问题
I want to compute the eigenvalues of a complex matrix in R. Comparing the values obtained in MATLAB, I don't get the same eigenvalues obtained in R computing it from the exact same matrix.
In R, I used eigen(). In MATLAB used eig() or eigs() (both functions give out the same eigenvalues but different to the R ones).
For real matrices, R and MATLAB are consistent. How to get in R the same result of MATLAB ?
Example in Matlab real matrix:
squeeze(Sigma_X(:,:,h+1)) %real matrix 10x10
ans =
1.0e+03 *
Columns 1 through 6
5.8706 5.9966 6.1225 6.2484 6.3744 6.5003
5.9966 6.1260 6.2554 6.3849 6.5143 6.6438
6.1225 6.2554 6.3884 6.5213 6.6543 6.7872
6.2484 6.3849 6.5213 6.6578 6.7942 6.9306
6.3744 6.5143 6.6543 6.7942 6.9341 7.0741
6.5003 6.6438 6.7872 6.9306 7.0741 7.2175
6.6263 6.7732 6.9201 7.0671 7.2140 7.3609
6.7522 6.9026 7.0531 7.2035 7.3539 7.5044
6.8782 7.0321 7.1860 7.3399 7.4939 7.6478
7.0041 7.1615 7.3190 7.4764 7.6338 7.7912
Columns 7 through 10
6.6263 6.7522 6.8782 7.0041
6.7732 6.9026 7.0321 7.1615
6.9201 7.0531 7.1860 7.3190
7.0671 7.2035 7.3399 7.4764
7.2140 7.3539 7.4939 7.6338
7.3609 7.5044 7.6478 7.7912
7.5079 7.6548 7.8017 7.9487
7.6548 7.8052 7.9557 8.1061
7.8017 7.9557 8.1096 8.2635
7.9487 8.1061 8.2635 8.4210
opt.disp = 0;
[P, D] = eigs(squeeze(Sigma_X(:,:,h+1)),q,'LM',opt) %compute the eigenvalues and eigenvectors
P =
-0.2872 0.5128
-0.2936 0.4029
-0.2999 0.2930
-0.3062 0.1830
-0.3125 0.0731
-0.3189 -0.0368
-0.3252 -0.1467
-0.3315 -0.2566
-0.3379 -0.3665
-0.3442 -0.4764
D =
1.0e+04 *
7.0984 0
0 0.0054
Example in R real matrix:
drop(Sigma_X[,,h+1]) #Same real matrix as before, 10x10
Columns 1 through 5
5870.610+0i 5996.552+0i 6122.495+0i 6248.438+0i 6374.381+0i
5996.552+0i 6125.994+0i 6255.435+0i 6384.876+0i 6514.317+0i
6122.495+0i 6255.435+0i 6388.375+0i 6521.314+0i 6654.254+0i
6248.438+0i 6384.876+0i 6521.314+0i 6657.752+0i 6794.190+0i
6374.381+0i 6514.317+0i 6654.254+0i 6794.190+0i 6934.127+0i
6500.324+0i 6643.759+0i 6787.194+0i 6930.629+0i 7074.063+0i
6626.267+0i 6773.200+0i 6920.133+0i 7067.067+0i 7214.000+0i
6752.210+0i 6902.641+0i 7053.073+0i 7203.505+0i 7353.937+0i
6878.152+0i 7032.083+0i 7186.013+0i 7339.943+0i 7493.873+0i
7004.095+0i 7161.524+0i 7318.952+0i 7476.381+0i 7633.810+0i
Columns 6 through 10
6500.324+0i 6626.267+0i 6752.210+0i 6878.152+0i 7004.095+0i
6643.759+0i 6773.200+0i 6902.641+0i 7032.083+0i 7161.524+0i
6787.194+0i 6920.133+0i 7053.073+0i 7186.013+0i 7318.952+0i
6930.629+0i 7067.067+0i 7203.505+0i 7339.943+0i 7476.381+0i
7074.063+0i 7214.000+0i 7353.937+0i 7493.873+0i 7633.810+0i
7217.498+0i 7360.933+0i 7504.368+0i 7647.803+0i 7791.238+0i
7360.933+0i 7507.867+0i 7654.800+0i 7801.733+0i 7948.667+0i
7504.368+0i 7654.800+0i 7805.232+0i 7955.663+0i 8106.095+0i
7647.803+0i 7801.733+0i 7955.663+0i 8109.594+0i 8263.524+0i
7791.238+0i 7948.667+0i 8106.095+0i 8263.524+0i 8420.952+0i
Decomp <- eigen(drop(Sigma_X[,,h+1])) #frequency 0
DD <- diag(Decomp$values[1:q])
PP <- Decomp$vectors[,1:q]
PP
[1,] -0.2872322+0i 0.5127886+0i
[2,] -0.2935595+0i 0.4028742+0i
[3,] -0.2998868+0i 0.2929598+0i
[4,] -0.3062141+0i 0.1830454+0i
[5,] -0.3125415+0i 0.0731310+0i
[6,] -0.3188688+0i -0.0367834+0i
[7,] -0.3251961+0i -0.1466978+0i
[8,] -0.3315234+0i -0.2566122+0i
[9,] -0.3378507+0i -0.3665266+0i
[10,] -0.3441780+0i -0.4764410+0i
DD
[1,] 70983.65 0.00000
[2,] 0.00 54.34878
AS YOU CAN SEE, WHEN THE MATRIX IS REAL, MATLAB AND R GIVE YOU BACK THE SAME EIGENVECTORS AND EIGENVALUES.
