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| dexvert
| LaTeX Document (document/latex)
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macFileType | [TEXT] |
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hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 73 65 63 74 69 6f 6e | 7b 65 69 67 20 2d 2d 2d |\section|{eig ---|
|00000010| 20 45 69 67 65 6e 20 44 | 65 63 6f 6d 70 6f 73 69 | Eigen D|ecomposi|
|00000020| 74 69 6f 6e 7d 0d 5c 6c | 61 62 65 6c 7b 66 6e 3a |tion}.\l|abel{fn:|
|00000030| 65 69 67 7d 0d 5c 69 6e | 64 65 78 7b 65 69 67 28 |eig}.\in|dex{eig(|
|00000040| 29 7d 0d 5c 69 6e 64 65 | 78 7b 65 69 67 65 6e 20 |)}.\inde|x{eigen |
|00000050| 64 65 63 6f 6d 70 6f 73 | 69 74 69 6f 6e 7d 0d 5c |decompos|ition}.\|
|00000060| 62 65 67 69 6e 7b 72 61 | 69 6c 7d 0d 45 69 67 65 |begin{ra|il}.Eige|
|00000070| 6e 20 3a 20 27 65 69 67 | 27 20 27 28 27 20 28 20 |n : 'eig|' '(' ( |
|00000080| 4d 61 74 72 69 78 20 7c | 20 4d 61 74 72 69 78 20 |Matrix || Matrix |
|00000090| 27 2c 27 20 4d 61 74 72 | 69 78 20 29 20 27 29 27 |',' Matr|ix ) ')'|
|000000a0| 20 3b 0d 5c 65 6e 64 7b | 72 61 69 6c 7d 0d 0d 5c | ;.\end{|rail}..\|
|000000b0| 76 65 72 62 2b 65 69 67 | 2b 20 77 69 74 68 20 61 |verb+eig|+ with a|
|000000c0| 20 73 69 6e 67 6c 65 20 | 73 71 75 61 72 65 20 61 | single |square a|
|000000d0| 72 67 75 6d 65 6e 74 20 | 6d 61 74 72 69 78 20 63 |rgument |matrix c|
|000000e0| 6f 6d 70 75 74 65 73 20 | 74 68 61 74 20 6d 61 74 |omputes |that mat|
|000000f0| 72 69 78 27 73 0d 65 69 | 67 65 6e 76 65 63 74 6f |rix's.ei|genvecto|
|00000100| 72 73 20 61 6e 64 20 65 | 69 67 65 6e 76 61 6c 75 |rs and e|igenvalu|
|00000110| 65 73 2e 20 54 68 65 20 | 72 65 73 75 6c 74 73 20 |es. The |results |
|00000120| 61 72 65 20 72 65 74 75 | 72 6e 65 64 20 61 73 20 |are retu|rned as |
|00000130| 61 20 6c 69 73 74 20 77 | 69 74 68 0d 65 6c 65 6d |a list w|ith.elem|
|00000140| 65 6e 74 73 20 5c 76 65 | 72 62 2b 76 61 6c 2b 20 |ents \ve|rb+val+ |
|00000150| 61 6e 64 20 5c 76 65 72 | 62 2b 76 65 63 2b 20 77 |and \ver|b+vec+ w|
|00000160| 68 69 63 68 20 61 72 65 | 20 74 68 65 20 65 69 67 |hich are| the eig|
|00000170| 65 6e 76 61 6c 75 65 73 | 20 61 6e 64 0d 72 69 67 |envalues| and.rig|
|00000180| 68 74 20 65 69 67 65 6e | 76 65 63 74 6f 72 73 20 |ht eigen|vectors |
|00000190| 72 65 73 70 65 63 74 69 | 76 65 6c 79 2e 20 49 6e |respecti|vely. In|
|000001a0| 20 67 65 6e 65 72 61 6c | 2c 20 74 68 65 73 65 20 | general|, these |
|000001b0| 77 69 6c 6c 20 62 65 0d | 63 6f 6d 70 6c 65 78 20 |will be.|complex |
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|000001d0| 76 65 72 62 2b 65 69 67 | 2b 20 77 69 74 68 20 74 |verb+eig|+ with t|
|000001e0| 77 6f 20 61 72 67 75 6d | 65 6e 74 73 20 63 6f 6d |wo argum|ents com|
|000001f0| 70 75 74 65 73 20 74 68 | 65 20 65 69 67 65 6e 76 |putes th|e eigenv|
