有关泰勒公式的复杂矩阵的解决方法外文翻译资料

 2023-01-12 09:01

有关泰勒公式的复杂矩阵的解决方法

Matthew X. Hea, Paolo E. Ricci b,_

关键字:矩阵方幂;矩阵不变量;解决方法

  1. 介绍

由于Hilbert矩阵元在[1]中的解决方法的非奇异矩阵 ( 表示单位矩阵)是一个在任何频谱空交中D域的参数的解析函数。因此,用泰勒展开在附近的任何固定的 ,我们可以发现在在[ 1 ]的一个利用所有的幂的表示公式。

在这篇文章中,运用一些之前的结论,例如[2]中的,我们写出一个只运用一个数量有限的的幂的表达式。只有一个是线性无关的似乎是很自然的。在这个框架的主要工具是由多变量多项式

根据中的这些不变量;在这里表示最小多项式的程度。

  1. 矩阵的幂和函数

我们回忆一下在这段幂矩阵的表示公式的一些结果(见例[2-6]和参考文献)。为简单起见,我们参考这样的情况,当矩阵是非减阶的,所以 。

命题2.1 让是一个的复杂矩阵,并且由表示,

的不变量,并且

的特征多项式(按照规定);然后,对于的幂用非负整数指数以下表示的公式是正确的。

(2.1)

函数是在(2.1)中作为系数出现的,被定义为递归关系:

(2.2)

并且其初始条件为:

(2.3)

此外,如果是非奇异的,那么公式(2.1)对于负值仍然成立,如果证明我们定义函数的负值如下:

  1. 泰勒展开的解决方案

我们考虑即将要解决的矩阵的定义如下:

(3.1)

注意到,有时这里有一个变化的符号在例(3.1)中,但是这当然是没有必要的。

众所周知,这个解决方法是一个的解析(理性的)函数,在每个域D的复平面包括频谱,并且此外,它会消失在无穷远处,所以这唯一的奇异点(杆)是的特征值。

在[6]中,证明了该中的不变量是通过以下的方程被联系了矩阵的这些量:

(3.2)

作为一个命题2.1的结果和方程(3.2),幂的积分可以被表示如下。

定理3.1 对于每一个和

(3.3)

这里的是通过方程(3.2)中被赋值的。由表示的的谱半径,对于任一个,例如Hilbert矩阵是正确的(见[1]):

(3.4)

然后对于任一,我们有

(3.5)

并且,大体上,

(3.6)

所以,对于任一可以在泰勒级数中被扩展

(3.7)

在D定义中,是绝对和一致收敛的。

(3.8)

(3.7)

其中是在中被定义的,我们能证明接下来的定理。

定理3.2 的解决方法的泰勒展开式(3.7)在任何一个常规点的领域可以被写成如下形式:

(3.10)

因此,我们可以得出一个结果:

推论3.1 对于任一个和该系列扩张

(3.11)

是收敛的。

证明:回顾(3.3),我们可以写

因此,考虑到的初始条件(2.3)我们可以写

所以(3.10)式子是成立的。级数展开收敛(3.11)是初始扩张(3.7)的收敛琐碎的结果。

  1. 结束语

值得一提的是,解决方法是一个代表了矩阵的解析函数的主题元素。事实上,通过函数的复变量,分析中的一个频谱,并且,通过与的多重鲜明特征值,拉格朗日 - 西尔维斯特公式(见[4])是由以下的式子被给出的:

这里的是与特征值的投影机相关联,并且

由JORDAN曲线表示,域的边界,从所有其他特征值中分离出一个固定的,再回顾里斯公式,可以得出:

当只是大约被知道时,这个投影不能利用残余定理导出。

在这种情况下,沿(作为可能是一个Gershgorin圈)结合是有必要的,通过使用已知代表性的解决方法(见[3])。

(4.1)

