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2024年4月16日发(作者：schedule近义词)
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
R
(A)
=
(
A)
1
of 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
A
of
A
.
Therefore, by using Taylor expansion in a neighborhood of 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 preceding results recalled, e.g., in [2], we
write down a representation formula using only afinite number of
powers of
R
0
(
A
)
. This seems to be natural since only the first powers of
R
0
(A)
are linearly main tool in this framework is given
by the multivariable polynomials
F
k,n
(
v
1
,
v
2
,...,
v
r
)
(
n1,0,1,...
;
k1,2,...,mr
) (see [2–6]), depending on the invariants
(v
1
,v
2
,...,v
r
)
of
R
(A)
); heremdenotes the degree of the
minimalpolynomial.
2. Powers of matrices and
F
k,n
functions
Werecall in this section some results on representation formulas for
powers of matrices ( [2–6] and the referencestherein). For
simplicity we refer to the case when the matrix is nonderogatory so that
mr
.
Proposition 2.1. Let
A
be an
rr(r2)
complex matrix, and denote
by
u
1
,u
2
,...,u
r
the invariants of
A
, and by
P(
)det(
A)
(1)
j
u
j
rj
.
j0
r
its characteristic polynomial (by convention
u
0
1
); then for the powers
of
A
with nonnegative integral exponents the following representation
formula holds true:
A
n
F
1,ni
(u
1
,...,u
7
)A
r1
F
2,n1
(u
1
,u
2
,...,u
r
)A
r2
F
r,n1
(u
1
,u
2
,u
r
)
. (2.1)
The functions
F
k,n
(
u
1
,
,
u
r
)
that appear as coefficients in (2.1) are defined
by the recurrence relation
F
k,n
(u
1
,,u
r
)u
1
F
k,n1
(u
1
,,u
r
)u
2
F
k,n2
(u
1,
,u
r
)(1)
r1
u
r
F
k,nr
(u
1,
,u
r
)
,
(k1,,r;n1)
(2.2)
and initial conditions:
F
rk1,h2
(u
1
,
,u
7
)
k,h
,
(k,h1,,r)
. (2.3)
Furthermore, if
A
is nonsingular
(
u
r
0)
, then formula (2.1) still holds for
negative values of n, provided that we define the
F
k,n
function for
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