|00000200| 65 63 74 6f 72 73 20 61 | 6e 64 20 0d 65 69 67 65 |ectors a|nd .eige|
|00000210| 6e 76 61 6c 75 65 73 20 | 6f 66 20 74 68 65 20 72 |nvalues |of the r|
|00000220| 65 61 6c 20 67 65 6e 65 | 72 61 6c 69 7a 65 64 20 |eal gene|ralized |
|00000230| 73 79 6d 6d 65 74 72 69 | 63 20 28 20 6f 72 20 63 |symmetri|c ( or c|
|00000240| 6f 6d 70 6c 65 78 20 67 | 65 6e 65 72 61 6c 69 7a |omplex g|eneraliz|
|00000250| 65 64 0d 48 65 72 6d 69 | 74 69 61 6e 29 20 64 65 |ed.Hermi|tian) de|
|00000260| 66 69 6e 69 74 65 20 65 | 69 67 65 6e 70 72 6f 62 |finite e|igenprob|
|00000270| 6c 65 6d 2e 20 54 68 65 | 20 65 69 67 65 6e 76 61 |lem. The| eigenva|
|00000280| 6c 75 65 73 20 61 6e 64 | 20 65 69 67 65 6e 76 65 |lues and| eigenve|
|00000290| 63 74 6f 72 73 20 61 72 | 65 0d 72 65 74 75 72 6e |ctors ar|e.return|
|000002a0| 65 64 20 69 6e 20 61 20 | 6c 69 73 74 20 61 73 20 |ed in a |list as |
|000002b0| 66 6f 72 20 74 68 65 20 | 66 69 72 73 74 20 66 6f |for the |first fo|
|000002c0| 72 6d 2e 20 20 49 66 20 | 74 68 65 20 74 77 6f 20 |rm. If |the two |
|000002d0| 61 72 67 75 6d 65 6e 74 | 73 20 61 72 65 0d 24 41 |argument|s are.$A|
|000002e0| 24 20 61 6e 64 20 24 42 | 24 2c 20 74 68 65 20 65 |$ and $B|$, the e|
|000002f0| 69 67 65 6e 70 72 6f 62 | 6c 65 6d 20 69 73 20 64 |igenprob|lem is d|
|00000300| 65 66 69 6e 65 64 20 62 | 79 0d 0d 5c 5b 20 41 78 |efined b|y..\[ Ax|
|00000310| 20 3d 20 5c 6c 61 6d 62 | 64 61 20 42 20 78 20 5c | = \lamb|da B x \|
|00000320| 5d 0d 0d 42 6f 74 68 20 | 66 6f 72 6d 73 20 6f 66 |]..Both |forms of|
|00000330| 20 5c 76 65 72 62 2b 65 | 69 67 2b 20 20 63 68 65 | \verb+e|ig+ che|
|00000340| 63 6b 20 66 6f 72 20 73 | 79 6d 6d 65 74 72 79 2c |ck for s|ymmetry,|
|00000350| 20 61 6e 64 20 75 73 65 | 20 61 6e 0d 61 70 70 72 | and use| an.appr|
|00000360| 6f 70 72 69 61 74 65 20 | 73 6f 6c 76 65 72 2e 0d |opriate |solver..|
|00000370| 0d 5c 70 61 72 61 67 72 | 61 70 68 7b 45 78 61 6d |.\paragr|aph{Exam|
|00000380| 70 6c 65 7d 0d 5c 69 6e | 64 65 78 7b 65 78 61 6d |ple}.\in|dex{exam|
|00000390| 70 6c 65 20 6f 66 20 65 | 69 67 28 29 7d 0d 5c 69 |ple of e|ig()}.\i|
|000003a0| 6e 64 65 78 7b 65 69 67 | 28 29 21 65 78 61 6d 70 |ndex{eig|()!examp|
|000003b0| 6c 65 7d 0d 5c 62 65 67 | 69 6e 7b 76 65 72 62 61 |le}.\beg|in{verba|
|000003c0| 74 69 6d 7d 0d 3e 20 41 | 20 3d 20 72 61 6e 64 28 |tim}.> A| = rand(|
|000003d0| 33 2c 33 29 0d 20 41 20 | 3d 0d 20 20 20 20 30 2e |3,3). A |=. 0.|
|000003e0| 31 30 38 20 20 20 20 20 | 20 30 2e 37 34 31 20 20 |108 | 0.741 |
|000003f0| 20 20 20 30 2e 30 36 34 | 36 20 20 0d 20 20 20 20 | 0.064|6 . |
|00000400| 30 2e 32 39 37 20 20 20 | 20 20 20 30 2e 33 32 39 |0.297 | 0.329|
|00000410| 20 20 20 20 20 30 2e 30 | 32 38 37 20 20 0d 20 20 | 0.0|287 . |
|00000420| 20 20 30 2e 32 35 34 20 | 20 20 20 20 20 30 2e 33 | 0.254 | 0.3|
