On Taylor’s formula for the resolvent of a complex matrix
杨香玉Matthew X. Hea, Paolo E. Ricci b,_
Article history:Received 25 June 2007
Received in revid form 14 March 2008
Accepted 25 March 2008
垃圾分类教案中班Keywords:Powers of a matrix
眉山三苏祠Matrix invariants
生态公园Resolvent
1. Introduction
旅夜书怀杜甫
As a conquence 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 interction with the spectrum of . Therefore, by using Taylor expansion in a neighborhood of any fixed , we can find in [1] a reprentation formula for using all powers of .
In this article, by using some preceding results recalled, e.g., in [2], we write down a reprentation formula using only afinite number of powers of . This ems to be natural since only the first powers of are linearly independent.The main tool in this framework is given by the multivariable polynomials (;) (e [2–6]), depending on the invariants of); heremdenotes the degree of the minimalpolynomial.
2. Powers of matrices andfunctions
Werecall in this ction some results on reprentation formulas for powers of matrices ( [2–6] and the referencestherein). For simplicity we refer to the ca 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 reprentation 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, ifis 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 follows:
. (3.1)
Note that sometimes there is a change of sign in Eq. (3.1), but this of cour is not esntial.
It is well known that the resolvent is an analytic (rational) function of in every domain D of the complex plane excludingthe spectrum of, and furthermore it is vanishing at infinity so the only singular points (poles) of are the eigenvaluesof 北京个人社保.
In [6] it is proved that the invariants of are linked with tho of by the equations
,. （3.2）
As a conquence of Proposition 2.1, and Eq. (3.2), the integral powers of can be reprented as follows.
For every and,
, (3.3)
where theare given by Eq.(3.2).Denoting by the spectral radius of , for every , such that the Hilbert identity holds true(e [1]):
. (3.4)
Therefore for every, we have
, (3.5)
and in general
,(3.6)
技能专长so, for every can be expanded in the Taylor ries
, (3.7)
which is absolutely and uniformly convergent in D.Defining
, (3.8)
假如给我三天光明好词好句摘抄, (3.9)
where the are defined by Eq. (3.2), we can prove the following theorem.
