BASIC MATRIX PERTURBATION THEORY

Transcription

1 BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity of the total eigenprojectors, existence and computation of one-sided directional derivatives of semi-simple eigenvalues, and Puiseux expansions of coalescing eigenvalues. These results are all classical, at least in the case of one-dimensional, analytical perturbations; a standard reference is the treatise of T. Kato, Perturbation theory for linear operators. In contrast with Kato, we consider perturbations which are not necessarily smooth, in arbitrary finite dimension, and in the coalescing case do not use the notion of multi-valued function. The proofs use Rouché s theorem, representations of projectors as contour integrals, and the description of conjugacy classes of connected covering maps of the punctured disk. Consider a family of matrices () M : x Ω M(x) C N N, where Ω is open in R d.. Continuity of the eigenvalues Proposition. If x M(x) is continuous, then the spectrum of M is continuous, in the following sense: given x 0 Ω, given λ 0 sp M(x 0 ) with multiplicity m as a root of the characteristic polynomial of M(x 0 ), for any small enough r > 0 there exists a neighborhood U of x 0 in Ω, such that for all x U, the matrix M(x) has m eigenvalues (counting multiplicites) in B(λ 0, r). Proof. This is a consequence of Rouché s theorem (see for instance the Corollary to Theorem 20 in chapter 4 of []), which states that if f, g are holomorphic in B(λ 0, r) C, and if f g < g in B(λ 0, r), then f and g have the same number of zeros (counting multiplicities) in the open ball B(λ 0, r). Let Π(λ, x) = det(λ M(x)), holomorphic in λ and continuous in x. By finiteness of the spectrum, if r > 0 is small enough, then Π(, x 0 ) has only one zero in the closed ball B(λ0, r). In particular, Π(λ, x 0 ) > 0 on the boundary B(λ 0, r), and the inequality (2) h(x) = max B(λ 0,r) Π(, x) Π(, x 0) Π(, x 0 ) < 0 holds at x = x 0. The map h is continuous. Hence (2) holds in some neighborhood U of x 0. By Rouché s theorem, applied with f = Π(, x) and g = Π(, x 0 ), with x fixed in U, the function Π(, x) has the same number of zeros as Π(, x 0 ) in B(λ 0, r), meaning that M(x) has exactly m eigenvalues in B(λ 0, r), for all x U.

2 2 BENJAMIN TEXIER We assume continuity of M in the following. In particular, Proposition applies. Let S := sp M(x) {x} = { (λ, x) C Ω, det(λ M(x)) = 0 } x Ω be the spectrum of the family of matrices M, and let the projection π : (λ, x) S x Ω, so that the spectrum of matrix M(x) is the fiber π ({x}). The multiplicity of a point (λ, x) S is the algebraic multiplicity of λ in sp M(x), that is the order of λ as a root of the characteristic polynomial of M(x). A point (λ 0, x 0 ) S is said to have constant multiplicity if locally around (λ 0, x 0 ), there exists only one eigenvalue of M(x), not counting multiplicity. Corollary 2. Around a point of constant multiplicity, the projection π is a local homeomorphism. If the whole spectrum of M(x 0 ) has constant multiplicity, then π is a covering map at x 0, and the number of sheets is equal to the number of distinct eigenvalues around x 0. Proof. If (λ 0, x 0 ) has constant multiplicity, the continuous branch of eigenvalues λ given by Proposition is a continuous section of the projection p, such that λ(x 0 ) = λ 0. Thus in restriction to a neighborhood of (λ 0, x 0 ), the projection p is a homeomorphism. If the whole spectrum {λ,..., λ p } of M(x 0 ) has constant multiplicity, then in addition the fibers have constant cardinal, equal to p, around x 0. Thus π is a covering map. If a point in S does not have constant multiplicity, it is said to be a coalescing point in the spectrum. The associated multiplicity is strictly greater than one. Coalescing points in the spectrum are not necessarily isolated, even if M is smooth. Consider for instance the case Ω = R, and let F be( a closed set ) in R. 0 There exists a smooth a 0 such that F = a ({0}). Then for every a(x) 0 point in {0} F is a coalescing point in the spectrum. Proposition 3. If Ω R, or if Ω is an open subset of C, and if M(x) is a polynomial in x Ω, then the spectrum has a finite number of coalescing points. Proof. We may work with irreducible components Π j of the characteristic polynomial Π (a polynomial in two variables, λ and x). For every such component, Π j and λ Π j are relatively prime. In particular (see for instance Theorem 3 in chapter 8 of []), there is a finite number of x such that Π j (, x) and λ Π j (, x) have a common root λ(x). These common roots (x, λ(x)) are precisely the coalescing points in the spectrum. We say that (λ, x) is a regular point in the spectrum (of M, ()) if either it has constant multiplicity or it is an isolated coalescing point, in the sense that there exists a neighborhood U of (λ, x) in C Ω such that ( U \ {(λ, x)} ) S comprises only points of constant multiplicity.

