Journal of Nanchang Institute of Technology ISSN1674-007608/16pp34-36 Voloume 32,Number 1,February 2013 An extension and application of Cauchy integral formula HUANG Xiaojie,JIN Benqing (Department of Science,Nanchang Institute of Technology,Nanchang 330099,China) Abstract:This essay tries to apply the Cauchy integral formula of the complex variable complex function to the study of the complex matrix value function.It follows that the inferred formula that is obtained from the study can testify the famous Hamil- ton-Cayley theorem in matrix theory. Key words:Cauchy integral formula;Cauchy theorem;entire function:matrix function:Hamilton-Cayley theorem 1 Introduction and main results By Lagrange-Sylvester theorem,we can define matrix function as follows: Definition 1 Assume the convergence radius of power series is R,for arbitrary I=<R,the sum func- k=0 tion ofri)=2a,.If matris ries2.a,converges andtm代W）=2a,A,henf代W）is a ma- k=0 =0 trix sum function of that series. Integral is the limit of integral sum.According to this definition,we can similarly define the integral of abstract function- Definition 2 Assume A(z)is a matrix function in complex matrix space C"x"and its domain is the complex region D.The simple curve L is contained in this domain D,4 denotes the partition of this curve L:zo(beginning point zo),a,a2,…,2…,zn-1'之n（ending point),g:is an arbitrary point on arcs2ri+i,‖△‖。max,{lzi+l- 0≤i≤n-1 if(()exists,then we define this limit as the integral of function (on this curve Denoting it by(2)dz. By Definition2,ifA(z）={a,(z）},then A()d=a( For complex analytical function,Cauchy proved the following results: Theorem A(Cauchy Theorem)Assume region D is the interior of Jordan curve L,function f()is analyti- cal in D,and continuous in closed region D,then f()d:=0. Theorem B(Cauchy formula)Assume region D is the interior of Jordan curve L,functionf()is analyti- cal in D,and continuous in closed region D,then in D, 0=岩l图m,=0123 For n=0 and L is a large circle,Ref.5]proved similar results of Theorem B for polynomial matrix function Received:2012-09-07 E-mail:359536229@qq.com ?1994-2018 China Academic Journal Electronic Publishing House.All rights reserved.http://www.cnki.net
Journal of Nanchang Institute of Technology ISSN 1674 － 0076 08 /16 pp 34 － 36 Voloume 32，Number 1，February 2013 Received: 2012 － 09 － 07 E-mail: 359536229@ qq． com An extension and application of Cauchy integral formula HUANG Xiaojie，JIN Benqing ( Department of Science，Nanchang Institute of Technology，Nanchang 330099，China) Abstract: This essay tries to apply the Cauchy integral formula of the complex variable complex function to the study of the complex matrix value function． It follows that the inferred formula that is obtained from the study can testify the famous Hamilton-Cayley theorem in matrix theory． Key words: Cauchy integral formula; Cauchy theorem; entire function; matrix function; Hamilton-Cayley theorem 