An explicit Jordan Decomposition of Companion matrices

Transcription

1 An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois Tououse Cedex 01 France E-mai: gratton@cerfacsfr Abstract We derive a cosed form for the Jordan decomposition of companion matrices with emphasis on properties of generaized eigenvectors As a by product we provide a formua for the inverse of confuent Vandermonde matrices and resuts on sensitivity of roots of poynomias 1 Introduction We are concerned with companion matrices of the form a a 1 C = a 2 (1) a m 1 where a i IC i = 0 m 1 Matrices ike this appear in a variety of areas in science and engineering [ ] There is a cose reationship between companion matrices and poynomias in a compex variabe Of course roots of poynomias can be computed as eigenvaues of C and vice versa This reies on the fact that the characteristic poynomia of C det(ti C) = π(t) is readiy proved to be π(t) = a 0 + a 1 t + + a m 1 t m 1 + t m We denote by λ 1 λ p the p distinct eigenvaues of C and by m 1 m p their respective agebraic mutipicities (ie π(t) = (t λ 1 ) m1 (t λ 2 ) m2 (t λ p ) mp with m m p = m) The fact that C is a non derogatory matrix [8 10] ensures thus that a particuar Jordan decomposition of C can be written as J λ1 J λp = L 1 L p C [ R 1 R p ] where for i = 1 p J λi = L 1 L p λ i 1 1 λ i ICmi m i [ R 1 R p ] = I (2) where I is the m m identity matrix R i IC m mi and L i IC mi m The coumns of R i (resp L i ) represent a right (resp eft ) Jordan chain associated with λ i the eading eigenvector being R i e [mi] 1 (resp L e [mi] m i ) The star symbo denotes conjugate transpose ie L = L T and e [mi] j is the j-th coumn of the m i m i identity matrix In this work we describe a cosed form for the Jordan decomposition of C concentrating on properties of generaized eigenvectors This eads to a formua for the inverse of Confuent Vandermonde matrices and resuts on the sensitivity of the roots of π(t) 2 Jordan Decomposition of Companion Matrices In this section we provide an expicit Jordan decomposition of C The foowing technica resut wi be needed Lemma 21 For arbitrary λ IC we set φ(λ) = [1 λ λ m 1 ] T and define by φ (m) (λ) the m-th derivative of φ(λ) with respect to λ Let H be the m m matrix

2 a 1 a 2 a m 1 1 a 2 a m 1 1 H = 1 (3) a m Then for any integers i and j there hods φ (i)t (λ) = π(i+j+1) (λ) (i + j + 1)! Proof The proof is done by induction on i For i = 0 an eementary computation shows that φ T (λ) Assume now that for a given i φ (i)t (λ) = π(j+1) (λ) (j + 1)! = π(i+j+1) (λ) (i + j + 1)! Taking the derivative of the above expression with respect to λ yieds φ (i+1)t (λ) φ (i+1)t (λ) which shows that φ (i+1)t (λ) + φ(i)t (λ) H φ(j+1) (λ) + (j + 1) π(i+j+2) (λ) (i + j + 2)! = π (i+j+2) (λ) (i + j + 1)! = π(i+j+2) (λ) (i + j + 1)! (j + 1) π(i+j+2) (λ) (i + j + 2)! = (i + 1)π(i+j+2) (λ) (i + j + 2)! which competes the proof Proposition 22 Define R = [r 1 r 2 r m ] where r i = H φ(i 1) (λ ) (i 1)! The set {r 1 r 2 r m } is a right Jordan chain of C associated with the eigenvaue λ and r 1 is the eading right eigenvector Simiary define L = [ 1 2 m ] where i = φ (m i) (λ ) (m i)! The set { 1 2 m } is a eft Jordan chain of C associated with the eigenvaue λ and m is the eading eft eigenvector The eft and right generaized Jordan chains satisfy 1 L R [ ] r 1 r m = m α 1 α 2 α m 1 α m α m 1 α 2 F IC m m α 1 = (4) where α i = π(m +i 