问题
First, don't be fooled by the long post, there is not a lot of code just an observation of results so there are few example matrices.
This is a bit related to this question: Matlab Codegen Eig Function - Is this a Bug?
I know that mex/C/C++ translated eig() function may not return the same eigenvectors when using the same function in MATLAB and that's fine, but i am puzzled with results I'm getting.
First this simple example:
Output
% c = diagonal matrix of eigenvalues
% b = matrix whose columns are the corresponding right eigenvectors
function [ b, c ] = eig_test(a)
[b, c] = eig(a);
end
Running as is for [b,c] = eig_test(magic(5)) this returns:
b =
-0.4472 0.0976 -0.6330 0.6780 -0.2619
-0.4472 0.3525 0.5895 0.3223 -0.1732
-0.4472 0.5501 -0.3915 -0.5501 0.3915
-0.4472 -0.3223 0.1732 -0.3525 -0.5895
-0.4472 -0.6780 0.2619 -0.0976 0.6330
c =
65.0000 0 0 0 0
0 -21.2768 0 0 0
0 0 -13.1263 0 0
0 0 0 21.2768 0
0 0 0 0 13.1263
Translating that to mex function and running eig_test_mex(magic(5)) returns:
b =
0.4472 + 0.0000i 0.0976 + 0.0000i -0.6330 + 0.0000i 0.6780 + 0.0000i -0.2619 + 0.0000i
0.4472 + 0.0000i 0.3525 + 0.0000i 0.5895 + 0.0000i 0.3223 + 0.0000i -0.1732 + 0.0000i
0.4472 + 0.0000i 0.5501 + 0.0000i -0.3915 + 0.0000i -0.5501 + 0.0000i 0.3915 + 0.0000i
0.4472 + 0.0000i -0.3223 + 0.0000i 0.1732 + 0.0000i -0.3525 + 0.0000i -0.5895 + 0.0000i
0.4472 + 0.0000i -0.6780 + 0.0000i 0.2619 + 0.0000i -0.0976 + 0.0000i 0.6330 + 0.0000i
c =
65.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i -21.2768 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i -13.1263 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 21.2768 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 13.1263 + 0.0000i
Now here the values are actually the same (with exception of first vector that differs only in - sign) but from where are these complex imaginary parts coming from?
Second example:
Using the same function as above, the input matrix is now this one below instead of magic(5):
input_matrix =
0.0440 -0.0000 0.0486 0.0752 0.0848 0.0881 0.0883 0.0874 0.0856 0.0832 0.0805 0.0775 0.0742 0.0709 0.0676 0.0644 0.0612 0.0580 0.0548 0.0518 0.0487
-0.0000 0 -0.0000 -0.0000 0 -0.0000 -0.0000 0 0 -0.0000 -0.0000 -0.0000 0 0 -0.0000 -0.0000 0 0 0 -0.0000 0
0.0486 -0.0000 0.1253 0.1231 0.1128 0.1028 0.0940 0.0867 0.0803 0.0747 0.0697 0.0651 0.0609 0.0570 0.0535 0.0503 0.0473 0.0446 0.0421 0.0397 0.0375
0.0752 -0.0000 0.1231 0.3049 0.2850 0.2641 0.2454 0.2297 0.2157 0.2034 0.1924 0.1820 0.1723 0.1636 0.1556 0.1482 0.1414 0.1351 0.1292 0.1237 0.1186
0.0848 0 0.1128 0.2850 0.3941 0.3685 0.3451 0.3253 0.3077 0.2921 0.2780 0.2646 0.2521 0.2407 0.2303 0.2207 0.2119 0.2036 0.1959 0.1887 0.1819