3 BASIC MATRIX PERTURBATION THEORY 3 Corollary 4. If (λ 0, x 0 ) is an isolated coalescing point in the spectrum, then if ε > 0 is small enough, the restriction of the projection π : π (B(x 0, ε) ) B(x 0, ε) is a covering map, where π (B(x 0, ε) ) is the inverse image of the ball B(x 0, ε). Proof. Identical to the proof of Corollary 2, since the fibers above the (connected) punctured ball have constant cardinal. At a coalescing point in the spectrum, eigenvalues may fail to be differentiable, even if M is smooth. The canonical example is ( ) 0 (3), x 0. x 0 2. Cauchy formulas We use notation S for the spectrum of the family of matrices M, and the notion of regular point in the spectrum, as defined in Section. Proposition 5 (Cauchy formula for total eigenprojectors). Let (λ 0, x 0 ) be a regular point in S, and a closed, positively oriented curve in C, which does not intersect sp M(x 0 ), and the interior of which intersects sp M(x 0 ) at λ 0 only. Then for x close to x 0, (4) P (x) = (λ M(x)) dλ is the sum of the projectors onto the generalized eigenspaces associated with eigenvalues of M(x) which lie in the interior of. In particular, P is as regular as M. Above and below, projectors onto generalized eigenspaces (equivalently, generalized eigenprojectors) are implicitly parallel to the direct sum of the other generalized eigenspaces. Proof. For fixed x near x 0, consider a spectral decomposition of M(x) : (5) M(x) = (λ j (x) + N j (x))p j (x), j J(x) where the λ j are distinct eigenvalues, the P j are projectors onto generalized eigenspaces, such that (6) Id = P j (x), j J(x) and the N j are the associated nilpotent components, such that N j P j = P j N j. The integer J(x) is the total number of distinct eigenvalues of M(x) near x 0. Since (λ 0, x 0 ) is regular, for some neighborhood U of x 0 the number of distinct eigenvalues near (λ 0, x 0 ) is constant on U \ {x 0 }. Let m J(x) be this number, and, for all x U \ {x 0 }, let λ (x),..., λ m (x) be the distinct eigenvalues of M(x) near λ 0. By (5) and (6), there holds for x U and λ / sp M(x) : (7) (λ M(x)) = ( λ λj (x) N j (x) ) Pj (x), j J(x)