1) (λ ) (m + i 1)! Proof For arbitrary λ of mutipicity q consider the vectors r 1 r q It is cear that these vectors are ineary independent Thus if we set r 0 = 0 we have to prove that r 1 is a right eigenvector of C associated with λ and that (C λi)r j = r j 1 1 j q (5) For this if x = [x 1 x m ] T is a right eigenvector of C associated with λ then a 0 x m = λx 1 x 1 a 1 x m = λx 2 Cx = λx x m 2 a m 2 x m = λx m 1 x m 1 a m 1 x m = λx m This shows that x m cannot be zero otherwise x woud be the 0 vector Setting x m = 1 it is easy to see x = Hφ(λ) Thus one has CHφ(λ) = λhφ(λ) (6) We now prove the conditions (5) Taking derivative with respect to λ in (6) we have CHφ (1) (λ) = Hφ(λ) + λhφ (1) (λ) (7) This shows that (5) hods for j = 2 and an inductive argument obtained by repeated differentiation of (7) concudes the proof in the case of the right generaized eigenvectors A simiar proof can be obtained for the generaized eft eigenvectors by starting with φ(λ) T C T = φ(λ) T λ instead of (6) and taking the derivatives of this equaity The normaization factors α i are a consequence of Lemma 21 To obtain the Jordan decomposition we transform the eft Jordan chain so that the normaization (2) hods Proposition 23 Define L = [ 1 2 m ] = L F The set { 1 2 m } is a eft Jordan chain of C associated with the eigenvaue λ m being the eading eft eigenvector The eft and right generaized Jordan chains are normaized so that L R 1 m [ r 1 r m ] = I IR m m (8) Simiary we define R = [ r 1 r 2 r m ] = [r 1 r m ]F 1 The set { r 1 r 2 r m } is a right Jordan chain of C associated with the eigenvaue λ and L R = I R m m

3 Proof Let γ i be defined by the recursion γ 1 = 1/α 1 γ i+1 = 1 α 1 i k=1 α i k+2γ k i = 1 m 1 in such a way that γ 1 γ 2 γ m 1 γ m γ m 1 G = γ 2 = F 1 γ 1 The set { m 1 } forms a right Jordan chain of C T associated with λ For any nonsinguar matrix X commuting with J λ [ m 1 ]X is a right Jordan chain C T associated with λ By definition of the i s [ ] [ ] m 1 = m 1 Ḡ A direct computation shows that G commutes with J λ which impies that { m 1 } is a right Jordan chain of C T associated with λ that is { m } is a eft Jordan chain of C associated with λ Since by definition of the γ i s GF = F G = I it foows 1 [r 1 r m ] = G 1 [r 1 r m ] = I m m and the first part of the proposition is proved The proof of the remaining part is a consequence of Eq (4) since [r 1 r m ]F 1 is a right Jordan chain of C associated with λ as we have seen that F 1 commutes with J λ 3 Eigenvector properties An immediate consequence of (8) is an expicit formua for computing the inverse of confuent Vandermonde matrices as described in the coroary beow Coroary 31 (Inversion formua) Let L be the confuent Vandermonde matrix defined by L = [ L 1 L p ] Then L 1 = [R 1 R p ]F 1 with F = diag(f 1 F p ) It is known that right eigenvectors of companion matrices ike the one we use here can be computed by appying the Eucidean agorithm to divide π(t) by (t λ ) (see eg Toh and Trefethen [6] or Bezerra and Bazán [3]) yieding in our notations a defated poynomia π 1 (t) φ(t) T r 1 = π(t)/(t λ ) This shows that the coefficients of the right eigenvector can be seen as coefficients of a defated poynomia The foowing proposition shows that the compete right Jordan chain [r 1 r m ] aso enjoy this property Proposition 32 Define π i (t) = φ(t) T r i (i = 1 m ) where r i are generaized right eigenvectors of C as introduced in Prop 22 Then π i is a monic poynomia of degree m i of the form π i (t) = (t λ ) m i p (t λ j ) mj (9) j=1 j Proof: It is cear that a π i are monic poynomias of degree m i The definition of the r i s and successive differentiation impy π 1 (t) = φ T (t) Hφ(λ ) π (1) 1 (t) = φ(1)t (t) Hφ(λ ) π (i) 1 (t) = φ(i)t (t) Hφ(λ ) π (m 1) 1 (t) = φ (m 1) T (t) Hφ(λ ) (10) If t = λ Prop 21 impies that (10) can be rewritten as π i (λ ) = π (i 1) 1 (λ ) = (i 1)!