0.0881 -0.0000 0.1028 0.2641 0.3685 0.4420 0.4161 0.3943 0.3746 0.3571 0.3413 0.3262 0.3120 0.2990 0.2871 0.2762 0.2661 0.2566 0.2478 0.2396 0.2318
0.0883 -0.0000 0.0940 0.2454 0.3451 0.4161 0.4705 0.4474 0.4266 0.4079 0.3910 0.3748 0.3595 0.3455 0.3326 0.3208 0.3098 0.2996 0.2900 0.2811 0.2726
0.0874 0 0.0867 0.2297 0.3253 0.3943 0.4474 0.4918 0.4701 0.4507 0.4330 0.4160 0.3999 0.3851 0.3716 0.3591 0.3474 0.3366 0.3265 0.3170 0.3080
0.0856 0 0.0803 0.2157 0.3077 0.3746 0.4266 0.4701 0.5067 0.4868 0.4686 0.4511 0.4343 0.4190 0.4049 0.3919 0.3798 0.3686 0.3580 0.3481 0.3387
0.0832 -0.0000 0.0747 0.2034 0.2921 0.3571 0.4079 0.4507 0.4868 0.5182 0.4997 0.4817 0.4645 0.4487 0.4343 0.4209 0.4084 0.3968 0.3859 0.3757 0.3660
0.0805 -0.0000 0.0697 0.1924 0.2780 0.3413 0.3910 0.4330 0.4686 0.4997 0.5269 0.5087 0.4911 0.4751 0.4603 0.4466 0.4339 0.4220 0.4108 0.4003 0.3905
0.0775 -0.0000 0.0651 0.1820 0.2646 0.3262 0.3748 0.4160 0.4511 0.4817 0.5087 0.5317 0.5140 0.4977 0.4827 0.4688 0.4559 0.4438 0.4325 0.4218 0.4117
0.0742 0 0.0609 0.1723 0.2521 0.3120 0.3595 0.3999 0.4343 0.4645 0.4911 0.5140 0.5337 0.5173 0.5021 0.4881 0.4750 0.4628 0.4514 0.4406 0.4304
0.0709 0 0.0570 0.1636 0.2407 0.2990 0.3455 0.3851 0.4190 0.4487 0.4751 0.4977 0.5173 0.5351 0.5199 0.5057 0.4926 0.4803 0.4687 0.4578 0.4475
0.0676 -0.0000 0.0535 0.1556 0.2303 0.2871 0.3326 0.3716 0.4049 0.4343 0.4603 0.4827 0.5021 0.5199 0.5362 0.5220 0.5087 0.4963 0.4847 0.4737 0.4634
0.0644 -0.0000 0.0503 0.1482 0.2207 0.2762 0.3208 0.3591 0.3919 0.4209 0.4466 0.4688 0.4881 0.5057 0.5220 0.5370 0.5237 0.5112 0.4995 0.4885 0.4781
0.0612 0 0.0473 0.1414 0.2119 0.2661 0.3098 0.3474 0.3798 0.4084 0.4339 0.4559 0.4750 0.4926 0.5087 0.5237 0.5376 0.5251 0.5134 0.5023 0.4918
0.0580 0 0.0446 0.1351 0.2036 0.2566 0.2996 0.3366 0.3686 0.3968 0.4220 0.4438 0.4628 0.4803 0.4963 0.5112 0.5251 0.5381 0.5263 0.5152 0.5047
0.0548 0 0.0421 0.1292 0.1959 0.2478 0.2900 0.3265 0.3580 0.3859 0.4108 0.4325 0.4514 0.4687 0.4847 0.4995 0.5134 0.5263 0.5384 0.5273 0.5167
0.0518 -0.0000 0.0397 0.1237 0.1887 0.2396 0.2811 0.3170 0.3481 0.3757 0.4003 0.4218 0.4406 0.4578 0.4737 0.4885 0.5023 0.5152 0.5273 0.5387 0.5281
0.0487 0 0.0375 0.1186 0.1819 0.2318 0.2726 0.3080 0.3387 0.3660 0.3905 0.4117 0.4304 0.4475 0.4634 0.4781 0.4918 0.5047 0.5167 0.5281 0.5389
Running for [b,c] = eig_test(input_matrix) returns:
b =
0.0000 -0.0085 -0.0117 -0.0166 -0.0251 -0.0442 0.1334 0.9260 -0.1493 0.0548 -0.0283 0.0382 0.0170 0.0977 0.1285 -0.1697 0.1100 -0.1149 0.0635 0.0881 0.0424
1.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000
-0.0000 0.0020 0.0029 0.0042 0.0067 0.0125 -0.0404 -0.2919 0.0495 -0.0178 0.0219 0.0179 0.1086 0.2330 0.4679 -0.5139 0.4286 -0.3220 0.2241 0.1454 0.0392