4 4 BENJAMIN TEXIER which we may rewrite, the matrix Id µn j being invertible for all µ : (λ M(x)) = j J(x) and, expanding in inverse powers of λ λ j (x), (8) ( λ λj (x) ) ( ) Pj Id (λ λ j (x)) N j (x) (x), (λ M(x)) = ( (λ λ j (x) ) + O(λ λj (x)) 2) P j (x). j J(x) Given now a closed curve as in the statement of the Proposition, for x close enough to x 0, and different from x 0, the spectrum of M(x) does not intersect, and intersects the interior of on exactly the set {λ (x),..., λ m (x)}. We compute residues: (λ λ j (x)) P j (x) dλ = P j (x), j m, (λ λ j (x)) P j (x) dλ = 0, m + j J(x), O(λ λ j (x)) 2 P j (x) dλ = 0, for all j. Thus P = j P j satisfies representation (4). Corollary 6. Around a point (λ 0, x 0 ) of constant multiplicity in the spectrum, the associated eigenvalue and generalized eigenprojector are as regular as M, and there holds (9) (λ(x) + N(x))P (x) = λ(λ M(x)) dλ, where x λ(x) is the local branch of eigenvalues such that λ(x 0 ) = λ 0, P is the associated projector, and N the associated nilpotent. Note that in the case m =, regularity of the eigenvalue follows directly from the implicit function theorem, since λ 0 is then a simple root of the characteristic polynomial of M. Proof. The constant multiplicity hypothesis implies that the total eigenprojector P (x) from Proposition 5 is the generalized eigenprojector onto the unique eigenvalue λ(x) of M(x) near λ 0. Thus P is as regular as M. Next we use a spectral decomposition of M(x) in order to express λ(λ M(x)), for λ C, as a sum of projectors, as we did in (7) in the proof of Proposition 5: λ(λ M(x)) = λ ( λ λ(x) N(x) ) P (x) + ( λ λj (x) N j (x) ) Pj (x), 2 j J(x)

5 BASIC MATRIX PERTURBATION THEORY 5 where λ(x) is the eigenvalue of M(x) which is equal to λ 0 at x 0, and the λ j (x) are the other eigenvalues of M(x). In particular, near x 0, the eigenvalues λ j (x) are far from λ 0. Then, if the interior of contains λ 0 and is small enough, there holds (0) λ(λ M(x)) dλ = λ(λ λ(x) N(x)) P (x) dλ. We now expand the integrand in the above right-hand side in powers of (λ λ(x)), going here one step further than in (8) because of the extra λ prefactor: λ(λ λ(x) N(x)) = λ(λ λ(x)) + λ(λ λ(x)) 2 N(x) + λo(λ λ(x)) 3. We compute residues: λ(λ λ(x)) P (x) dλ = λ(x)p (x), λ(λ λ(x)) 2 N(x)P (x) dλ = N(x)P (x), and λo(λ λ(x)) 3 P (x) dλ = 0. With (0), this implies representation (9), from which we deduce that the map x (λ(x)+n(x))p (x) is as regular as M. Taking the trace, we find that x mλ(x) is as regular as M, where m is the multiplicity of λ. Corollary 7. If (λ 0, x 0 ) is an isolated coalescing point in the spectrum, with multiplicity m >, then there holds () (λ j (x) + N j (x))p j (x) = λ(λ M(x)) dλ, j m where x λ j (x), for m m, are the distinct branches of eigenvalues such that λ j (x 0 ) = λ 0, the matrices P j are the associated projectors, and N j the associated nilpotents. Proof. For all x U \ {x 0 }, where U is some neighborhood of x 0, the matrix M(x) has the same number of distinct eigenvalues in a neighborhood of λ 0. This number is less than or equal to m, the multiplicity of λ 0. Let λ,..., λ m be these eigenvalues. It suffices to reproduce the computations of the proof of Corollary 6, where each λ j plays the same role as λ in the proof of Corollary 6, to arrive at (). 3. Regularity of the eigenvalues Proposition 8. If M is differentiable at x 0, then for any local branch λ of eigenvalues of M around x 0, there holds the bound (2) λ(x) λ(x 0 ) C(M) x x 0 /m, locally around x 0, with C(M) > 0, where m is the multiplicity of (λ(x 0 ), x 0 ).