φ T (λ )H φ(i 1) (λ ) (i 1)! = (i 1)!π i 1 (λ ) i = 1 m But since λ is a mutipe root of π this equaity impies that λ is a root of π i (i = 1 m 1) and a recursive argument shows that this root is of mutipicity m i If t = λ k λ a simiar procedure and the existing biorthogonaity condition between eft and right generaized eigenvectors eads to π i (λ k ) which concudes the proof = π (i 1) 1 (λ k ) = (i 1)!φ T (λ )H φ(i 1) (λ k ) (i 1)! = 0 i = 1 m Remark A comment concerning the meaning of this proposition is in order Let C i (i = 1 m 1) denote the (m i) (m i) companion matrix associated with the poynomia π i and for i = 1 m et ř i be the vector formed by taking the first m i+1 components of r i Then with the convention that C 0 = C the proposition ensures that C i 1 ř i = λ ř i i = 1 m (11) and λ is a simpe eigenvaue of the companion matrix C m 1 For future reference the eft eigenvector of C m 1 wi be denoted by ψ(λ ) It is defined by ψ(λ ) = [1 λ λ m m ] T (12) 31 Numerica iustration: Jordan decomposition We present an iustration of the above notions for m = 5 (λ 1 m 1 ) = (1 2) (λ 2 m 2 ) = (2 2)

4 (λ 3 m 3 ) = (3 1) in which case π(t) = (t 1) 2 (t 2) 2 (t 3) We show how to obtain easiy a Jordan form of the companion matrix associated with π Note that π(t) = t 5 9t t 3 51t t 12 Case of λ = 1 m 1 = 2 From (t 2) 2 (t 3) = t 3 7t t 12 and (t 1)(t 2) 2 (t 3) = t 4 8t t 2 28t + 12 foows using Proposition 32 and the definition of the i s that R 1 = and L 1 = From π (2) (1)/2 = 2 and π (3) (1)/6 = 5 we obtain ( ) 2 5 F 1 = F = 1 ( ) and L 1 = L 1 F = The same cacuation for the two remaining roots gives and R = [R 1 R 2 R 3 ] = L = [L 1 L 2 L 3] = yieding RJL = C where J (a Jordan matrix) and C are of the form J = C = which is a Jordan decomposition as expected 4 Condition estimation We sha anayze the sensitivity of the roots of π(t) to perturbations in the coefficients a j viewing the roots as eigenvaues of the associated companion matrix C Let π(t) denote the perturbed monic poynomia with coefficients ã j = a j + a j and et C denote its associated companion matrix Then depending on the way the perturbations ã j are measured different condition numbers for λ can be obtained Suppose for instance that the a j s are assumed to satisfy the componentwise inequaities a j ɛα j j = 1 m 1 (13) where α j are arbitrary non negative rea numbers Simiary we denote by λ j j = 1 d the eigenvaues of C corresponding to λ for ɛ sma enough We set λ = max j=1d λ λ j For the so-caed componentwise mode of perturbations defined by (13) we have the definition beow where for simpicity λ and its corresponding mutipicity m wi be denoted by λ and d respectivey Definition 41 ([4]) The componentwise reative condition number of the root λ of mutipicity d is defined by κ C (λ) = im ɛ 0 sup a j ɛα j λ (14) ɛ1/d A precise description of this condition number is the subject of the foowing proposition Proposition 42 