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
>> 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 - 0.0000i5、 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