6
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
“ ”设 || X k X 0 || 0 (k ) 则
max | x x | 0
(k ) 1 i n i i
(k) (k) (k) k 1 2 n
k k 0
lim x
k
(k ) i
x , i 1,2, , n
i
从而 max | x x | 0
(k ) i i i
即|| X X || 0(k )
k 0
由范数之间的等价性质知结论成立.
School of Math. & Phys.
13
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
定义 设m n 的复变量z 的一元函数矩
阵 A( z ) (a ( z )) 在 z 的一个邻域内有定义,
ij mn
0
即 A( z )中 a ( z ) (i 1,2,, m; j 1,2,, n) 都有定义,
1 0 E. 0 1
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Mathematical Methods & its Applications
2015/10/20
J. G. Liu
二、函数矩阵的极限
设 a z 是一元复变函数,
J. G. Liu
一、矩阵序列的极限
1. 定义
1 2 k
A C
k
m n
, A (a ) ,( k 1,2,)
(k ) k ij mn
A , A ,, A ,, 记作 {A } 为矩阵序列.
k
若 lim a a (有限) (i 1,2,, m, j 1,2,, n)
k
School of Math. & Phys.
8
North China Elec. P.U.
Mathematical Methods & its Applications
k
2015/10/20
J. G. Liu
nn A , B C , lim Ak A, lim Bk B ② k k k
0
区域 D的每一点连续, 则称 A( z )在区域 D上连
续.
定义 若 n阶方阵 A( z )在区域 D中每一点
都可逆, 称 A( z ) 在 D 上可逆, 记
为A
1
A( z ) 的逆矩阵
(z)
. 显然满足:
1 1
z D 有 A( z ) A ( z ) A ( z ) A( z ) E
k k k
k k
1 lim(1 ) e 1 k , ( 1) 1 0 lim k
k k
lim( 1) 不存在， lim B
k
(k )
不存在.
North China Elec. P.U.
School of Math. & Phys.
-1
School of Math. & Phys.
10
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
例1：求如下矩阵序列的极限
1 (1 ) (k ) k A 1
ij
称
a z a z a z a z A( z ) a z a z
11 12 21 22 m1 m2
a z a z a z
1n 2n mn
为函数矩阵.
School of Math. & Phys.
1 A ( z ) ②若 为连续可逆方阵, 则 A ( z )也连续.
School of Math. & Phys.
16
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
例3
求 lim A( t ) .
k
1 k 1 (1 ) 1 (k ) k k , B ( 1) ( 1)k k
k k k
1 1 k ( 1) k
k
解：
1 lim(1 ) (k ) k lim A lim 1
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
矩阵分析方法
高等数学中函数的极限运算、微分运
算、积分运算、幂级数运算 矩阵理
论中.
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1
North China Elec. P.U.
Mathematical Methods & its Applications
k k k
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9
North China Elec. P.U.
Mathematical Methods & its Applications
证明 对任何一种范数 || ||
k k k k k
2015/10/20
J. G. Liu
|| A B AB |||| A B A B+A B AB ||
2015/10/20
m n
J. G. Liu
也即
lim A lim(a )
(k ) k k k ij (k) k ij
(lim a )
(a )
ij m n
m n
特别:
m n－方阵序列收敛;
n 1 －列矩阵(向量)序列收敛; m 1 －行矩阵(向量)序列收敛;
（ k ,i 1,2,, n） 于是 | x x | 0
(k) i i
从而 A A (k )
k
证毕.
②考虑方阵 A 的F-范数似①可证.
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North China Elec. P.U.
Mathematical Methods & its Applications
n k k k 0 1 2 n
对任意一种n维向量范数 || || , 有
|| X X || 0, (k )
k 0
② { A } C , lim A A 对任意一种
nn k k k
方阵范数|| || , 有 || A A || 0,(k ).
2015/10/20
J. G. Liu
注 范数序列实际上是数列. 向量序列的收敛性, 方阵序列的收敛性可
通过向量范数序列, 方阵范数序列的收敛
性来讨论!
3. 方阵序列收敛的性质 ① { A } C ,lim A A
nn k k k
任一方阵范数 || || , {|| A ||} 有界.
School of Math. & Phys.
4
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
2. 向量序列、方阵序列收敛的判定条件
定理1 ①{ X } C , lim X X ( x , x ,, x )
A
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14
North China Elec. P.U.
Mathematical Methods & its Applications
z z0 ij ij
2015/10/20
0
J. G. Liu
定义 若 lim a ( z ) a ( z )( i 1, 2, , m; j 1, 2,, n) 则称 A( z ) 在 z 连续. 又若 A( z )在复平面的一个
k
School of Math. & Phys.
5
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
证明 “ ” 设 X ( x , x ,, x ) , 由 lim X X 知
k
|| A B A B || || A B AB ||
k k k k
|| A || || B B || || B || || A A ||
k k k
联系性质①可证! ④ 若
lim A A ,且 Ak1 , A1都存在
k k
lim A =A
1 k k
2 t t 2 t 2
2
1 0 e e E. 0 1

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t