Suppose the perturbations a j satisfy (13) Then the componentwise reative condition number of the root λ of mutipicity d κ C (λ) is κ C (λ) = 1 m 1 d! λ j α j j=0 π (d) (λ) 1/d (15) Proof A sketch of the proof is as foows Let { λ r} be a right eigenpair of C = C + C with λ = λ + λ r = r + r Then (C + C)(r + r) = (λ + λ)(r + r) iff C r + Cr + C r = λ r + λr + λ r Using the fact that the right eigenvector r of C is (see Prop 22)r+ r = Hφ( λ) where H = H + H has the same structure as H but with entries ã j = a j + a j it can be proved that the m-th component of r equas zero Next use this observation and the fact that C = ae [m] m where a = [ a 0 a 1 a m 1 ] T and e [m] m is the m-th canonica vector in IR m to prove that C r = 0 (16) Some agebraic manipuations ead then to the foowing first order resut

5 from which the proof foows λ d = d! φt (λ) a (17) π (d) (λ) Remark 1 Another condition number for λ can be readiy obtained if the perturbations are assumed to satisfy a 2 δα (18) where α is an arbitrary positive rea number (eg α = a 2 ) This gives rise to the so-caed normwise reative condition number κ(λ) It is immediate that κ(λ) = 1 ( ) 1/d d! φ(λ) α (19) π (d) (λ) Remark 2 When the perturbations are measured in a normwise absoute sense ie when α = 1 in (18) a simiar procedure eads to the so-caed normwise absoute condition κ a (λ) which can be shown to be κ a (λ) = [ ] 1/d d! φ(λ) 2 (20) π (d) (λ) Proposition 43 Assume the same hypothesis as in the previous proposition Let sec Θ λ be the secant of the ange between the eading eft generaized eigenvector and the ast right generaized eigenvector associated with the eigenvaue λ as defined in Prop 22 ie and define sec Θ λ = φ(λ) 2 r d 2 φ T (λ)r d (21) ω k = 2 m d + k + 2 cos( ) k = 1 d m d + k + 1 Assume aso that the perturbation a satisfy the mode (18) with α = a 2 Then the condition number κ(λ) given in (19) satisfies κ(λ) 1+ ( a a 2 2 ) 1 2d [sec Θ λ ] 1/d (22) Proof: Omitted because of space imitation There exists a cose reationship between sec Θ λ and the Wikinson number of λ when regarded as eigenvaue (simpe) of the companion matrix C d 1 in (11) (or equivaenty as simpe root of π d 1 (t) see the remark after Prop 32) This can be seen as foows Since λ is an eigenvaue simpe of this matrix the Wikinson condition number of λ is [12] κ W (λ) = ψ(λ) 2 ř d 2 ψ (λ)ř d From this because φ T (λ)r d = ψ (λ)ř d (see (11) again) it is cear that sec Θ λ = κ W (λ) φ(λ) 2 ψ(λ) 2 (23) It turns out that if the the mutipicity d is not arge and is a moderate number then sec Θ λ κ W (λ) in which case κ(λ) essentiay depends on κ W (λ) Thus if λ is a we conditioned eigenvaue of the defated companion matrix C d 1 (or equivaenty a we conditioned simpe root of π d 1 (t) ) and the ratio φ(λ) 2 / ψ(λ) 2 is rather sma then moderate vaues for κ(λ) may be expected However even if κ(λ) is sma the error in λ wi be determined by the mutipicity d and the size of a j In genera if the perturbations a j are sma enough the reative error in λ can be estimated by the rue λ whie the absoute error by κ(λ)δ 1/d (24) λ κ a (λ)δ 1/d (25) 41 Numerica iustration: Condition estimation We consider the poynomia π(t) of degree m = 20 defined by π(t) = (t λ) 5 (1 + t + + t 15 ) with λ = (1 + 9s) + si 0 s 2 This exampe is designed to iustrate the roe of the defated poynomia π d 1 (in this case d = 5 see Prop 32) in estimating the sensitivity of