长沙理工大学：《矩阵论》课程教学资源（课件讲稿，打印版）第三章 矩阵分析

第三章 矩阵分析 一、矩阵序列 二、矩阵级数 三、矩阵函数 四、矩阵的微分和积分 五、矩阵分析应用举例 理工大学 This document is produced by trial version of Print2Flash. Visit www.print2flash.com for more information

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矩阵序列与矩阵级数 矩阵的谱与谱半径 ·矩阵的谱半径：矩阵范数在特征值估计中的应用 设AeC”,方，元为A的个特征值，称集合 {02…2,} 为矩阵A的谱，记作G(4) 称 p(A)=max 为矩阵A的谱半径 This documentis produced bytrialversinofPVisit www.prinashcmformoreinfomio

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3.1.矩阵序列 定义1设矩阵序列{A)}∈Cm”，其中 A=(a,)m,如果mn个数列 1ima,=a,i=1,2,…,mj=1,2,,n k)0 则称矩阵序列{A}收敛于A=(a,)mn或称A 为矩阵序列{A)}的极限。记为 IimA)=A或A→A(k→0) 否则称为发散。 This docur at is produced by trial versi on of Print2Flash.Visit www.print2flash.com for mo nformatio

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例 如果设4=[a,]eC22,其中 ,=+1 ,a:=r01),a=- k2+k 那么 s作 Print2Flash

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定理3.1矩阵序列{A}收敛于A的充分 必要条件是 lim44)4=0 k)0 其中为任意一种矩阵范数。 证明取矩阵范数 14L=2a i=1i=1 必要性：设 lim 4()=A=(an) →00 his doc uced by trial ver of Print2Flash.Visit w

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那么由定义可知对每一对i,i都有 ingo0 (i=1,2,…,;j=1,2,…,n) 从而有 m22a,-o0 k0 i=1j=1 上式记为 lim4-=0 d by trial of Print2Flash Visit

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充分性：设 ml4-A-m22a,“-a,=0 k0=1j=l 那么对每一对i,j都有 imaa0 (i=1,2,…,m;j=1,2,…,n) 即 lim d,=d (i=1,2,…,m;j=1,2,…,n) This docur t is produced by trial version of Print2Flash.Visit www.print2flash.com for more information

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故有 limA)=A=(a） k)00 现在已经证明了定理对于所选取的范数成 立，如果4是另外任意一种范数，那么 由范数的等价性可知 dA-A≤4-A≤d,A-A 这样，当m4-A=0 时同样可得im4-A-0 因此定理对于任意一种范数都成立

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推论：若imA)=A,则m4=4, 其中为任意一种矩阵范数。反之不成立。 同数列的极限运算一样，关于矩阵序列的极 限运算也有下面的性质。 (1)一个收敛的矩阵序列的极限是唯一的。 (2)设1imA)=A,limB)=B →0 →00

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则 limaA(k)+bB(k)=aA+bB,a,bEC →00 (3)im 4()=4,lim B()=B, 其中A)∈Cm,B)∈Cg那么 lim A(R)B()=AB k0 (4)设1imA)=A,其中 k→0 A)∈Cmx",P∈Cmwm,Q∈CmxM ed by trial v of Print2Flash Visit

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