泰勒公式及应用翻译(原文)

On Taylor’s formula for the resolvent of a complex matrix

Matthew X. He a , 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 )(A R λ= 1)(--E A λof a nonsingular square matrix A (E denoting the identity matrix) is shown to be an analytic function of the parameter λ in any domain D with empty intersection with the spectrum ∑A of A . Therefore, by using Taylor expansion in a

neighborhood of any fixed D ∈0λ, we can find in [1] a representation formula for )(A R λ using all powers of )(0A R λ.

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 )(0A R λ. This seems to be natural since only the first powers of )(0A R λ are linearly independent.The main tool in this framework is given by the multivariable polynomials ),...,,(21,r n k v v v F (,...1,0,1-=n ;r m k ≤=,...,2,1) (see [2–6]), depending on the invariants ),...,,(21r v v v of )(A R λ); here m denotes the degree of the minimal polynomial.

2. Powers of matrices a nd n k F , 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 r m =.

Proposition 2.1. Let A be an )2(≥⨯r r r complex matrix, and denote by r u u u ,...,,21 the invariants of A , and by

∑=--=-E =r

j j r j j u A P 0)1()det()(λλλ.

its characteristic polynomial (by convention 10-=u ); then for the powers of A with nonnegative integral exponents the following representation formula holds true: E +++=-----),,(),...,,(),...,(211,2211,2171,1r n r r r n r i n n u u u F A u u u F A u u F A . (2.1) The functions ),,(1,r n k u u F that appear as coefficients in (2.1) are defined by the recurrence relation

),()1(),(),,(),,(,1,1,12,211,11,r r n k r r r n k r n k r n k u u F u u u F u u u F u u u F -----++-=,

)1;,,1(-≥=n r k (2.2)

and initial conditions:

,),,(,12,17h k h k r u u F σ=-+- ),,1,(r h k =. (2.3)

Furthermore, if A is nonsingular )0(≠r u , then formula (2.1) still holds for negative values of n, provided that we define the n k F , function for negative values of n as follows:

)1,,,(),,(7

112,171,u u u u u F u u F r r r r n k r n k --+-+-=，)1;,,1(-= n r k . 3. Taylor expansion of the resolvent

We consider the resolvent matrix )(A R λ defined as follows:

1)()(--E =≡A A R R λλλ. (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 λ 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 )(A R λ are the eigenvalues of A .

In [6] it is proved that the invariants r v v v ,,,21 of )(A R λ are linked with those of A by the equations

∑=-⎪⎪⎭⎫ ⎝⎛---=l j j l j j

l u j l j r v 0()1()(λ

λ,),,2,1(r l =. （3.2） As a consequence of Proposition 2.1, and Eq. (3.2), the integral powers of )(A R λ