LET'S TRY THE SAME CODE BUT WITH A COMPLEX MATRIX.
Example in Matlab complex matrix:
j=1
squeeze(Sigma_X(:,:,j))
ans =
1.0e+02 *
Columns 1 through 5
3.4601+0.0000i 3.5075-0.0304i 3.5548-0.0607i 3.6022-0.0911i 3.6496-0.1215i
3.5075+0.0304i 3.5562+0.0000i 3.6049-0.0304i 3.6535-0.0607i 3.7022-0.0911i
3.5548+0.0607i 3.6049+0.0304i 3.6549+0.0000i 3.7049-0.0304i 3.7549- 0.0607i
3.6022+0.0911i 3.6535+0.0607i 3.7049+0.0304i 3.7562+0.0000i 3.8075- 0.0304i
3.6496+0.1215i 3.7022+0.0911i 3.7549+0.0607i 3.8075+0.0304i 3.8602+ 0.0000i
3.6970+0.1518i 3.7509+0.1215i 3.8049+0.0911i 3.8588+0.0607i 3.9128+ 0.0304i
3.7444+0.1822i 3.7996+0.1518i 3.8549+0.1215i 3.9102+0.0911i 3.9654+ 0.0607i
3.7917+0.2126i 3.8483+0.1822i 3.9049+0.1518i 3.9615+0.1215i 4.0181+ 0.0911i
3.8391+0.2429i 3.8970+0.2126i 3.9549+0.1822i 4.0128+0.1518i 4.0707+ 0.1215i
3.8865+0.2733i 3.9457+0.2429i 4.0049+0.2126i 4.0641+0.1822i 4.1234+ 0.1518i
Columns 6 through 10
3.6970-0.1518i 3.7444-0.1822i 3.7917-0.2126i 3.8391-0.2429i 3.8865- 0.2733i
3.7509-0.1215i 3.7996-0.1518i 3.8483-0.1822i 3.8970-0.2126i 3.9457- 0.2429i
3.8049-0.0911i 3.8549-0.1215i 3.9049-0.1518i 3.9549-0.1822i 4.0049- 0.2126i
3.8588-0.0607i 3.9102-0.0911i 3.9615-0.1215i 4.0128-0.1518i 4.0641- 0.1822i
3.9128-0.0304i 3.9654-0.0607i 4.0181-0.0911i 4.0707-0.1215i 4.1234- 0.1518i
3.9668+0.0000i 4.0207-0.0304i 4.0747-0.0607i 4.1286-0.0911i 4.1826- 0.1215i
4.0207+0.0304i 4.0760+0.0000i 4.1313-0.0304i 4.1865-0.0607i 4.2418- 0.0911i
4.0747+0.0607i 4.1313+0.0304i 4.1878+0.0000i 4.2444-0.0304i 4.3010- 0.0607i
4.1286+0.0911i 4.1865+0.0607i 4.2444+0.0304i 4.3023+0.0000i 4.3602- 0.0304i
4.1826+0.1215i 4.2418+0.0911i 4.3010+0.0607i 4.3602+0.0304i 4.4195+ 0.0000i
[P, D] = eigs(squeeze(Sigma_X(:,:,j)),q,'LM',opt);
P =
-0.1206 - 0.2711i 0.0471 + 0.5052i
-0.1199 - 0.2760i 0.0384 + 0.3955i
-0.1192 - 0.2810i 0.0297 + 0.2859i
-0.1186 - 0.2859i 0.0210 + 0.1762i
-0.1179 - 0.2908i 0.0124 + 0.0666i
-0.1172 - 0.2957i 0.0037 - 0.0430i
-0.1165 - 0.3006i -0.0050 - 0.1527i
-0.1159 - 0.3055i -0.0137 - 0.2623i
-0.1152 - 0.3104i -0.0224 - 0.3720i
-0.1145 - 0.3153i -0.0311 - 0.4816i
D =
1.0e+03 *
3.9211 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0029 - 0.0000i
Example in R complex matrix:
j=1
drop(Sigma_X[,,j])
Columns 1 through 4
346.0094+0.0000i 350.7470-3.0368i 355.4846-6.0736i 360.2222-9.1104i
350.7470+3.0368i 355.6162+0.0000i 360.4854-3.0368i 365.3546-6.0736i
355.4846+6.0736i 360.4854+3.0368i 365.4862+0.0000i 370.4870-3.0368i
360.2222+9.1104i 365.3546+6.0736i 370.4870+3.0368i 375.6194+0.0000i
364.9598+12.1472i 370.2238+9.1104i 375.4878+6.0736i 380.7518+3.0368i
369.6974+15.1839i 375.0930+12.1472i 380.4886+9.1104i 385.8842+6.0736i
374.4350+18.2207i 379.9622+15.1839i 385.4894+12.1472i 391.0166+9.1104i
379.1726+21.2575i 384.8314+18.2207i 390.4902+15.1839i 396.1490+12.1472i
383.9102+24.2943i 389.7006+21.2575i 395.4910+18.2207i 401.2814+15.1839i
388.6478+27.3311i 394.5698+24.2943i 400.4918+21.2575i 406.4138+18.2207i
Columns 5 through 7
364.9598-12.1472i 369.6974-15.1839i 374.4350-18.2207i
370.2238-9.1104i 375.0930-12.1472i 379.9622-15.1839i
