Matlab中矩阵的分解
1、常见的分解方法
（1）三角分解（LU分解）
（2）正交分解（QR）
（3）特征值分解（eig分解）
（4）奇异值分解（svd）
（5）Chollesky分解
2、三角分解（LU分解）
>> A = [1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]
A =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
>> [l,u] = lu(A)
l =
0.0769 1.0000 0 0
0.3846 0.6667 -0.5000 1.0000
0.6923 0.3333 1.0000 0
1.0000 0 0 0
u =
13.0000 14.0000 15.0000 16.0000
0 0.9231 1.8462 2.7692
0 0 -0.0000 -0.0000
0 0 0 -0.0000
>> [l,u,p] = lu(A)
l =
1.0000 0 0 0
0.0769 1.0000 0 0
0.6923 0.3333 1.0000 0
0.3846 0.6667 -0.5000 1.0000
u =
13.0000 14.0000 15.0000 16.0000
0 0.9231 1.8462 2.7692
0 0 -0.0000 -0.0000
0 0 0 -0.0000
p =
0 0 0 1
1 0 0 0
0 0 1 0
0 1 0 0
说明：p是A交换矩阵
3、正交分解（QR）
>> A = [1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]
A =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
>> [q,r] = qr(A)
q =
-0.0602 -0.8345 0.2702 -0.4765
-0.3010 -0.4576 -0.0051 0.8366
-0.5417 -0.0808 -0.8003 -0.2439
-0.7825 0.2961 0.5352 -0.1163
r =
-16.6132 -18.2986 -19.9841 -21.6695
0 -1.0768 -2.1535 -3.2303
0 0 0.0000 0.0000
0 0 0 0.0000
>> [q,r,e] = qr(A)
q =
-0.1826 -0.8165 0.5068 0.2078
-0.3651 -0.4082 -0.8306 0.1006
-0.5477 -0.0000 0.1409 -0.8247
-0.7303 0.4082 0.1829 0.5163
r =
-21.9089 -16.4317 -18.2574 -20.0832
0 2.4495 1.6330 0.8165
0 0 0.0000 -0.0000
0 0 0 -0.0000
e =
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
说明：e是A交换矩阵
4、 特征值分解（eig分解）
>> A = magic(10)
A =
92 99 1 8 15 67 74 51 58 40
98 80 7 14 16 73 55 57 64 41
4 81 88 20 22 54 56 63 70 47
85 87 19 21 3 60 62 69 71 28
86 93 25 2 9 61 68 75 52 34
17 24 76 83 90 42 49 26 33 65
23 5 82 89 91 48 30 32 39 66
79 6 13 95 97 29 31 38 45 72
10 12 94 96 78 35 37 44 46 53
11 18 100 77 84 36 43 50 27 59
>> [v,d] = eig(A)
v =
Columns 1 through 5
0.3162 0.2500 -0.3162 0.3322 -0.4512
0.3162 0.2500 -0.3162 0.3725 0.3964
0.3162 -0.5000 -0.3162 0.3581 -0.4078
0.3162 0.2500 -0.3162 0.2660 0.1077
0.3162 0.2500 -0.3162 0.2098 0.1692
0.3162 -0.2500 0.3162 -0.3014 -0.4266
0.3162 -0.2500 0.3162 -0.2612 0.4211
0.3162 0.5000 0.3162 -0.1845 -0.1352
0.3162 -0.2500 0.3162 -0.3677 0.1324
0.3162 -0.2500 0.3162 -0.4238 0.1939
Columns 6 through 10
0.0336 0.1227 0.4554 -0.3542 - 0.0179i -0.3542 + 0.0179i
-0.0586 0.0781 0.1344 -0.4556 + 0.0000i -0.4556 - 0.0000i
0.4509 0.2997 0.4554 -0.3542 - 0.0179i -0.3542 + 0.0179i
-0.2334 0.2875 -0.0319 -0.0419 + 0.0952i -0.0419 - 0.0952i
-0.1674 -0.3274 -0.2572 -0.0177 - 0.1725i -0.0177 + 0.1725i
0.3178 0.1407 -0.4554 0.3542 + 0.0179i 0.3542 - 0.0179i
0.2256 0.0961 -0.1344 0.4556 0.4556
-0.7359 -0.6936 -0.4554 0.3542 + 0.0179i 0.3542 - 0.0179i
0.0508 0.3055 0.0319 0.0419 - 0.0952i 0.0419 + 0.0952i
0.1168 -0.3094 0.2572 0.0177 + 0.1725i 0.0177 - 0.1725i
d =
1.0e+02 *
Columns 1 through 5
5.0500 0 0 0 0
0 0.7500 0 0 0
0 0 -0.7500 0 0
0 0 0 -0.4255 0
0 0 0 0 -0.2625
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Columns 6 through 10
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0.4255 0 0 0 0
0 0.2625 0 0 0
0 0 -0.0000 0 0
0 0 0 0.0000 + 0.0000i 0
0 0 0 0 0.0000 - 0.0000i
>> [v,d] = eig(A,'nobalance')
v =
Columns 1 through 5
1.0000 0.5000 -1.0000 0.7839 -1.0000
1.0000 0.5000 -1.0000 0.8788 0.8785
1.0000 -1.0000 -1.0000 0.8449 -0.9038
1.0000 0.5000 -1.0000 0.6276 0.2388
1.0000 0.5000 -1.0000 0.4951 0.3751
1.0000 -0.5000 1.0000 -0.7112 -0.9454
1.0000 -0.5000 1.0000 -0.6163 0.9332
1.0000 1.0000 1.0000 -0.4353 -0.2995
1.0000 -0.5000 1.0000 -0.8676 0.2934
1.0000 -0.5000 1.0000 -1.0000 0.4297
Columns 6 through 10
0.0456 0.1769 1.0000 -0.1120 - 0.6408i -0.1120 + 0.6408i
-0.0796 0.1126 0.2952 -0.1852 - 0.8148i -0.1852 + 0.8148i
0.6127 0.4321 1.0000 -0.1120 - 0.6408i -0.1120 + 0.6408i
-0.3172 0.4145 -0.0700 -0.1872 - 0.0362i -0.1872 + 0.0362i
-0.2275 -0.4720 -0.5648 0.3013 - 0.1017i 0.3013 + 0.1017i
0.4318 0.2029 -1.0000 0.1120 + 0.6408i 0.1120 - 0.6408i
0.3066 0.1386 -0.2952 0.1852 + 0.8148i 0.1852 - 0.8148i
-1.0000 -1.0000 -1.0000 0.1120 + 0.6408i 0.1120 - 0.6408i
0.0690 0.4405 0.0700 0.1872 + 0.0362i 0.1872 - 0.0362i
0.1587 -0.4461 0.5648 -0.3013 + 0.1017i -0.3013 - 0.1017i
d =
1.0e+02 *
Columns 1 through 5
5.0500 0 0 0 0
0 0.7500 0 0 0
0 0 -0.7500 0 0
0 0 0 -0.4255 0
0 0 0 0 -0.2625
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Columns 6 through 10
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0.4255 0 0 0 0
0 0.2625 0 0 0
0 0 -0.0000 0 0
0 0 0 0.0000 + 0.0000i 0
0 0 0 0 0.0000 - 0.0000i
5、
Chollesky分解
>> A = [4,-1,1;-1,4.25,2.75;1,2.75,3]
A =
4.0000 -1.0000 1.0000
-1.0000 4.2500 2.7500
1.0000 2.7500 3.0000
>> chol(A)
ans =
2.0000 -0.5000 0.5000
0 2.0000 1.5000
0 0 0.7071