6 6 BENJAMIN TEXIER If (λ(x 0 ), x 0 ) has constant multiplicity and M is locally Lipschitz, then by Corollary 6 the eigenvalues are actually Lipschitz, locally around x 0, which of course is much better than (2) in the case m >. Estimate (2) however accurately describes the eigenvalue behavior in the canonical coalescing case (3). Proof of Proposition 8. We denote λ 0 = λ(x 0 ). The characteristic polynomial Π(λ, x) = det ( λ M(x) ) factorizes into Π = Π 0 Π, where Π (λ 0, x 0 ) 0, and Π 0 (λ, x 0 ) = (λ λ 0 ) m. The degree of Π 0 is equal to m, the multiplicity of (λ 0, x 0 ), and Π 0 is unitary. We may focus on Π 0 in the following. Let λ be a branch of eigenvalues such that λ(x 0 ) = λ 0. Expanding Π 0 in powers of λ(x) λ 0, we find, since j λ Π 0(λ 0, x 0 ) = 0 for 0 j m : Π 0 (λ(x), x 0 ) = m! (λ(x) λ 0 ) m + O( λ(x) λ 0 ) m+. Besides, the matrices M being differentiable at x 0, the characteristic polynomial Π is differentiable in x at x 0, and so is Π 0 : Thus which implies (2). Π 0 (λ(x), x) = Π 0 (λ(x), x 0 ) + O( x x 0 ) 0. m! (λ(x) λ 0 ) m + O( λ(x) λ 0 ) m+ = O( x x 0 ), The estimate of Proposition 8 is much improved in the semi-simple case: Proposition 9. If M is differentiable at x 0, and if (λ 0, x 0 ) is an isolated coalescing point such that λ 0 is a semi-simple eigenvalue of M(x 0 ), any local branch λ of eigenvalues of M around x 0 has a one-sided directional derivative in every direction, and there holds, for all e R d, λ(x 0 + t e ) λ(x 0 ) lim t 0 t t>0 In particular, the eigenvalue are Lipschitz: locally around x 0, with C(M) > 0. sp P (λ 0, x 0 )M (x 0 ) e P (λ 0, x 0 ). λ(x) λ(x 0 ) C(M) x x 0, Proof. We denote λ 0 = λ(x 0 ). Let m be the multiplicity of λ 0, and λ,..., λ m, 2 m m, the distinct eigenvalues that coalesce at x 0 with value λ 0. By Corollary 7, there holds λ ( λ M(x 0 + h) ) dλ = ( λj (h) + Ñj(h) ) Pj (h), j m where h R d is small and is a suitable curve in C. Above, λ j (h) := λ j (x 0 + h), and Ñj and P j are the corresponding nilpotent and projector. By Proposition 5,

7 there holds Thus (3) BASIC MATRIX PERTURBATION THEORY 7 λ 0 (λ M(x 0 + h) dλ = P (h) = (λ λ 0 ) ( λ M(x 0 + h) ) dλ = j m j m Pj (h). ( λj (h) λ 0 + Ñj(h) ) Pj (t), By differentiability of M at x 0 : ( λ M(x0 + h) ) = (λ M(x0 )) + O(h), the O(h) term being precisely (λ M(x 0 )) M (x 0 ) h(λ M(x 0 )) + o(h). Since λ 0 is semi-simple, the spectral decomposition at x 0 is M(x 0 ) = λ 0 P (λ 0, x 0 ) + M(x 0 )(Id P (λ 0, x 0 )), where P (λ 0, x 0 ) is the generalized eigenprojector. Thus (λ M(x 0 )) = (λ λ 0 ) P (λ 0, x 0 ) + (λ M(x 0 )) (Id P (λ 0, x 0 )), and, computing residues, we find for a suitable curve : (λ λ 0 )(λ M(x 0 )) dλ = 0, and (λ λ 0 )(λ M(x 0 )) (Id P (λ 0, x 0 )) dλ = 0. From (3) and the above, we deduce ( ) λ j (x 0 + h) λ 0 + Ñj(h) P j (h) h j m = P (λ 0, x 0 )M (x 0 ) h h P (λ 0, x 0 ) + o(), Equating spectra, evaluating at h = t e, for t > 0, and taking the limit t 0 implies the result. We could trade in Proposition 9 the assumption that (λ 0, x 0 ) is an isolated coalescing point for an assumption of regularity of M, by reasoning as in the proof of the upcoming Proposition.