375.4878-6.0736i 380.4886-9.1104i 385.4894-12.1472i
380.7518-3.0368i 385.8842-6.0736i 391.0166-9.1104i
386.0158+ 0.0000i 391.2798- 3.0368i 396.5438- 6.0736i
391.2798+ 3.0368i 396.6754+ 0.0000i 402.0710- 3.0368i
396.5438+ 6.0736i 402.0710+ 3.0368i 407.5982+ 0.0000i
401.8078+ 9.1104i 407.4666+ 6.0736i 413.1254+ 3.0368i
407.0718+12.1472i 412.8622+ 9.1104i 418.6526+ 6.0736i
412.3358+15.1839i 418.2578+12.1472i 424.1798+ 9.1104i
Columns 8 through 10
379.1726-21.2575i 383.9102-24.2943i 388.6478-27.3311i
384.8314-18.2207i 389.7006-21.2575i 394.5698-24.2943i
390.4902-15.1839i 395.4910-18.2207i 400.4918-21.2575i
396.1490-12.1472i 401.2814-15.1839i 406.4138-18.2207i
401.8078- 9.1104i 407.0718-12.1472i 412.3358-15.1839i
407.4666- 6.0736i 412.8622- 9.1104i 418.2578-12.1472i
413.1254- 3.0368i 418.6526- 6.0736i 424.1798- 9.1104i
418.7842+ 0.0000i 424.4430- 3.0368i 430.1018- 6.0736i
424.4430+ 3.0368i 430.2334+ 0.0000i 436.0237- 3.0368i
430.1018+ 6.0736i 436.0237+ 3.0368i 441.9457+ 0.0000i
As you can see, the matrix is the same as those computed before in Matlab. Let's compute eigenvalues and eigenvectors.
PP
[1,] -0.2967359+0.0000000i 0.50734838+0.00000000i
[2,] -0.3009476-0.0026119i 0.39737421-0.00152766i
[3,] -0.3051593-0.0052239i 0.28740004-0.00305533i
[4,] -0.3093709-0.0078358i 0.17742587-0.00458299i
[5,] -0.3135826-0.0104478i 0.06745170-0.00611065i
[6,] -0.3177943-0.0130597i -0.04252247-0.00763831i
[7,] -0.3220060-0.0156717i -0.15249664-0.00916598i
[8,] -0.3262177-0.0182836i -0.26247080-0.01069364i
[9,] -0.3304294-0.0208955i -0.37244497-0.01222130i
[10,] -0.3346411-0.0235075i -0.48241914-0.01374896i
DD
3921.066 0.000000
0.000 2.917833
As you can see, the eigenvalues are the same as those computed with MATLAB, but the eigenvectors are different. How can I get the same eigenvectors as those of MATLAB?
回答1:
The answer is simple but difficult to see because the vectors are complex. If you enter both your matrices in MATLAB, and do this
P(:,1)./PP(:,1)
ans =
0.4064 + 0.9136i
0.4063 + 0.9136i
0.4063 + 0.9139i
0.4065 + 0.9138i
0.4064 + 0.9138i
0.4063 + 0.9138i
0.4063 + 0.9138i
0.4065 + 0.9137i
0.4064 + 0.9137i
0.4063 + 0.9137i
you see that they are linearly dependent, same for the other one.
I am afraid, that there is no way that I know of, that guarantees you the same result within both programs. Computing eigenvalues and vectors in general is not easy, and slight differences in implementation result in the differences that you see.
To get the same result you could try normalizing the vectors with their first component, meaning
P(:,1)/P(1,1) and PP(:,1)/PP(1,1)
That gives me the same vectors, modulo small differences, as you provide a different number of digits for your examples.
Edit: Another test shows, that this is indeed what Matlab does, in addition to normalizing the length to 1. So
tmp=P(:,1)/P(1,1);
tmp/norm(tmp)
returns the same vector as your Matlab Example.
来源：https://stackoverflow.com/questions/45648626/eigenvectors-complex-nonsymmetric-matrix-in-r-different-from-matlab-how-to-solv