泰勒公式及应用翻译(原文).docVIP

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 = of a nonsingular square matrix ( denoting the identity matrix) is shown to be an analytic function of the parameter in any domain D with empty intersection with the spectrum of . Therefore, by using Taylor expansion in a neighborhood of any fixed , we can find in [1] a representation formula for using all powers of . 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 . This seems to be natural since only the first powers of are linearly independent.The main tool in this framework is given by the multivariable polynomials (;) (see [2–6]), depending on the invariants of ); here m denotes the degree of the minimal polynomial. 2. Powers of matrices and 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 . Proposition 2.1. Let be an complex matrix, and denote by the invariants of , and by . its characteristic polynomial (by convention ); then for the powers of with nonnegative integral exponents the following representation formula holds true: . (2.1) The functions that appear as coefficients in (2.1) are defined by the recurrence relation , (2.2) and initial conditions: . (2.3) Furthermore, if is nonsingular , then formula (2.1) still holds for negative values of n, provided that we define the function for negative values of n as follows: ，. 3. Taylor expansion of the resolvent We consider the resolvent matrix defined as foll