8 8 BENJAMIN TEXIER 4. Puiseux expansions We describe eigenvalues around a coalescing point, following the approach of [9]. Consider a point (λ 0, x 0 ) with multiplicity m in the spectrum, and suppose that M is q m times differentiable at x 0, so that M(x) = M 0 (x x 0 ) + o(x x 0 ) q, where M 0 is a polynomial of degree q. In particular, M 0 has an extension to C d. Let e R d be a fixed spatial direction, and consider S 0 := { (λ, z) C B(0, ε), det(m 0 (x 0 + z e) λid) = 0 }, where B(0, ε) C is the open disk centered at 0 and with radius ε > 0 in the complex plane. We denote π 0 the projection π 0 : (λ, z) S 0 z B(0, ε). By Proposition 3, if ε is small enough then S 0 has only (λ 0, 0) has a coalescing point. Thus by Corollary 4, the restriction of π 0 to π0 (B(0, ε) ) is a covering map if ε is small enough. Let V be a connected component of S 0 which intersects R +. We denote π 0 the restriction π 0 : (λ, z) V π 0 (B(0, ε) ) z B(0, ε). Then π 0 is a connected covering of the punctured disk B(0, ε), in the sense that the total space V π 0 (B(0, ε) ) is connected. Lemma 0. The covering map π 0 is conjugated to the covering p : z z m of B(0, ε) for some m m. That is, there exists a homeomorphism ψ such that the following diagram is commutative: V π 0 (B(0, ε) ) ψ p (B(0, ε) ) π 0 p B(0, ε) Proof. Let λ (x 0 + z e),..., λ m (x 0 + z e) be the eigenvalues of M 0 which takes values in V 0 for z B(0, ε). The numbers of these eigenvalues is constant over the connected punctured disk B(0, ε). In particular, π 0 is an m -sheeted covering of B(0, ε). Connected coverings of a punctured ball in C are determined, up to isomorphism, by their numbers of sheets (see for instance [6], chapter V, Theorem 6.6). Thus π 0 is conjugated to p, by a homeomorphism ψ. Based on Lemma 0, we may give Puiseux expansions of eigenvalues around a coalescing point: Proposition. If (λ 0, x 0 ) is a coalescing point in the spectrum of M, with multiplicity m, and if M is q m times differentiable at x 0, then for any local branch λ of eigenvalues of M which coalesce at x 0 with value λ 0, any e R d, there exists

9 BASIC MATRIX PERTURBATION THEORY 9 a smooth map φ defined in [0, t 0 ], for some t 0 > 0, and a positive integer m m, where m is the multiplicity of λ 0, such that (4) λ(x 0 + t e ) = φ(t /m ) + o(t q/m ), for 0 t t 0. Proof. Let M(x) = M 0 (x x 0 ) + x x 0 q R(x 0, x), be the Taylor expansion of M around x 0 at order q. The family of matrices R(x 0, x) converges to 0 as x x 0. Associated with the family of matrix polynomials M 0, consider the spectrum S 0 and the connected covering π 0, as described above Lemma 0. Let µ be a section of π 0, that is a branch of eigenvalues of M 0 (x 0 + z e). There holds π 0 (µ) Id, hence, by Lemma 0, p ψ µ Id. Thus ψ µ is a section of p, meaning an m -th root of unity: (5) µ(z) = φ(ωz /m ), where φ is the first component of ψ, and ω is a given m -th root of unity. Equality (5) holds for all z π 0 (V ). By definition, V is a connected component of the total space p (B(0, ε) ) which intersects R +. If t 0 > 0 belongs to π 0 (V ), then all of the arc { (µ(t), t), 0 < t < t 0 } belongs to V, by connectedness of V and continuity of µ. Thus t belongs to π 0 (V ), if t > 0 is small enough, and equality (5) holds for small enough t > 0. In particular, µ(t m ) = φ(ωt), for 0 < t t 0. implying that φ is as regular as µ, hence analytical. Thus φ is analytical in 0 < t < t 0 and bounded around t = 0, implying that φ is analytical in [0, t 0 ], so that (5) holds for all 0 t t 0, with µ(0) = φ(0). Let now M(x, y) = M 0 (x x 0 ) + y, y C N 2, and let λ be a local branch of eigenvalues of M, such that, for small t, λ(x 0 + t e, 0) = µ(t), and for small x x 0 : ( ) λ x 0 + t e, x x 0 q R(x 0, x) = λ(x 0 + t e ), where λ is a local branch of coalescing eigenvalues of M. By Proposition 8 (where the variable y C N 2 is seen as a real variable y R 2N 2 ), there holds λ(x 0 + t e, y) λ(x 0 + t e, 0) = O( y ) /m. Thus λ(x 0 + t e ) = µ(t) + O ( x x 0 q R(x 0, x ) ) /m, which implies (4), via (5), since R = o().