|00000430| 34 35 20 20 20 20 20 20 | 30 2e 34 39 39 20 20 0d |45 |0.499 .|
|00000440| 3e 20 65 69 67 65 6e 3d | 20 65 69 67 28 41 29 0d |> eigen=| eig(A).|
|00000450| 20 65 69 67 65 6e 20 3d | 0d 20 20 20 76 61 6c 20 | eigen =|. val |
|00000460| 20 20 20 20 20 20 20 20 | 20 76 65 63 20 20 20 20 | | vec |
|00000470| 20 20 20 20 20 20 0d 3e | 20 41 2a 20 65 69 67 65 | .>| A* eige|
|00000480| 6e 2e 76 65 63 5b 3b 31 | 5d 20 2d 20 65 69 67 65 |n.vec[;1|] - eige|
|00000490| 6e 2e 76 61 6c 5b 31 5d | 2a 65 69 67 65 6e 2e 76 |n.val[1]|*eigen.v|
|000004a0| 65 63 5b 3b 31 5d 0d 20 | 20 20 20 20 20 20 20 20 |ec[;1]. | |
|000004b0| 20 20 31 2e 33 39 65 2d | 31 36 20 2b 20 30 69 0d | 1.39e-|16 + 0i.|
|000004c0| 20 20 20 20 20 20 20 20 | 20 20 2d 31 2e 31 31 65 | | -1.11e|
|000004d0| 2d 31 36 20 2b 20 30 69 | 0d 20 20 20 20 20 20 20 |-16 + 0i|. |
|000004e0| 20 20 20 20 34 2e 38 36 | 65 2d 31 37 20 2b 20 30 | 4.86|e-17 + 0|
|000004f0| 69 0d 3e 20 41 2a 20 65 | 69 67 65 6e 2e 76 65 63 |i.> A* e|igen.vec|
|00000500| 5b 3b 32 5d 20 2d 20 65 | 69 67 65 6e 2e 76 61 6c |[;2] - e|igen.val|
|00000510| 5b 32 5d 2a 65 69 67 65 | 6e 2e 76 65 63 5b 3b 32 |[2]*eige|n.vec[;2|
|00000520| 5d 0d 20 20 20 20 20 20 | 20 20 20 20 20 31 2e 31 |]. | 1.1|
|00000530| 31 65 2d 31 36 20 2b 20 | 30 69 0d 20 20 20 20 20 |1e-16 + |0i. |
|00000540| 20 20 20 20 20 2d 33 2e | 33 33 65 2d 31 36 20 2b | -3.|33e-16 +|
|00000550| 20 30 69 0d 20 20 20 20 | 20 20 20 20 20 20 2d 31 | 0i. | -1|
|00000560| 2e 31 31 65 2d 31 36 20 | 2b 20 30 69 0d 3e 20 41 |.11e-16 |+ 0i.> A|
|00000570| 2a 20 65 69 67 65 6e 2e | 76 65 63 5b 3b 33 5d 20 |* eigen.|vec[;3] |
|00000580| 2d 20 65 69 67 65 6e 2e | 76 61 6c 5b 33 5d 2a 65 |- eigen.|val[3]*e|
|00000590| 69 67 65 6e 2e 76 65 63 | 5b 3b 33 5d 0d 20 20 20 |igen.vec|[;3]. |
|000005a0| 20 20 20 20 20 20 20 20 | 39 2e 37 31 65 2d 31 37 | |9.71e-17|
|000005b0| 20 2b 20 30 69 0d 20 20 | 20 20 20 20 20 20 20 20 | + 0i. | |
|000005c0| 2d 31 2e 33 39 65 2d 31 | 36 20 2b 20 30 69 0d 20 |-1.39e-1|6 + 0i. |
|000005d0| 20 20 20 20 20 20 20 20 | 20 2d 31 2e 31 31 65 2d | | -1.11e-|
|000005e0| 31 36 20 2b 20 30 69 0d | 5c 65 6e 64 7b 76 65 72 |16 + 0i.|\end{ver|
|000005f0| 62 61 74 69 6d 7d 0d 0d | 53 65 65 20 61 6c 73 6f |batim}..|See also|
|00000600| 3a 20 50 61 67 65 7e 5c | 70 61 67 65 72 65 66 7b |: Page~\|pageref{|
|00000610| 66 6e 3a 65 69 67 6e 7d | 20 5c 76 65 72 62 2b 65 |fn:eign}| \verb+e|
|00000620| 69 67 6e 28 29 2b 2c 0d | 50 61 67 65 7e 5c 70 61 |ign()+,.|Page~\pa|
|00000630| 67 65 72 65 66 7b 66 6e | 3a 65 69 67 73 7d 20 5c |geref{fn|:eigs} \|
|00000640| 76 65 72 62 2b 65 69 67 | 73 28 29 2b 2e 0d 0d 5c |verb+eig|s()+...\|
|00000650| 6e 65 77 70 61 67 65 0d | 0d 0d 0d 0d |newpage.|.... |
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