泰勒公式及应用翻译(原文)
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1 On Taylor’s formula for the resolvent of a complex matrix Matthew X. Hea, Paolo E. Ricci b,_ Article history:Received 25 June 2007 Received in revised form 14 March 2008 Accepted 25 March 2008 Keywords: Powers of a matrix Matrix invariants Resolvent
1. Introduction As a consequence of the Hilbert identity in [1], the resolvent )(AR=
1)(Aof a nonsingular square matrix A( denoting the identity matrix) is
shown to be an analytic function of the parameter in any domain D with empty intersection with the spectrum Aof A. Therefore, by using Taylor expansion in a
neighborhood of any fixed D0, we can find in [1] a representation formula for )(AR using all powers of )(0AR.
In this article, by using some preceding results recalled, e.g., in [2], we write down a representation formula using only a finite number of powers of )(0AR. This seems
to be natural since only the first powers of )(0AR are linearly independent.The main tool in this framework is given by the multivariable polynomials ),...,,(21,rnkvvvF (,...1,0,1n;rmk,...,2,1) (see [2–6]), depending on the
invariants ),...,,(21rvvv of )(AR); here m denotes the degree of the minimal polynomial. 2. Powers of matrices and nkF, functions We recall in this section some results on representation formulas for powers of matrices (see e.g. [2–6] and the references therein). For simplicity we refer to the case when the matrix is nonderogatory so that rm.
Proposition 2.1. Let A be an )2(rrr complex matrix, and denote by
ruuu,...,,21 the invariants of A, and by rjjrjjuAP0)1()det()(. 2
its characteristic polynomial (by convention 10u); then for the powers of A with nonnegative integral exponents the following representation formula holds true: ),,(),...,,(),...,(211,2211,2171,1rnrrrnrinnuuuFAuuuFAuuFA. (2.1)
The functions ),,(1,rnkuuF that appear as coefficients in (2.1) are defined by the recurrence relation ),()1(),(),,(),,(,1,1,12,211,11,rrnkrrrnkrnkrnkuuFuuuFuuuFuuuF, )1;,,1(nrk (2.2) and initial conditions: ,),,(,12,17hkhkruuF ),,1,(rhk. (2.3)
Furthermore, if A is nonsingular )0(ru, then formula (2.1) still holds for negative values of n, provided that we define the nkF, function for negative values of n as follows: )1,,,(),,(7112,171,uuuuuFuuFrrrrnkrnk，)1;,,1(nrk. 3. Taylor expansion of the resolvent We consider the resolvent matrix )(AR defined as follows:
1)()(AARR
. (3.1)
Note that sometimes there is a change of sign in Eq. (3.1), but this of course is not essential. It is well known that the resolvent is an analytic (rational) function of in every domain D of the complex plane excluding the spectrum of A, and furthermore it is
vanishing at infinity so the only singular points (poles) of )(AR are the eigenvalues of A. In [6] it is proved that the invariants rvvv,,,21 of )(AR are linked with those of A by the equations ljjljjlujljrv0()1()(,),,2,1(rl. （3.2）
As a consequence of Proposition 2.1, and Eq. (3.2), the integral powers of )(AR 3
can be represented as follows. Theorem 3.1 For every Aand Nn,
10211,)())(,),(),(()(rkkrnkrnARvvvFAR, (3.3)
where the )(lv ),,2,1(rlare given by Eq.(3.2). Denoting by )(A the spectral radius of A, for every , such that ),,min()(A the Hilbert identity holds true(see [1]): )()()()()(ARARARAR. (3.4)
Therefore for every A, we have
)()(2ARdAdR, (3.5)
and in general )()1()(1AkRdARdkkkk
,);,2,1(Ak (3.6)
so, for every )(,0ARDcan be expanded in the Taylor series kkkkAkRAR))(()1()(0010, (3.7)
which is absolutely and uniformly convergent in D. Defining )(,),(000110rrvvvv, (3.8) ),,(010,,0rnknkvvFF, (3.9)
where the )(lv are defined by Eq. (3.2), we can prove the following theorem. Theorem 3.2 The Taylor expansion (3.7) of the resolvent )(AR in a neighborhood of any regular point 0 can be written in the form
)()()1()(01000,0ARFARnrhkkknrk. (3.10) Therefore we can derive as a consequence: Corollary 3.1 For every A0and rL,2,1 the series expansions