10 0 BENJAMIN TEXIER Bibliographical note. The Cauchy formula of Proposition 5 is found in equation (.6), paragraph.4, chapter 2, in [3]. The existence of directional derivatives (Proposition 9) is found in Theorem 2.3, paragraph 2.3, chapter 2, in [3]. Kato refers to Knopp [4], without proof, for details on Puiseux expansions (see [3], chapter 2, paragraph.2). So do Reed and Simon ([7], XII.). Knopp s discussion is limited to polynomials in two variables, the roots of which are described as multivalued analytical functions; here eigenvalues around a coalescing point are seen as (perturbations of) sections of a ramified covering of a disk in the complex plane. Remark 2 (On hyperbolic polynomials). If the spectrum of M(x) is real for all x Ω, then the family M is said to be hyperbolic. The eigenvalues are then locally Lipschitz; see Brohnstein [2], or Kurdyka and Paunescu [5]. In one space dimension, Rellich s theorem [8] states that analytic family of hermitian matrices have analytic eigenvalues and eigenvectors. Remark 3 (On geometric optics). An important consequence of Proposition 9 is that the amplitude of a wave-packet is transported by a hyperbolic system at group velocity; this is a crucial step in the derivation of amplitude equations in geometric optics, see [9] and references therein. Similar formulas exist for higher derivatives (see [9], Proposition 2.6 and Remark 2.7, and Kato [3], paragraphs 2. and 2.2, chapter 2). The corresponding identity for second-order derivatives describes the Schrödinger correction to the transport along rays for distances of propagation equal to the inverse of the wavelength. References [] L. Ahlfors. Complex analysis. Second edition. McGraw-Hill Book Co. New York-Toronto- London 966 xiii+37 pp. [2] M. D. Brohnstein. Smoothness of roots depending on parameter, Siberian Math. J. 20 (979), [3] T. Kato. Perturbation theory for linear operators. Reprint of the 980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 995. xxii+69 pp. [4] K. Knopp. Theory of Functions. II. Applications and Continuation of the General Theory. Dover Publications, New York, 947. x+50 pp. [5] K. Kurdyka, L. Paunescu. Hyperbolic polynomials and multiparameter real analytic perturbation theory, Duke Math. J. 4 (2008), [6] W. S. Massey. A basic course in algebraic topology. Graduate Texts in Mathematics, 27. Springer-Verlag, New York, 99. xvi+428 pp [7] M. Reed, B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 978. xv+396 pp. [8] F. Rellich. Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz. Gordon and Breach Science Publishers, New York-London- Paris 969 x+27 pp. [9] B. Texier. The short-wave limit for nonlinear symmetric hyperbolic systems. Adv. Diff. Eq. 9 (2004), no., -52. Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR CNRS 7586, Université Paris-Diderot address: benjamin.texier@imj-prg.fr