1 Introduction and main results By Lagrange-Sylvester theorem ［1］，we can define matrix function as follows: Definition 1 Assume the convergence radius of power series ∑ ∞ k = 0 ak z k is R，for arbitrary |z| ＜ R，the sum function of series is f( z) = ∑ ∞ k = 0 ak z k ． If matrix series ∑ ∞ k = 0 akAk converges and its sum is f( A) = ∑ ∞ k = 0 akAk ，then f( A) is a matrix sum function of that series． Integral is the limit of integral sum． According to this definition，we can similarly define the integral of abstract function ［2 － 3］． Definition 2 Assume A( z) is a matrix function in complex matrix space Cn × n and its domain is the complex region D． The simple curve L is contained in this domain D，Δ denotes the partition of this curve L: z0 ( beginning point z0 ) ，z1，z2，…，zi…，zn － 1，zn ( ending point) ，ζi is an arbitrary point on arcs zizi + 1，‖Δ‖ max 0≤i≤n － 1 { | zi + 1 － zi | } ，if lim ‖Δ‖0 ∑ n － 1 i = 0 A( ξi ) ( zi + 1 － zi ) exists，then we define this limit as the integral of function A( z) on this curve L． Denoting it by ∫L A( z) dz ． By Definition 2，if A( z) = { aij ( z) } ，then ∫L A( z) dz = ∫L aij { } ( z) dz ． For complex analytical function，Cauchy proved the following results: Theorem A( Cauchy Theorem) ［4］ Assume region D is the interior of Jordan curve L，function f( z) is analytical in D，and continuous in closed region D，then ∫L f( z) dz = 0． Theorem B( Cauchy formula) ［4］ Assume region D is the interior of Jordan curve L，function f( z) is analytical in D，and continuous in closed region D，then in D， f ( n) ( z) = n! 2πi∫L f( ξ) ( ξ － z) n+1 dξ， n = 0，1，2，3…． For n = 0 and L is a large circle，Ref．［5］proved similar results of Theorem B for polynomial matrix function
Journal of Nanchang Institute of Technology,2013,32 (1): 35 p(A)=coE+cA+.+c,A': pW=2(1-A0s This paper generalized above results. Theorem 1 Assume f(z)is an entire function on the complex plane C,A={a}EC"x",the matrix norm IA‖叁（∑glagl2)÷,Lis a eircle for which the radius is large enough(r≥IAI),hen )((Ag holds For convenience of application,we rephrased Theorem 1. Theorem 1'Assume f(z)is an entire function on the complex plane C,A=(aC"x",the matrix norm IA‖叁（∑glagI2)÷,isa circle for which the radius is large enough(r≥IAI),(H-A)-1={b,()},hen w)=aE6,e holds. Remark When r≥‖A‖，(EH-A)-exists.. 2 Examples Example 1 (Cayley-Hamilton theorem)Assume A=(a}EC"x",p(2)=I2/-A1,then p(A)=0 holds. fsince(A)Misminor which is polynomial of and an entire funetio then by Theoem .wehavedy nicin p()A. According to Theorem 1': pw=9g-s=品因告9n-{,-o=0 3 Proof of Theorem 1 In order to prove Theorem 1,we first present the following lemmas: Lemma 1 Assume function f(z)is analytical in circle C:Iz-al<R,then f(z)can be expanded by a pow- er series in C =包e-a Lemma 2 Assume the integral of f(g)along curve L exits,A=(aECx',then (f(g)dg)A"=f(g)A"dg. Proof Denote A"=(a},then (r()de)A"=(f(e)e)a=r()a e)= (g)a)dg =f(g)(a)ag =f(g)Adg Proof is finished. Lemma Assume((n1,2,is continuous on the curve L,series()uniformly converges to function f(z),then 业et=Akot Lemma4 Assume A.(z）,A(a）∈C,matrix norm‖A∥4（∑gla,I2)立，An(z)(n=l,2,…）is continuous on the curve L,series uniformly converges to function A()on L,then n=1 ?1994-2018 China Academic Journal Electronic Publishing House.All rights reserved.http://www.cnki.net