a mutipe root In fact as in in this case the defated poynomia π d 1 (t) = (t λ)(1+t+ +t 15 ) reduces to the poynomia t 16 1 when λ = 1 a roots of which are known to be extremey we-conditioned [7 Exampe 43] sma condition condition numbers for the mutipe root λ (as root of π(t)) can be expected provided that λ 1 the conditioning being more favorabe for the (simpe) roots of π(t) (the roots of 1 + t + + t 15 ) Indeed if the simpe roots of π(t) are denoted by λ k it is not difficut to prove that κ a (λ) = ( ) k=1 λ λ k 02 whereas for simpe roots one has 5 λ k 1 κ a ( λ k ) = 8 λ k λ 5 Some numerica resuts dispayed in Tabes 1 and 2 corresponding to severa λ s confirm the theoretica prediction The tabes incude condition numbers the predicted eigenvaue errors described in

6 (24) and (25) and the ratio ρ = φ(λ) 2 / ψ(λ) 2 Aso and mainy to verify the theoretica prediction of the error approximate roots obtained from poynomias with coefficients ã j = a j + a j where a j are random numbers satisfying a normwise reative error δ = are dispayed in Figure 1 A computations were performed using MATLAB λ κ(λ) κ a(λ) κ W ı 13169e e e ı 10724e e e ı 74747e e e ı 39220e e e ı 15800e e e e e e + 0 Tabe 1: Condition numbers [4] F Chaitin-Chatein and V Frayssé Lectures on Finite Precision Computations SIAM Phiadephia 1996 [5] M I Friswe U Pres and S D Garvey Lowrank damping modifications and defective systems Journa of Sound and Vibration Vo 279 pp January 2005 [6] K-C Toh and Loyd N Trefethen Pseudozeros of poynomias and pseudospectra of companion matrices Numer Math [7] W Gautschi Questions of numerica condition reated to poynomias in MAAA Studies in Mathematics Vo 24 Studies in Numerica Anaysis G H Goub ed USA 1984 The Mathematica Association of America pp λ ρ λ / λ ı 13322e e e ı 51642e e e ı 10201e e e ı 63756e e e ı 44310e e e e e e 2 Tabe 2: Ratio ρ and predicted errors The resuts confirm that moderate vaues of κ(λ) do not necessariy impy sma eigenvaue errors when the mutipicity is rather arge and that reasonaby sma errors can be expected when both the Wikinson condition number κ W (λ) and the ratio ρ are sma Further the theoretica prediction of the eigenvaue error according to (25) dispayed in coumns 3 and 4 of Tabe 2 is verified to be consistent with the numerica resuts as dispayed in Figure 1 The reative insensitivity of simpe roots is aso apparent as predicted [8] G H Goub and C F Van Loan Matrix Computations The Johns Hopkins University Press Batimore 1996 [9] P de Groen and B de Moor The fit of a sum of exponentias to noisy data Comput App Math [10] R Horn and Ch R Johnson Matrix Anaysis Cambridge University Press 1999 [11] H M Möer and J Stetter Mutivariate poynomia equations with mutipe zeros soved by matrix eigenprobems Numer Math Vo 70 pp [12] J H Wikinson The Agebraic Eigenvaue Probem Oxford University Press Oxford UK References [1] F S V Bazán and Ph L Toint Error anaysis of signa zeros from a reated companion matrix eigenvaue probem Appied Mathematics Letters 14 (2001) [2] F S V Bazán Error anaysis of signa zeros: a projected companion matrix approach Linear Agebra App 369 (2003) [3] L H Bezerra and F S V Bazán Eigenvaue ocations of generaized predictor companion matrices SIAM J Matrix Ana App 19 (4) October 1998 pp Figure 1: Case 1: λ = ı : Exact eigenvaue : Approximate eigenvaue Case 2: λ = 10 + ı : Exact eigenvaue + : Approximate eigenvaue