0.0000 0.0001 0.0001 0.0002 0.0003 0.0005 -0.0010 -0.0102 -0.0017 -0.0143 -0.0484 -0.1305 -0.2829 -0.4495 -0.4237 0.0949 0.2592 -0.3986 0.4161 0.3145 0.1072
-0.0000 0.0001 0.0002 0.0003 0.0006 0.0005 -0.0076 -0.0267 0.0241 0.0641 0.1763 0.3391 0.4702 0.3538 -0.0858 0.3847 -0.2103 -0.1113 0.3607 0.3698 0.1531
0.0000 0.0001 0.0001 0.0003 -0.0001 0.0044 0.0173 -0.0280 -0.0740 -0.2070 -0.3832 -0.4693 -0.2586 0.1790 0.3224 0.0909 -0.3636 0.1799 0.2003 0.3654 0.1849
0.0000 0.0001 0.0001 -0.0000 0.0029 -0.0168 -0.0758 0.0155 0.2178 0.4038 0.4549 0.1676 -0.2686 -0.2836 0.1914 -0.2398 -0.2029 0.3295 0.0210 0.3288 0.2079
-0.0000 0.0001 0.0000 0.0015 -0.0114 0.0623 0.1958 -0.0857 -0.3987 -0.4488 -0.1154 0.3136 0.2223 -0.2549 -0.1457 -0.2976 0.0628 0.3254 -0.1323 0.2755 0.2255
-0.0000 0.0001 0.0005 -0.0058 0.0419 -0.1658 -0.3782 0.0966 0.4538 0.1163 -0.3266 -0.1826 0.2893 0.0922 -0.2891 -0.1140 0.2566 0.2114 -0.2394 0.2128 0.2385
0.0000 0.0001 -0.0020 0.0224 -0.1169 0.3338 0.4695 -0.1012 -0.1655 0.3142 0.1810 -0.3027 -0.0583 0.2923 -0.1581 0.1266 0.3013 0.0487 -0.2951 0.1468 0.2482
-0.0000 -0.0003 0.0085 -0.0675 0.2572 -0.4725 -0.2826 -0.0124 -0.2658 -0.2251 0.2927 0.0580 -0.2952 0.1782 0.0838 0.2657 0.2098 -0.1089 -0.3025 0.0811 0.2552
0.0000 0.0016 -0.0282 0.1667 -0.4258 0.4018 -0.1483 0.0743 0.2952 -0.2511 -0.1046 0.3011 -0.1748 -0.0861 0.2409 0.2463 0.0445 -0.2242 -0.2699 0.0183 0.2593
-0.0000 -0.0060 0.0796 -0.3248 0.4873 -0.0371 0.3582 -0.0471 0.1556 0.1904 -0.3026 0.1397 0.1135 -0.2564 0.2258 0.1072 -0.1212 -0.2793 -0.2087 -0.0393 0.2613
-0.0000 0.0202 -0.1839 0.4791 -0.2780 -0.3330 -0.0369 -0.0733 -0.2823 0.2701 -0.0569 -0.1694 0.2707 -0.2224 0.0779 -0.0674 -0.2333 -0.2741 -0.1307 -0.0908 0.2618
0.0000 -0.0569 0.3438 -0.4801 -0.1422 0.2530 -0.3217 0.0344 -0.1598 -0.0783 0.2433 -0.2745 0.1829 -0.0400 -0.1011 -0.1994 -0.2669 -0.2193 -0.0462 -0.1354 0.2611
0.0000 0.1366 -0.4999 0.2085 0.3815 0.1931 0.0665 0.0563 0.2298 -0.2886 0.2299 -0.1025 -0.0438 0.1536 -0.2162 -0.2443 -0.2244 -0.1314 0.0362 -0.1730 0.2594
-0.0000 -0.2750 0.5187 0.2125 -0.1234 -0.2906 0.3002 -0.0330 0.2275 -0.0976 -0.0389 0.1510 -0.2225 0.2429 -0.2221 -0.1981 -0.1272 -0.0288 0.1102 -0.2036 0.2569
0.0000 0.4544 -0.2791 -0.3911 -0.2949 -0.1393 -0.0141 -0.0736 -0.1126 0.2031 -0.2494 0.2586 -0.2356 0.1924 -0.1303 -0.0881 -0.0057 0.0716 0.1710 -0.2273 0.2536
-0.0000 -0.5901 -0.1434 0.0927 0.2187 0.2750 -0.2874 -0.0090 -0.2727 0.2455 -0.2023 0.1516 -0.0947 0.0451 0.0094 0.0433 0.1096 0.1561 0.2158 -0.2445 0.2496
0.0000 0.5431 0.4134 0.3143 0.2317 0.1604 -0.0958 0.0563 -0.0617 0.0119 0.0294 -0.0645 0.0962 -0.1161 0.1372 0.1548 0.1950 0.2157 0.2434 -0.2554 0.2451
-0.0000 -0.2286 -0.2287 -0.2288 -0.2283 -0.2265 0.2263 0.0338 0.2163 -0.2220 0.2231 -0.2228 0.2210 -0.2154 0.2075 0.2171 0.2365 0.2458 0.2538 -0.2607 0.2401