p( A) = c0E + c1A + … + crAr : p( A) = 1 2πi∫L p( ξ) ( ξI － A) －1 dξ． This paper generalized above results． Theorem 1 Assume f( z) is an entire function on the complex plane C，A = { aij} ∈Cn × n ，the matrix norm ‖A‖( ∑ij | aij | 2 ) 1 2 ，L is a circle for which the radius is large enough ( r≥‖A‖) ，then f( A) = 1 2πi∫L f( ξ) ( ξI － A) －1 dξ holds． For convenience of application，we rephrased Theorem 1． Theorem 1' Assume f( z) is an entire function on the complex plane C，A = { aij} ∈Cn × n ，the matrix norm ‖A‖( ∑ij | aij | 2 ) 1 2 ，L is a circle for which the radius is large enough ( r≥‖A‖) ，( ξI － A) － 1 = { bij ( ξ) } ，then f( A) = 1 2πi∫L f( ξ) bij { } ( ξ) dξ holds． Remark When r≥‖A‖，( ξI － A) － 1 exists． 2 Examples Example 1 ［5］( Cayley-Hamilton theorem) Assume A = { aij } ∈Cn × n ，p( z) = |zI － A |，then p( A) = 0 holds． Proof since ( ξI － A) － 1 = Mij ( ξ) { } | ξI － A | ，Mij ( ξ) is a minor which is a polynomial of ξ and an entire function， then by Theorem A，we have 1 2πi∫L Mij ( ξ) dξ = 0 ; by noticing p( ξ) = | ξI － A | ． According to Theorem 1': p( A) = 1 2πi∫L p( ξ) ( ξI － A) －1 dξ = 1 2πi∫L p( ξ) Mij ( ξ) | ξI － A | { } dξ = 1 2πi∫L Mij { } ( ξ) dξ = { 0} = 0． 3 Proof of Theorem 1 In order to prove Theorem 1，we first present the following lemmas: Lemma 1 ［4］ Assume function f( z) is analytical in circle C: |z － a | ＜ R，then f( z) can be expanded by a power series in C f( z) = ∑ ∞ n = 0 f ( n) ( a) n! ( z － a) n ． Lemma 2 Assume the integral of f( ξ) along curve L exits，A{ aij } ∈Ct × t ，then ∫L ( ) f( ξ) dξ An = ∫L f( ξ) An dξ ． Proof Denote An = { a( n) ij } ，then ∫L ( ) f( ξ) dξ An = ∫L ( ) f( ξ) dξ a( n) { } ij = ∫L f( ξ) a( n) { } ij dξ = ∫L { f( ξ) a( n) ij } dξ = ∫L f( ξ) { a( n) ij } dξ = ∫L f( ξ) An dξ ． Proof is finished． Lemma 3 ［4］ Assume fn ( z) ( n = 1，2，…) is continuous on the curve L，series ∑ ∞ n = 1 fn ( z) uniformly converges to function f( z) ，then ∫L f( z) dz = ∑ ∞ n = 1 ∫L fn ( z) dz． Lemma 4 Assume An ( z) ，A( z) ∈Ct × t ，matrix norm ‖A‖( ∑ij | aij | 2 ) 1 2 ，An ( z) ( n = 1，2，…) is continuous on the curve L，series ∑ ∞ n = 1 An ( z) uniformly converges to function A( z) on L，then Journal of Nanchang Institute of Technology，2013，32( 1) : 35
