矩阵论第四章 矩阵分解.pdf

第四章矩阵分解 1
第四章矩阵分解
§ 三角分解
目的:将 An×n 分解为下三角矩阵与上三角矩阵的乘积.
一、分解原理:以 n = 4 为例.
ai1
①∆1 ()A = a11 : a11 ≠ 0 ⇒ ci1 = ()i = 2,3,4
a11
 1  1 
c 1 − c 1 
 21 −1  21 
L1 = , L1 =
c31 0 1 − c31 0 1 

c 0 0 1 − c 0 0 1
 41  41 
a11 a12 a13 a14 
(1) ()1 ()1 
a a a ∆
−1  22 23 24 ()1
L1 A = (1) ()1 ()1 = A
 a32 a33 a34 

a (1) a ()1 a ()1
 42 43 44 
a (1)
②()1 ()1 (1) i 2
∆2 ()A = ∆2 ()A = a11a22 : a22 ≠ 0 ⇒ ci 2 = ()1 ()i = 3,4
a22
1 1 
0 1 0 1 
−1 
L2 = , L2 =
0 c32 1 0 − c32 1 

0 c42 0 10 − c42 0 1
a11 a12 a13 a14 
()1 ()1 ()1 
a a a ∆
−1 ()1  22 23 24 (2)
L2 A = ()2 (2) = A
 a33 a34 
()2 (2)
 a43 a44 
a (2)
③()2 (1) ()2 ()2 43
∆3 ()A = ∆3 ()A = a11a22 a33 : a33 ≠ 0 ⇒ c43 = ()2
a33
第四章矩阵分解 2
1 1 
0 1 0 1 
L = , L−1 = 
3 0 0 1  3 0 0 1 

0 0 c 1 0 0 − c 1
 43  43 
a11 a12 a13 a14 
()1 ()1 ()1 
a a a ∆
−1 ()2  22 23 24 (3)
L3 A = ()2 ()2 = A
 a33 a34 

a ()3
 44 
−1 −1 −1 (3) (3)
即 L3 L2 L1 A = A ⇒ A = L1 L2 L3 A
 1 

∆ c 1
 21 (3)
令 L= L1 L2 L3 = ,则 A = LA .
c31 c32 1 

c41 c42 c43 1
a11 1 ∗∗∗
 a ()1  1 ∗∗
分解()3  22 则.
A = ()2 = DU , A = LDU
 a33  1 ∗
()3 
 a44  1
唯一.
Th1 An×n , ∆k ()A ≠ 0 (k = 1,2,L,n − 1)⇒ A = LDU
∆~
二、紧凑格式算法: A = LDU = LU (Crout 分解)
l11 1 u12 L u1n 

~ l l 1 u
L =  21 22 , U =  L 2n 
 M M O  O M 

ln1 ln2 L lnn  1 
i,1 元:a = l ⋅1 ⇒ l = a i = 1, ,n
() i1 i1 i1 i1 ( L )
a
()1, j 元:a = l ⋅ u ⇒ u = 1 j ()j = 2, ,n
1 j 11 1 j 1 j l L
11
元:
()i,k aik = li1 ⋅ u1k +L+ li,k−1 ⋅ uk−1,k + lik ⋅1 (i ≥ k)
⇒ lik = aik −(li1 ⋅ u1k +L+ li,k−1 ⋅ uk−1,k )
第四章矩阵分解 3
元
()k, j : akj = lk1 ⋅ u1 j +L+ lk ,k−1 ⋅ uk−1, j + lkk ⋅ uk j ( j > k)
1

⇒ ukj = []akj −(lk1 ⋅ u1 j +L+ lk ,k−1 ⋅ uk−1, j )