c =
-0.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0.0063 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0.0076 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.0089 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0.0105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0.0123 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.0144 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0.0157 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0.0171 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0.0203 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0.0244 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0.0298 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0.0368 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0.0464 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0581 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0762 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1063 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1746 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3324 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0781 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7.1136
And running mex version for [b,c] = eig_test_mex(input_matrix) returns:
b=
0.0424+0.0000i 0.0881+0.0000i 0.0635+0.0000i -0.1149+0.0000i 0.11+0.0000i -0.1697+0.0000i 0.1285+0.0000i 0.0977+0.0000i 0.017+0.0000i 0.0382+0.0000i -0.0283+0.0000i 0.0548+0.0000i 0.1493+0.0000i -0.926+0.0000i -0.1334+0.0000i 0.0085+0.0000i -0.0442+0.0000i -0.0117+0.0000i -0.0251+0.0000i -0.0166+0.0000i 0+0.0000i
+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i -1+0.0000i
0.0392+0.0000i 0.1454+0.0000i 0.2241+0.0000i -0.322+0.0000i 0.4286+0.0000i -0.5139+0.0000i 0.4679+0.0000i 0.233+0.0000i 0.1086+0.0000i 0.0179+0.0000i 0.0219+0.0000i -0.0178+0.0000i -0.0495+0.0000i 0.2919+0.0000i 0.0404+0.0000i -0.002+0.0000i 0.0125+0.0000i 0.0029+0.0000i 0.0067+0.0000i 0.0042+0.0000i 0+0.0000i
0.1072+0.0000i 0.3145+0.0000i 0.4161+0.0000i -0.3986+0.0000i 0.2592+0.0000i 0.0949+0.0000i -0.4237+0.0000i -0.4495+0.0000i -0.2829+0.0000i -0.1305+0.0000i -0.0484+0.0000i -0.0143+0.0000i 0.0017+0.0000i 0.0102+0.0000i 0.001+0.0000i -0.0001+0.0000i 0.0005+0.0000i 0.0001+0.0000i 0.0003+0.0000i 0.0002+0.0000i 0+0.0000i
0.1531+0.0000i 0.3698+0.0000i 0.3607+0.0000i -0.1113+0.0000i -0.2103+0.0000i 0.3847+0.0000i -0.0858+0.0000i 0.3538+0.0000i 0.4702+0.0000i 0.3391+0.0000i 0.1763+0.0000i 0.0641+0.0000i -0.0241+0.0000i 0.0267+0.0000i 0.0076+0.0000i -0.0001+0.0000i 0.0005+0.0000i 0.0002+0.0000i 0.0006+0.0000i 0.0003+0.0000i 0+0.0000i
