基础矩阵分析

∞

)µ A
A
m∞
1
= A
= 0.6
§
= 0.4 A m1 = 1 √ A F = 0.18 ≈ 0.4243
§
§ § u´
5 / 33
ρ(A) ≤ 0.4 4.
Nanjing University of Science and Technology
Matrix Analysis and Computing
Matrix Analysis and Computing
§2.3
. Ý
'Ì»
êA^
Nanjing University of Science and Technology
1 / 33
Matrix Analysis and Computing
§2.3
. Ý
'Ì» 1. ½Âµ A ∈ C § λ , λ , · · · , λ A ' n A§ ¡
≤ =
A A
∞,
1 1
∞
≤ max{ A
A 1}
Ónk
A
2
≤ A F, A
m1 ,
A
m∞
Nanjing University of Science and Technology
4 / 33
Matrix Analysis and Computing
~ 1µÁO A ' Ì»

0.2 0.1 0 A = −0.2 0 0.2 −0.1 −0.2 0
4.
½nµ A ∈ C §é?¿ ε > 0§ 3Ý
ê ¦ ·
n×n m
A
m
≤ ρ(A) + ε
Nanjing University of Science and Technology
6 / 33
Matrix Analysis and Computing
y²µ P ∈ C ¦&
n×n n

Nanjing University of Science and Technology
5 / 33
Matrix Analysis and Computing
~ 1µÁO A ' Ì»

0.2 0.1 0 A = −0.2 0 0.2 −0.1 −0.2 0
2 / 33
Matrix Analysis and Computing
3.
½nµρ(A) ≤
A .
Nanjing University of Science and Technology
3 / 33
Matrix Analysis and Computing
3.
½nµρ(A) ≤ A . y²µ λ ´ A 'A§ · C þ · N'þê"d Ax = λx &
ê · N'þê§K
1.
n×n n×n (0) (§ ^gñ"
v
= Px (0)
v
≤ P
x (0)
v
Nanjing University of Science and Technology
10 / 33
Matrix Analysis and Computing
n×n n n×n n n×n −1
δx x
v v
≤
v
A 1− A
n
A−1 A−1
δA A
δb v δA + A b v ·
Ù¥ ê"
·
´ C þÝ
ê
N'þ
15 / 33
Nanjing University of Science and Technology
Matrix Analysis and Computing
Nanjing University of Science and Technology
13 / 33
Matrix Analysis and Computing
íص A ∈ C § δA ∈ C . eé C þ,Ý
ê · k A δA < 1§ K £éØ ¤
n×n n −1 n×n n×n A A A−1 δA A−1 − (A + δ A)−1 ≤ A A−1 1 − A A−1 δA δA
n−1

λn
Nanjing University of Science and Technology 7 / 33
Matrix Analysis and Computing
u´
D −1 P −1 APD
∞
≤ max(|λj | + ε) = ρ(A) + ε
j
S½
B
m
= D −1 P −1 BPD
Nanjing University of Science and Technology
11 / 33
Matrix Analysis and Computing
y²µ(1) A + δA = A(I + A δA) (2) d (A + δ A)(A + δ A) = I § &
−1 −1
A(A + δ A)−1 = I − δ A(A + δ A)−1 (A + δ A)−1 = A−1 − A−1 δ A(A + δ A)−1 (A + δ A)−1 ≤ A−1 + A−1 δ A (A + δ A)
Nanjing University of Science and Technology
2 / 33
Matrix Analysis and Computing
2.
½nµ A ∈ C §K
n×n
(1) ρ(Ak ) = (ρ(A))k (2) ρ(AH A) = ρ(AAH ) = A
2 2
Nanjing University of Science and Technology
A A−1 ≤ 1 − A−1 δ A
v
= Ax
≤ A
x
v
§ \&
δA δb v + A A x
v
Nanjing University of Science and Technology
16 / 33
Matrix Analysis and Computing
4.
½Âµ A ∈ C § ¡
n×n n
2 / 33
Matrix Analysis and Computing
2.
½nµ A ∈ C §K
n×n
(1) ρ(Ak ) = (ρ(A))k (2) ρ(AH A) = ρ(AAH ) = A (3)
¨ A ´SÝ
§ρ(A) =
2 2
A 2.
Nanjing University of Science and Technology
n×n 1 2 n
êA^
ρ(A) = max |λj |
A 'Ì»"
j
Nanjing University of Science and Technology
1 / 33
Matrix Analysis and Computing
2.
½nµ A ∈ C §K
n×n
(1) ρ(Ak ) = (ρ(A))k
Matrix Analysis and Computing
~^^êµ
cond∞ (A) = A cond2 (A) = A µ1 , µ n
∞ 2
A−1
2
∞
A−1
H
=
©O A A '¢A λ A SÝ
§cond (A) = λ λ , λ ©O´ A 'UÚ¢A
1 2 n 1 n
Nanjing University of Science and Technology 18 / 33
µ1 µn
Matrix Analysis and Computing
1.
A ∈ C ÷v a = 1 (j = 1, 2, · · · , n)), a ≥ 0 § y² ρ(A) = 1 1.
λ1 P AP = J =
2 −1
δ1 λ2
... . .. δ
n−1 )
n−1
,
δi = 0
½1

- D = diag (1, ε, ε , · · · , ε
−1 −1
λn
) ),
Kk
λ1 εδ1
λ2
D P APD = D JD =
−1
... . . . εδ
n×n
∞,
B ∈ C n×n
K
B
m
´ B ∈ C þ'«Ý
ê k
A
m
= D −1 P −1 APD
∞
≤ ρ(A) + ε
8 / 33
Nanjing University of Science and Technology
Matrix Analysis and Computing
. Ý
'^ê A ∈ C , b ∈ C . 3¦ A ½)§| Ax = b § éuØ δA Ú δb § ïÄ a. A Ú (A + δ A) 'Cq§Ý§ b. x δ x 'Ø ¢"
v n
|λ| x
v
= λx
v
= Ax
v
≤ A
x