36 Journal of Nanchang Institute of Technology,2013,32(1): LA()d:A( The above formula states that the series can be integrated term by term on L. Proof Denote含A,(a)=(g.a9(ea)},A(a）=fa,(e),series言A.(e）convergesifomly to function A(2)on L,then for arbitrary s>0,exists N>0,if n >N,for arbitrary zL, I含Aa-A园I=(E,l2园-aa)<e, each column(converges uniformly to),by Lemma 3,we have( =1 a(a止，lhen A(d)()1AC)d Proof is finished. Lemma5 Let g be a complex number,A=(a)C"",matrix norm Al andAl, then (H-A)-1= and series converges uniformly for lg|≥‖A∥. Proof of Theorem 1 f()is an entire function and analytical in the whole plane.Let the radius of L be large enough.By Lemma 1 f(2)=∑（oi· n! then by Definition 1 f(A)=∑OAN, n! then by Theorem B o=恩 then w=含(， then apply Lemma 2,4,5,we have fA)=】 (恩r-盒(a六)县e越 e名六=阳-0 Proof is finished. References [1]Chen Z M,Zhou J S.The matrix theory [M].Beijing:Beihang University press,2000:314-342. 2]Wang S W,Zheng W X.Real function and functional analysis (In second volumes)[M].Beijing:higher education press,2005: 199-204. B]Zhang G Q,Lin Y Q.Functional analysis [M].Beijing:Peking University Press,2003:126-127. 4]Tan X J,Wu S J.The complex variable function concise tutorial [M].Beijing:Peking University Press,2007. 5]McCarthy C A.The Cayley-Hamilton Theorem ]Amer Math Monthly,1975,82:390-391. (下转第40页)】 ?1994-2018 China Academic Journal Electronic Publishing House.All rights reserved.http://www.cnki.net
∫L A( z) dz = ∑ ∞ n = 1 ∫L An ( z) dz． The above formula states that the series can be integrated term by term on L． Proof Denote ∑ n k = 1 Ak ( z) = { ∑ n k = 1 a( k) ij ( z) } ，A( z) = { aij ( z) } ，series ∑ ∞ n = 1 An ( z) converges uniformly to function A( z) on L，then for arbitrary ε ＞ 0，exists N ＞ 0，if n ＞ N，for arbitrary z∈L， ‖∑n k = 1 Ak ( z) － A( z) ‖ = ( ∑ij | ∑ n k = 1 a( k) ij ( z) － aij ( z) | 2 ) 1 2 ＜ ε， each column ∑ n k = 1 a( k) ij ( z) converges uniformly to aij ( z) ，by Lemma 3，we have ∫L aij ( z) dz = ∑ ∞ n = 1 ∫L a( n) ij ( z) dz，then ∑ ∞ n = 1 ∫L An ( z) dz = ∑ ∞ n = 1 ∫L a( n) ij { } ( z) dz = ∑ ∞ n = 1 ∫L a( n) ij { } ( z) dz = ∫L aij { } ( z) dz = ∫L A( z) dz． Proof is finished． Lemma 5 ［2 － 3］ Let ξ be a complex number，A = { aij } ∈Cn × n ，matrix norm ‖A‖( ∑ij |aij | 2 ) 1 2 and |ξ |≥‖A‖， then ( ξI － A) －1 = ∑ ∞ n = 0 An ξ n+1 and series converges uniformly for | ξ |≥‖A‖． Proof of Theorem 1 f( z) is an entire function and analytical in the whole plane． Let the radius of L be large enough． By Lemma 1 f( z) = ∑ ∞ n = 0 f ( n) ( 0) n! z n ， then by Definition 1 f( A) = ∑ ∞ n = 0 f ( n) ( 0) n! An ， then by Theorem B f ( n) ( 0) = n! 2πi∫L f( ξ) ξ n+1 dξ ， then f( A) = ∑ ∞ n = 0 1 2πi∫L f( ξ) ξ n+1 d ( ) ξ An ， then apply Lemma 2，4，5，we have f( A) = ∑ ∞ n = 0 1 2πi∫L f( ξ) ξ n+1 d ( ) ξ An = ∑ ∞ n = 0 1 2πi∫L f( ξ) An ξ n+1 d ( ) ξ = 1 2πi∫L∑ ∞ n = 0 f( ξ) An ξ n+1 dξ = 1 2πi∫L f( ξ) ∑ ∞ n = 0 An ξ n+1 dξ = 1 2πi∫L f( ξ) ( ξI － A) －1 dξ． Proof is finished． References ［1］Chen Z M，Zhou J S． The matrix theory［M］． Beijing: Beihang University press，2000: 314 － 342． ［2］Wang S W，Zheng W X． Real function and functional analysis ( In second volumes) ［M］． Beijing: higher education press，2005: 199 － 204． ［3］Zhang G Q，Lin Y Q． Functional analysis［M］． Beijing: Peking University Press，2003: 126 － 127． ［4］Tan X J，Wu S J． The complex variable function concise tutorial［M］． Beijing: Peking University Press，2007． ［5］McCarthy C A． The Cayley-Hamilton Theorem［J］． Amer Math Monthly，1975，82: 390 － 391． ( 下转第 40 页) 36 Journal of Nanchang Institute of Technology，2013，32( 1) :