0.1849+0.0000i 0.3654+0.0000i 0.2003+0.0000i 0.1799+0.0000i -0.3636+0.0000i 0.0909+0.0000i 0.3224+0.0000i 0.179+0.0000i -0.2586+0.0000i -0.4693+0.0000i -0.3832+0.0000i -0.207+0.0000i 0.074+0.0000i 0.028+0.0000i -0.0173+0.0000i -0.0001+0.0000i 0.0044+0.0000i 0.0001+0.0000i -0.0001+0.0000i 0.0003+0.0000i 0+0.0000i
0.2079+0.0000i 0.3288+0.0000i 0.021+0.0000i 0.3295+0.0000i -0.2029+0.0000i -0.2398+0.0000i 0.1914+0.0000i -0.2836+0.0000i -0.2686+0.0000i 0.1676+0.0000i 0.4549+0.0000i 0.4038+0.0000i -0.2178+0.0000i -0.0155+0.0000i 0.0758+0.0000i -0.0001+0.0000i -0.0168+0.0000i 0.0001+0.0000i 0.0029+0.0000i 0+0.0000i 0+0.0000i
0.2255+0.0000i 0.2755+0.0000i -0.1323+0.0000i 0.3254+0.0000i 0.0628+0.0000i -0.2976+0.0000i -0.1457+0.0000i -0.2549+0.0000i 0.2223+0.0000i 0.3136+0.0000i -0.1154+0.0000i -0.4488+0.0000i 0.3987+0.0000i 0.0857+0.0000i -0.1958+0.0000i -0.0001+0.0000i 0.0623+0.0000i 0+0.0000i -0.0114+0.0000i 0.0015+0.0000i 0+0.0000i
0.2385+0.0000i 0.2128+0.0000i -0.2394+0.0000i 0.2114+0.0000i 0.2566+0.0000i -0.114+0.0000i -0.2891+0.0000i 0.0922+0.0000i 0.2893+0.0000i -0.1826+0.0000i -0.3266+0.0000i 0.1163+0.0000i -0.4538+0.0000i -0.0966+0.0000i 0.3782+0.0000i -0.0001+0.0000i -0.1658+0.0000i 0.0005+0.0000i 0.0419+0.0000i -0.0058+0.0000i 0+0.0000i
0.2482+0.0000i 0.1468+0.0000i -0.2951+0.0000i 0.0487+0.0000i 0.3013+0.0000i 0.1266+0.0000i -0.1581+0.0000i 0.2923+0.0000i -0.0583+0.0000i -0.3027+0.0000i 0.181+0.0000i 0.3142+0.0000i 0.1655+0.0000i 0.1012+0.0000i -0.4695+0.0000i -0.0001+0.0000i 0.3338+0.0000i -0.002+0.0000i -0.1169+0.0000i 0.0224+0.0000i 0+0.0000i
0.2552+0.0000i 0.0811+0.0000i -0.3025+0.0000i -0.1089+0.0000i 0.2098+0.0000i 0.2657+0.0000i 0.0838+0.0000i 0.1782+0.0000i -0.2952+0.0000i 0.058+0.0000i 0.2927+0.0000i -0.2251+0.0000i 0.2658+0.0000i 0.0124+0.0000i 0.2826+0.0000i 0.0003+0.0000i -0.4725+0.0000i 0.0085+0.0000i 0.2572+0.0000i -0.0675+0.0000i 0+0.0000i
0.2593+0.0000i 0.0183+0.0000i -0.2699+0.0000i -0.2242+0.0000i 0.0445+0.0000i 0.2463+0.0000i 0.2409+0.0000i -0.0861+0.0000i -0.1748+0.0000i 0.3011+0.0000i -0.1046+0.0000i -0.2511+0.0000i -0.2952+0.0000i -0.0743+0.0000i 0.1483+0.0000i -0.0016+0.0000i 0.4018+0.0000i -0.0282+0.0000i -0.4258+0.0000i 0.1667+0.0000i 0+0.0000i
0.2613+0.0000i -0.0393+0.0000i -0.2087+0.0000i -0.2793+0.0000i -0.1212+0.0000i 0.1072+0.0000i 0.2258+0.0000i -0.2564+0.0000i 0.1135+0.0000i 0.1397+0.0000i -0.3026+0.0000i 0.1904+0.0000i -0.1556+0.0000i 0.0471+0.0000i -0.3582+0.0000i 0.006+0.0000i -0.0371+0.0000i 0.0796+0.0000i 0.4873+0.0000i -0.3248+0.0000i 0+0.0000i
0.2618+0.0000i -0.0908+0.0000i -0.1307+0.0000i -0.2741+0.0000i -0.2333+0.0000i -0.0674+0.0000i 0.0779+0.0000i -0.2224+0.0000i 0.2707+0.0000i -0.1694+0.0000i -0.0569+0.0000i 0.2701+0.0000i 0.2823+0.0000i 0.0733+0.0000i 0.0369+0.0000i -0.0202+0.0000i -0.333+0.0000i -0.1839+0.0000i -0.278+0.0000i 0.4791+0.0000i 0+0.0000i