或者通过用泰勒展开式取代,并且,假设任意一个初始点在内部。

一个最好的公式是依赖于相关的稳定性和计算成本的公式。从理论角度,式(3.7),(3.10)及(4.1)似乎是从稳定性来看是等效的,因为矩阵的所有需要给定的不变量的知识。然而,在我们看来,在考虑这样的情况下,似乎是相对于(3.7)是较不重要的,因为,它需要一个近似一系列的基本功能,而不是一个无穷级数的矩阵。

  1. 致谢

我们感谢所有匿名审稿并提出意见的学者,来帮助我们改善此文章。

参考文献:

[1] I. Glazman, Y. Liubitch, Analyse lineacute;aire dans les espaces de dimension finies: Manuel et problegrave;mes, in: H. Damadian (Ed.), Traduit du russe par, Mir,

Moscow, 1972.

[2] M. Bruschi, P.E. Ricci, Sulle potenze di una matrice quadrata della quale sia noto il polinomio minimo, Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stud. Roma,

Quad. 13 (1979) 9–18.

[3] V.N. Faddeeva, Computational Methods of Linear Algebra, Dover Pub. Inc., New York, 1959.

[4] F.R. Gantmacher, The Theory of Matrices, Vols. 1, 2 (K.A. Hirsch, Trans.), Chelsea Publishing Co., New York, 1959.

[5] M. Bruschi, P.E. Ricci, Sulle funzioni Fk,n e i polinomi di Lucas di seconda specie generalizzati, Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stud. Roma, Quad. 14

(1979) 49–58.

[6] M. Bruschi, P.E. Ricci, An explicit formula for f (A) and the generating function of the generalized Lucas polynomials, SIAM J. Math. Anal. 13 (1982)

162–165.

附件

文献原文:

On Taylorrsquo;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 follows:

. (3.1)

Note that sometimes there is a change of sign in Eq. (3.1), but this of course is not ess

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On Taylorrsquo;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 follows:

. (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 , and furthermore it is vanishing at infinity so the only singular points (poles) of are the eigenvalues of .

In [6] it is proved that the invariants of are linked with those of by the equations

,. (3.2)

As a consequence of Proposition 2.1, and Eq. (3.2), the integral powers of can be represented as follows.

Theorem 3.1 For every and ,

, (3.3)

where the are given by Eq.(3.2). Denoting by the spectral radius of , for every , such that the Hilbert identity holds true(see [1]):

. (3.4)

Therefore for every , we have

, (3.5)

and in general

, (3.6)

so, for every can be expanded in the Taylor series

, (3.7)

which is absolutely and uniformly convergent in D. Defining

, (3.8)

, (3.9)

where the are defined by Eq. (3.2), we can prove the following theorem.

Theorem 3.2 The Taylor expansion (3.7) of the resolvent in a neighborhood of any regular point can be written in the form

. (3.10)

Therefore we can derive as a consequence:

Corollary 3.1 For every and the series expansions

(3.11)

are convergent.

Proof. Recalling (3.3), we can write

,,

Therefore, taking into account the initial conditions (2.3) we can write

,

so (3.10) holds true. The convergence of series expansions (3.11) is a trivial consequence of the convergence of the initial expansion (3.7).

4. Concluding remarks

It is worth noting that the resolvent is a keynote element for representing analytic functions of a matrix . In fact,denoting by a function of the complex variable , analytic in a domain containing the spectrum of , and denoting by the distinct eigenvalues of with multiplicities , the Lagrange–Sylvester formula (see [4]) is given by

,

where is the projector associated with the eigenvalue , and

.

Denoting by a Jordan curve, the boundary of the domain , separating a fixed from all other eigenvalues, recalling the Riesz formula, it follows that

.

When is only known approximately, this projector cannot be derived by using the residue theorem.

In this case it is necessary to integrate along (being possibly a Gershgorin circle), by using the known representation of the resolvent (see [3])

, (4.1)

or by substituting with its Taylor expansion, and assuming as initial point any inside .

Which is the best formula depends on the relevant stability and computational cost. From the theoretical point of view,formulas (3.7), (3.10) and (4.1) seem to be equivalent from the stability point of view, since all require knowledge of

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