泰勒公式及應用翻譯(原文)

1、精品文檔精品文檔On Taylor s formula for the resolvent of a complex matrixMatthew X. He a, Paolo E. Riccib,_Article history:Received 25 June 2007Received in revised form 14 March 2008Accepted 25 March 2008Keywords:Powers of a matrixMatrix inv aria ntsResolve ntIntroductionAs a con seque nee of the Hilbert id

2、e ntity in 1, the resolve nt R,(A)= (，；. _A)of a non sin gular square matrix A(上 deno ti ng the ide ntity matrix) is show n to be an an alytic fun ctio n of the parameter - in any doma in D with empty intersection with the spectrum v A of A. Therefore, by using Taylor expansion in a neighborhood of

3、any fixed 0 D , we can find in 1 a representation formula for R.(A) using all powers of R.0(A).In this article, by using some precedi ng results recalled, e.g., i n 2, we write dow n a representation formula using only a finite number of powers of R.0( A). This seems to be natural since only the fir

4、st powers of R.0(A) are linearly independent.The ma in tool in this framework is give n by the multivariable poly no mials Fk,n (v1, v2,.,vr) (n= -1,0,1，；k = 1,2,，m 玄 r ) (see 2-6), depe nding on the invariants (w,v2,.,vr) of R.(A); here m denotes the degree of the minimal polyno mial.Powers of matr

5、ices ad Fk,n functionsWe recall in this sect ion some results on represe ntati on formulas for powers of matrices (see e.g. 2-6 and the references therein). For simplicity we refer to the case whe n the matrix is non derogatory so that m = r.Propositi on 2.1. Let A be a nr r(r 2) complex matrix, and

6、 denote by u1,u2,.,ur the invariants of A, and byrP() =det(； - A) f (1)jujrj .j=0its characteristic polynomial (by convention u - -1); then for the powers of A with nonn egative in tegral exp onents the followi ng represe ntati on formula holds true: An = F1,nl(u1,.,U7)ArJ 卩2心(5山2,Ur)Ar，F(xiàn)gUg, u)： (2

7、.1)The functions Fk(u，u)that appear as coefficients in (2.1) are defined by the recurre nee relati onr 1Fk,n(u1,u)汕氐(u1, ,uj -上卩心(6,u(-1) ur Fk,n_c (u,ur)J(k =1,r; n 一 -1)(2.2)and in itial con diti ons:Fr 上 1,h_2 (u1 , ,u7)=Ok,h, (k, 1, ,r).(23)Furthermore, if A is nonsingular (u= 0) , then formula

8、(2.1) still holds for negative values of n, provided that we define the Fk,n function for negative values of n as follows:Fk,n(u1,，u7)=Fr1,2(S,里，丄),(k =1,，r; n -1).ur ur u7Taylor expansion of the resolventWe con sider the resolve nt matrix R.(A) defi ned as follows:R =R.(A)*A).(3.1)Note that sometim

9、es there is a cha nge of sig n in Eq. (3.1), but this of course is not esse ntial.It is well known that the resolve nt is an an alytic (rati on al) function ofin everydomain D of the complex plane excluding the spectrum of A, and furthermore it is vanishing at infinity so the only singular points (p

10、oles) of R. (A) are the eigenvalues of A.In 6 it is proved that the invariants v1,v2/ ,vr of R.(A) are linked with those of A by the equationsi(r jW 仏)=送(-1)(hi ,(l=12 ，r).(3.2)uJ - J JAs a con seque nceof Propositi on 2.1, and Eq. (3.2), the in tegral powers of R. (A)can be represe nted as follows.

