1 向量范数与方阵范数(1)

nn nn
)
则称实函数 || A ||为方阵 A 的范数. 注: 注意方阵范数与向量范数定义的区别.
School of Math. & Phys. 16 North China Elec. P.U.
Mathematical Methods & its Applications
n n
2015/10/20
p p 1 i n i
记为 || X || lim || X || max | x |
p p 1 i n i
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
记|| A || ,满足以下四个条件: ①正定条件: || A || 0, 且 || A || =0 A 0 ②齐次条件: || aA || | a | || A || (a C ) ③三角不等式: || A B || || A| | || B || ， (A, B C ④相容条件: || A B || || A || || B || (A, B C )
2015/10/20
J. G. Liu
注:
C 上的任意两个范数 || X || ,|| X || 等价
n
a b
即
c , c R , s.t .X C , 有
n 1 2
c || X || || X || c || X ||
1 b a 2
b
School of Math. & Phys.
J. G. Liu
② a R, 都有
|| aX || ((aX ) A(aX ))
T A
2 T 1 2
1 2
(a X AX ) | a | || X ||A
③ 因为 A正定, 所以存在矩阵 B 使
A B B
T
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North China Elec. P.U.
X V ,有一个实数|| X || 与之对应, 满足:
①正定性: || X || 0, 且 X =0 X =0,X V ; ②齐次性:
|| aX ||| a | || X ||, a C , X V ;
③三角不等式： || X Y |||| X || || Y ||, X ,Y V . 则称 || X ||是 V 中向量 X 的范数,
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
于是有
|| X || ( X AX )
T A
T
1 2
( X B BX )
T T 1 2
1 2
[( BX ) ( BX )]
|| BX ||
2
从而 || X Y || || B( X Y ) || || BX BY ||
2015/10/20
p 1 p
J. G. Liu
p
x 1 x
k k
1 n 1 xi 1p ( x ) ( ) np , i 1 x xk
n i 1 i k
令 p 由两面夹定理知
x lim( ) 1 x
n i 1 p p i 1 k p
从而 lim || X || max | x |
2015/10/20
J. G. Liu
向量范数
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
1. 定义: V是 n维复数域C上的线性空间,
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
-范数:
|| A || max | a |
1 i n j 1 ij
n
即 A 的各行向量的1-范数的最大值, 称为行范数.
School of Math. & Phys.
p
2015/10/20
J. G. Liu
定理 p -范数 || X || , 取 p 1, p 2, p 即得1 -范数, 2 -范数及 -范数.
证明
设
p 1, p 2 时结论显然.
p
1 2 n
以下看
情况.
n k i i
X ( x , x , , x ) C , X 0, | x | max | x | 0
2015/10/20
J. G. Liu
1-范数：
X C , X ( x , x ,, x ) 规定
n 1 2 n
|| X ||1 | xi |
i 1
n
则|| X || 是 C 上的范数, 称为 1-范数.
n
1
School of Math. & Phys.
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North China Elec. P.U.
p
n p 1 p
x || X || ( | x | ) | x | ( ) x
n i
p
i 1 i
1 p
k
i 1
k
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
例1
设
1 X 3 , 2
求 X , X , X
1 2

解:
X 6, X 14, X
1 2
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
2-范数:
X C , X ( x , x ,, x ) 规定
n 1 2 n
|| X ||2 XX | x1 | | x2 | | xn |
H 2 2
n
20
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
注:
C 上的任意两个范数 || A || ,|| A || 等价,
nn
a b
即
c , c R , s.t .A C , 有

3
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
n
J. G. Liu
例2 设 A是 n阶正定矩阵, X R 列向量,
证明|| X || ( X AX ) 是向量范数(称为加权范
|| A || ( A A)
H 2
H
这里 ( A A) 表示A A特征值中模最大者(即
H
A A 的谱半径)
H
注:
( A) A
19
p
North China Elec. P.U.
School of Math. & Phys.
Mathematical Methods & its Applications
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
向量范数与矩阵范数
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
n 1 2 n
|| X || p ( | xi | ) , (1 p )
i 1
n || X || C 则 是 中的范数,称为 p -范数.
p
n
p
1 p
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
18
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
2-范数: 定义 设 A , A 的所有特征值 , ,,
nn 1 2
n
的模最大者称为 A 的谱半径，记作 ( A) . 称
T A 1 2
数或椭圆范数). 证明 ①X R , X 0 由A正定知 || X || 0
n
A
显然成立.
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
p 1 p
J. G. Liu
|| A || p ( | aij | )
i 1 j 1
——矩阵的p-范数 2.几个常见的方阵范数 1-范数:
|| A || 1 max | a | ij 1 j n
i 1
n
即 A 的各列向量的1-范数的最大值, 称为列范数.
School of Math. & Phys. 17 North China Elec. P.U.
1 2 n
|| X || max | x |
1 i n i
n || X || 则 是 C 上的一种范数, 称为 -范数.

可以证明, || X || 满足向量范数的定义.

School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
nn 1 2
c || A || || A || c || A ||
1 b a 2
b
School of Math. & Phys.
Leabharlann Baidu
21
North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
例3
2 1 A 0 4
1 2 F
求 A , A , A , A
解:
A 5,
1
A 4,

A ( A A) 4.1594,( 3.7, 17.3)
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
2. 几个常见的范数 范数:
X C ,
n
X ( x , x ,, x ) 规定
A 2 2 2
A
2
|| BX || || BY || || X || || Y ||
A
由知① ② ③ || X || 是向量范数.
A
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
Fribonius范数: 设 A C , 定义
nn
|| A ||F ( | aij | )
2 i 1 j 1
n
n
1 2
称 || A || 为
F
A
的Frobeius范数.
简称为F-范数. 注: 为向量2-范数的自然推广!
School of Math. & Phys.
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2
则|| X || 是C 上的一种向量范数, 称为向量的2范数.
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
2015/10/20
J. G. Liu
p-范数:
X C , X ( x , x ,, x ) , 规定
2015/10/20
J. G. Liu
矩阵范数
School of Math. & Phys.
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North China Elec. P.U.
Mathematical Methods & its Applications
1. 矩阵范数的概念
2015/10/20
J. G. Liu
定义1
AC
nn
, 规定一个非负实函数,