizations

Hyperbolic Matrix Polynomialsand Definite LinearizationsD. Steven MackeyWestern Michigan UniversityKalamazoo, MI, U.S.A.Joint work withNick Higham and Françoise TisseurThe University of ManchesterBMW-3Workshop on Nonlinear Eigenvalue ProblemsManchester, 22 March 2007Manchester, 22March07 – p. 1/18

Definite PencilsA standard way to define definite pencils –A Hermitian pencil L(λ) = λX + Y is definite if the Crawfordnumber γ(X,Y ) is nonzero, i.e. ifγ(X,Y ) := minz∈C n‖z‖ 2 =1√(z ∗ Xz) 2 + (z ∗ Y z) 2 > 0.This suffices to imply that all eigenvalues of L(λ) are real.Manchester, 22March07 – p. 3/18

Hyperbolic Quadratic PolynomialsStandard definition (goes back at least to Duffin, 1955, foroverdamped systems)For Hermitian Q(λ) = λ 2 A + λB + C and x ∈ C n , definea x := x ∗ Ax , b x := x ∗ Bx , c x := x ∗ Cx ,and consider D x (Q) := b 2 x − 4a x c x .Then Q is hyperbolic if A > 0 and D x (Q) > 0 for all x ∈ C n .Manchester, 22March07 – p. 4/18

Hyperbolic Quadratic PolynomialsStandard definition (goes back at least to Duffin, 1955, foroverdamped systems)For Hermitian Q(λ) = λ 2 A + λB + C and x ∈ C n , definea x := x ∗ Ax , b x := x ∗ Bx , c x := x ∗ Cx ,and consider D x (Q) := b 2 x − 4a x c x .Then Q is hyperbolic if A > 0 and D x (Q) > 0 for all x ∈ C n .The discriminant condition D x (Q) > 0 implies:x ∗ Q(λ)x = 0 has two real roots for any x ∈ C n ,hence all eigenvalues of Q are real.Manchester, 22March07 – p. 4/18

Extension to Higher DegreeVarious ways to extend [Markus, 1988] —Strongly Hyperbolic, .... , QuasihyperbolicHermitian P(λ) of degree k is strongly hyperbolic ifA k > 0the scalar polynomial p x (λ) = x ∗ P(λ)xhas k real roots for every x ∈ C n .Manchester, 22March07 – p. 5/18

Key Common PropertyAll eigenvalues are real and semisimple.Suggests that unification of notions may be possible.Manchester, 22March07 – p. 6/18

Other CharacterizationsLead us further towards a unified understanding.Definite Pencil L(λ) = λX + YL(µ) is a definite matrix for some µ ∈ RHyperbolic Quadratic Q(λ) = λ 2 A + λB + CA > 0 and Q(γ) < 0 for some γ ∈ R.Manchester, 22March07 – p. 7/18

Other CharacterizationsLead us further towards a unified understanding.Definite PencilL(λ) = λX + YL(µ) is a definite matrix for some µ ∈ RHyperbolic QuadraticQ(λ) = λ 2 A + λB + CA > 0 and Q(γ) < 0 for some γ ∈ R.Any two of the following implies the third1. Q(µ) > 0 for some µ ∈ R2. D x (Q) > 0 for all x ∈ C n3. Q(γ) < 0 for some γ ∈ RManchester, 22March07 – p. 7/18

Other CharacterizationsLead us further towards a unified understanding.Definite PencilL(λ) = λX + YL(µ) is a definite matrix for some µ ∈ RHyperbolic QuadraticQ(λ) = λ 2 A + λB + CA > 0 and Q(γ) < 0 for some γ ∈ R.Any two of the following implies the third1. Q(µ) > 0 for some µ ∈ R2. D x (Q) > 0 for all x ∈ C n3. Q(γ) < 0 for some γ ∈ RStrongly Hyperbolic [Markus] P(λ) with A k > 0 andµ k−1 < µ k−2 < ... < µ 1 in R such that(−1) j P(µ j ) > 0.Manchester, 22March07 – p. 7/18

Homogeneous VariablesReplace λ by (α,β) ∈ R 2 such that α 2 + β 2 = 1.Correspondence: (α,β) ∈ R 2 ↔ λ = α/β when β ≠ 0(±1, 0) ↔ λ = ”∞”L(λ) = λX + Y L(α,β) = αX + βYQ(λ) = λ 2 A + λB + C Q(α,β) = α 2 A + αβB + β 2 C“Circle picture” in α,β-planeManchester, 22March07 – p. 8/18

Definiteness DiagramsDiagrammatic renderings of previously known resultsDuffin(1955)Lancaster(1966, 1991)Markus(1988)Guo/Lancaster(2005)And many others ...ExamplesManchester, 22March07 – p. 9/18