11、Theorem 3.1 For every 扎貳：A and n 二 N,Ar _1Rn(A)八J,v()Rk(A),(3.3)k 30where the V|(丸)(|=1,2,，r)are given by Eq.(3.2). Denoting by P(A) the spectral radius of A, for every ，二 such that(A) min(卜|,|! |), the Hilbert iden tity holds true(see 1):R,(A) -R(A)- )R,(A)Rt(A).(3.4)Therefore for every - A , we ha

12、veAdR(A)d (3.5)(3.6)(3.7)(3.8)(3.9)r 4R(A)八h=0- 0 (-1)kF.k =05( 0)k R；(A).(3.10)and in gen eralk=(-1)kkRk1(A),(k =1,2,、a )so, for every -0 D, R. (A) can be expanded in the Taylor seriesoCiR.(A)八(_1)kkR；1(A)(- 0)k,k=0which is absolutely and uniformly convergent in D. Defining0 0v1 二 v, 0), ,vr 二 v( 0

13、),0 0 0F k，n 二 Fk,n(w，vr),where the vl () are defined by Eq. (3.2), we can prove the following theorem.Theorem 3.2 The Taylor expa nsion (3.7) of the resolve nt R. (A) in a neighborhood of any regular point 0 can be written in the formTherefore we can derive as a con seque nee:Corollary 3.1 For ever

14、y 1 A and L=1,2, r the series expansions: 0、(-1)k Fi,k(- o)k(3.11)are conv erge nt.Proof. Recalli ng (3.3), we can write0 0 0Rj 二 Fi,k R；-F2,k R；2 -F r,-. , k N),0 0 0R.(A) = JF1,1 Rf0j - F2,1 Fr,1 ；：(Fr,2 ；： (一，0)2-0 0r 1r 2F1,2R； F2,2R/(-1)k0 0F1,kR； F2,k R：-F r,kTherefore, tak ing in to acco unt

15、the in itial con diti ons (2.3) we can write: 0R,(A)二 M)kFr,k(._kz91 i O0- J V (-1)k Fr,k(一 .0)kILk =0oO0+ Z (1)kF1,k仏尢0)k rR1,2 一so (3.10) holds true. The convergenee of series expansions (3.11) is a trivial con seque nee of the con verge nee of the in itial expa nsion (3.7).4. Concluding remarksIt

16、 is worth noting that the resolve ntR.(A) is a keynote element for representinganalytic functions of a matrixA . In fact,denoting by f(z) a function of thecomplex variable z , analytic in a domain containing the spectrum of A, and denoting by k(k=1,2,s) the distinet eigenvalues of A with multiplicit

17、ies Jk,the Lagra nge-Sylvester formula (see 4) is give n bysf(A)八 -k j=0kwhere k = *(k =12 ,s) is the projector associated with the eigenvalue kand=( kl -A)j,(k =1,2,s; j 71；,% -1).Denoting by k a Jordan curve, the boundary of the domain Dk, separating a fixedk from all other eigenvalues, recalling

18、the Riesz formula, it follows thatWhen kis only known approximately, this projector cannot be derived by using the residue theorem.In this case it is necessary to integrate R.(A) along k (being possibly aGershgori n circle), by using the known represe ntati on of the resolve nt (see 3)1 rJ rR(A)= 環(huán)瓦

19、 正(一1)5憶口Ak,(4.1)P(丸)k=0 - j=0or by substituting R. (A) with its Taylor expansion, and assuming as initial pointany o /.k in side Dk.Which is the best formula depe nds on the releva nt stability and computatio nal cost. From the theoretical point of view,formulas (3.7), (3.10) and (4.1) seem to be e

20、quivale nt from the stability point of view, since all require kno wledge of inv aria nts of the given matrix A. However, in our opinion, in the situation considered, Eq. (3.10) seems to be less expensive with respect to (3.7), since it requires one to approximate r series of eleme ntary fun cti ons

21、 in stead of an infin ite series of matrices.AcknowledgementsWe are grateful to the anonym ous referees for comme nts that led us to improve this paper.ReferencesI. Glazman, Y. Liubitch, Analyse lin aire dans lesespaces de dimension finies: Manuel et probl mes, in: H. Damadian (Ed.), Traduit du russe par, Mir, Moscow, 1972.M. Bruschi