0.2611+0.0000i -0.1354+0.0000i -0.0462+0.0000i -0.2193+0.0000i -0.2669+0.0000i -0.1994+0.0000i -0.1011+0.0000i -0.04+0.0000i 0.1829+0.0000i -0.2745+0.0000i 0.2433+0.0000i -0.0783+0.0000i 0.1598+0.0000i -0.0344+0.0000i 0.3217+0.0000i 0.0569+0.0000i 0.253+0.0000i 0.3438+0.0000i -0.1422+0.0000i -0.4801+0.0000i 0+0.0000i
0.2594+0.0000i -0.173+0.0000i 0.0362+0.0000i -0.1314+0.0000i -0.2244+0.0000i -0.2443+0.0000i -0.2162+0.0000i 0.1536+0.0000i -0.0438+0.0000i -0.1025+0.0000i 0.2299+0.0000i -0.2886+0.0000i -0.2298+0.0000i -0.0563+0.0000i -0.0665+0.0000i -0.1366+0.0000i 0.1931+0.0000i -0.4999+0.0000i 0.3815+0.0000i 0.2085+0.0000i 0+0.0000i
0.2569+0.0000i -0.2036+0.0000i 0.1102+0.0000i -0.0288+0.0000i -0.1272+0.0000i -0.1981+0.0000i -0.2221+0.0000i 0.2429+0.0000i -0.2225+0.0000i 0.151+0.0000i -0.0389+0.0000i -0.0976+0.0000i -0.2275+0.0000i 0.033+0.0000i -0.3002+0.0000i 0.275+0.0000i -0.2906+0.0000i 0.5187+0.0000i -0.1234+0.0000i 0.2125+0.0000i 0+0.0000i
0.2536+0.0000i -0.2273+0.0000i 0.171+0.0000i 0.0716+0.0000i -0.0057+0.0000i -0.0881+0.0000i -0.1303+0.0000i 0.1924+0.0000i -0.2356+0.0000i 0.2586+0.0000i -0.2494+0.0000i 0.2031+0.0000i 0.1126+0.0000i 0.0736+0.0000i 0.0141+0.0000i -0.4544+0.0000i -0.1393+0.0000i -0.2791+0.0000i -0.2949+0.0000i -0.3911+0.0000i 0+0.0000i
0.2496+0.0000i -0.2445+0.0000i 0.2158+0.0000i 0.1561+0.0000i 0.1096+0.0000i 0.0433+0.0000i 0.0094+0.0000i 0.0451+0.0000i -0.0947+0.0000i 0.1516+0.0000i -0.2023+0.0000i 0.2455+0.0000i 0.2727+0.0000i 0.009+0.0000i 0.2874+0.0000i 0.5901+0.0000i 0.275+0.0000i -0.1434+0.0000i 0.2187+0.0000i 0.0927+0.0000i 0+0.0000i
0.2451+0.0000i -0.2554+0.0000i 0.2434+0.0000i 0.2157+0.0000i 0.195+0.0000i 0.1548+0.0000i 0.1372+0.0000i -0.1161+0.0000i 0.0962+0.0000i -0.0645+0.0000i 0.0294+0.0000i 0.0119+0.0000i 0.0617+0.0000i -0.0563+0.0000i 0.0958+0.0000i -0.5431+0.0000i 0.1604+0.0000i 0.4134+0.0000i 0.2317+0.0000i 0.3143+0.0000i 0+0.0000i
0.2401+0.0000i -0.2607+0.0000i 0.2538+0.0000i 0.2458+0.0000i 0.2365+0.0000i 0.2171+0.0000i 0.2075+0.0000i -0.2154+0.0000i 0.221+0.0000i -0.2228+0.0000i 0.2231+0.0000i -0.222+0.0000i -0.2163+0.0000i -0.0338+0.0000i -0.2263+0.0000i 0.2286+0.0000i -0.2265+0.0000i -0.2287+0.0000i -0.2283+0.0000i -0.2288+0.0000i 0+0.0000i
c=
7.1136 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1.0781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0.3324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.1746 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0.1063 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0.0762 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.0581 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0.0464 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0.0368 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0.0298 