Unifying DefinitionThese diagrams suggest a definition that(mildly) generalizes (strong) hyperbolicity,and unifies definite pencils with hyperbolic polynomials.Manchester, 22March07 – p. 10/18

Unifying DefinitionThese diagrams suggest a definition that(mildly) generalizes (strong) hyperbolicity,and unifies definite pencils with hyperbolic polynomials.Definition A Hermitian polynomial P(λ) of degree k is saidto be completely definite if there exist µ 1 < µ 2 < ... < µ k inR ∪ {∞} such thateach P(µ i ) is a definite matrix,the definiteness “parity” is strictly alternating.Completely definite polynomials have the same spectralproperties as strongly hyperbolic polynomials:eigenvalues all real and semisimple, .....Manchester, 22March07 – p. 10/18

Example[ ] [ ]Q(λ) = λ 2 A+λB +C = λ 2 −3 −1 6 3+λ +−1 2 3 −10[0]−2−2 9is completely definite even though A,B,C are all indefinite,since Q(1) > 0 and Q(3) < 0.Manchester, 22March07 – p. 11/18

L(λ) = λX + Y,LinearizationsX,Y ∈ C kn×knis a linearization of P(λ) = ∑ ki=0 λi A i ifE(λ)L(λ)F(λ) =for some unimodular E(λ) and F(λ).[ ] P(λ) 00 I (k−1)nManchester, 22March07 – p. 12/18

L(λ) = λX + Y,LinearizationsX,Y ∈ C kn×knis a linearization of P(λ) = ∑ ki=0 λi A i ifE(λ)L(λ)F(λ) =for some unimodular E(λ) and F(λ).[ ] P(λ) 00 I (k−1)nExampleFirst companion form linearization for quadratic P :( [ ] [ ])A2 0 A1 AE(λ) λ + 0F(λ) =0 I −I 0[λ 2 A 2 + λA 1 + A 0 00 I].Manchester, 22March07 – p. 12/18

Why a Definite Linearization?Why we might want a definite linearization for a completelydefinite matrix polynomial:Structure-preservingGood Algorithms are readily available for definitepencilsManchester, 22March07 – p. 13/18

A Good Place to LookL 1 (P) — a large vector space of pencils associated to PL(λ)(Λ ⊗ I n ) = v ⊗ P(λ)where Λ = [λ k−1 ,...,λ, 1] T and v ∈ C k . [MMMM, 2006]Generalizes the first companion form.Almost all pencils in L 1 (P) are linearizations for P .L(λ) ∈ L 1 (P) is a linearization for P iff L is regular.For Hermitian P , all the Hermitian pencils in L 1 (P) forma real k-dim’l subspace H(P) ⊂ L 1 (P). (k = deg P )Each pencil in H(P) is uniquely determined by thevector v ∈ R k . [HMMT, 2006]Manchester, 22March07 – p. 14/18

A Good Place to LookAre there any definite pencils in H(P)?Any such pencil will be regular, hence a (definite)linearization for P .DefineD(P) = {All definite pencils in H(P)}Is D(P) non-empty?Manchester, 22March07 – p. 15/18

Definite Linearization TheoremFor any completely definite P , D(P) is non-empty.Manchester, 22March07 – p. 16/18

Definite Linearization TheoremFor any completely definite P , D(P) is non-empty.Surprise: non-emptiness of D(P) characterizes theproperty of complete definiteness of P .TheoremA Hermitian matrix polynomial P(λ) is completely definiteif and only ifP(λ) has a definite linearization in L 1 (P) .Manchester, 22March07 – p. 16/18

Characterization of D(P)Recall: pencils in H(P) uniquely determined by v ∈ R k .Define scalar polynomial p(λ;v) := v T Λ = ∑ ki=1 v iλ k−i ofdegree k − 1; by convention p(λ;v) has a “root at ∞”whenever v 1 = 0.TheoremSuppose P is completely definite of degree k, and v ∈ R kcorresponds to L(λ) = λX + Y ∈ H(P). ThenL ∈ D(P) iff the k − 1 roots of p(x;v) are real (incl. ∞),simple, and lie in distinct definiteness intervals for P .The matrix αX + βY is definite iffL ∈ D(P)(α,β) lies in the unique definiteness interval for P notoccupied by any root of p(λ;v).Manchester, 22March07 – p. 17/18

CommentsProofs of these theorems are long.Strategy: Exploit the special block structure of pencilsin H(P) to reduce αX + βY to block-diagonal form.(Use nonsingular ∗-congruence to preserve theHermitian structure.)Unification of notions of definite and hyperbolicUnified definition via a common generalizationStructure-preserving linearizationManchester, 22March07 – p. 18/18