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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