40 Journal of Nanchang Institute of Technology,2013,32 (1): 3]Zhai Fuju,Zhang Chuanzhou.The boundedness of Banach-space martingale transform operator Journal of Math,2004,3:341-346. 4]Zhai Fuju,Wu Haiyan.The boundedness of operator-valued martingale transforms and the application].Joural of Qing dao Uni- versity of Science and Technology,2008,4:374-376. 5]Yu Lin,Jin Yanming.Boundedness of Banach-space-valued martingale transform and its applications Mathematica Applicata, 2006,2:407-413. 6]Zhang Yong,Lu Lin,Wang Tian.Boundedness of Banach-space-valued martingale transform operator ]Journal of China Three Gorges University,2007,1:73-75. Liu Peide.Martingale and geometry in Banach spaces [M].Beijing:Science Press,2007. 算子值鞅变换的凸Φ不等式及其应用 王丽娜，崔红新2 (1.南昌工程学院理学系，江西南昌330099：2.河南中医学院数理学科，河南郑州450008) 摘要：建立了算子值鞅变换的凸Φ不等式，并且通过算子值鞅变换进一步研究了极大算子和均方算子的 性质，讨论了鞅在其中取值的Banach空间的几何性质. 关键词：算子值鞅变换；极大算子；均方算子 中图分类号：0211.6 文献标识码：A 基金项目：南昌工程学院青年基金项目(2012KJ028) 作者简介：王丽娜(1980-)，女，硕士，讲师，echo.wwei@163.com. (上接第36页) 柯西积分公式的一个推广及其应用 黄小杰，金本清 (南昌工程学院理学系，江西南昌330099) 摘要：将关于复变复值函数的Cauchy积分公式推广到了复变矩阵值函，数的情况，这一推广的结论可用于 证明矩阵论中著名的Hamilton-Cayley定理. 关键词：Cauchy积分公式：Cauchy定理：整函数；矩阵函数；Hamilton-Cayley定理 中图分类号：0174.5 文献标识码：A 作者简介：黄小杰(1983-)，男，硕士，讲师，359536229@qq.c0m. ?1994-2018 China Academic Journal Electronic Publishing House.All rights reserved.http://www.cnki.net
［3］Zhai Fuju，Zhang Chuanzhou． The boundedness of Banach-space martingale transform operator［J］． Journal of Math，2004，3: 341 －346． ［4］Zhai Fuju，Wu Haiyan． The boundedness of operator-valued martingale transforms and the application［J］． Journal of Qing dao University of Science and Technology，2008，4: 374 － 376． ［5］Yu Lin，Jin Yanming． Boundedness of Banach-space-valued martingale transform and its applications［J］． Mathematica Applicata， 2006，2: 407 － 413． ［6］Zhang Yong，Lu Lin，Wang Tian． Boundedness of Banach-space-valued martingale transform operator［J］． Journal of China Three Gorges University，2007，1: 73 － 75． ［7］Liu Peide． Martingale and geometry in Banach spaces［M］． Beijing: Science Press，2007． 算子值鞅变换的凸 Φ 不等式及其应用 王丽娜1 ，崔红新2 ( 1． 南昌工程学院 理学系，江西 南昌 330099; 2． 河南中医学院 数理学科，河南 郑州 450008) 摘 要: 建立了算子值鞅变换的凸 Φ 不等式，并且通过算子值鞅变换进一步研究了极大算子和均方算子的 性质，讨论了鞅在其中取值的 Banach 空间的几何性质． 关键词: 算子值鞅变换; 极大算子; 均方算子 中图分类号: O211． 6 文献标识码: A 基金项目: 南昌工程学院青年基金项目( 2012KJ028) 作者简介: 王丽娜( 1980 － ) ，女，硕士，讲师，echo． wwei@ 163． com． ( 上接第 36 页) 柯西积分公式的一个推广及其应用 黄小杰，金本清 ( 南昌工程学院 理学系，江西 南昌 330099) 摘 要: 将关于复变复值函数的 Cauchy 积分公式推广到了复变矩阵值函数的情况，这一推广的结论可用于 证明矩阵论中著名的 Hamilton-Cayley 定理． 关键词: Cauchy 积分公式; Cauchy 定理; 整函数; 矩阵函数; Hamilton-Cayley 定理 中图分类号: O174． 5 文献标识码: A 作者简介: 黄小杰( 1983 － ) ，男，硕士，讲师，359536229@ qq． com． 40 Journal of Nanchang Institute of Technology，2013，32( 1) :