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0.0244 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0.0203 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0.0171 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0.0157 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0144 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0063 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0123 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0076 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0105 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0089 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ok, so the complex imaginary part is here again (i removed it from matrix c for clarity) but notice something. In eigenvalue matrix c there are the same values again but in reverse order. Why is that? At first glance is seems that values in b matrices are also the same just columns are reversed, but there are some differences. But as i understand this is ok.
Finally
I'm confused manly with two things. One is this transformation to complex numbers and the other one is this reverse ordering.
Does anyone have some insight on whats going on. Could first phenomena be resolved with updated MATLAB version or tool-chain? (I'm using MATLAB 2013, GCC 4.6 on Ubuntu 12.04). Second phenomena can probably be explained with math and implemented algorithms to compute eigenvectors/values. Any possible explanation?
回答1:
There is not only one valid answer for a eigenvalue problem. Whenever dealing with such cases, best practice is to use or define a canonical form to convert equivalent answers to identical answers. To convert any answer into such a canonical form, I would apply the following steps:
- Normalize all eigenvectors to length 1 (example:
[-0.4472,-0.4472,-0.4472,0.4472,0.4472]'instead of[-1,-1,-1,1,1]'). Could be achieved usingb=bsxfun(@rdivide,b,sqrt(sum(b.^2,1))) - For each eigenvectors with a negative value in the first component, take the negative value. (example:
[0.4472,0.4472,0.4472,-0.4472,-0.4472]'instead of[-0.4472,-0.4472,-0.4472,0.4472,0.4472]'). Could be achieved usingb=bsxfun(@times,sign(b(1,:)),b) - Sort eigenvectors and eigenvalues in ascending order of the eigenvalues. Could be achieved using this code
Please note that this is just one canonical form, not the canonical form. There might be canonical forms established but I have not found any reference so I basically put together what we discussed above to define a canonical form. Mathematically, applying such a canonical form results in a unique solution. In practice you will observe minor differences because of floating point precision errors.
来源：https://stackoverflow.com/questions/32205736/matlab-codgen-eig-function-strange-behaviour