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+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 73 65 63 74 69 6f 6e | 7b 73 63 68 75 72 20 2d |\section|{schur -|
|00000010| 2d 2d 20 53 63 68 75 72 | 20 64 65 63 6f 6d 70 6f |-- Schur| decompo|
|00000020| 73 69 74 69 6f 6e 7d 0d | 5c 6c 61 62 65 6c 7b 66 |sition}.|\label{f|
|00000030| 6e 3a 73 63 68 75 72 7d | 0d 5c 69 6e 64 65 78 7b |n:schur}|.\index{|
|00000040| 73 63 68 75 72 28 29 7d | 0d 5c 69 6e 64 65 78 7b |schur()}|.\index{|
|00000050| 53 63 68 75 72 20 64 65 | 63 6f 6d 70 6f 73 69 74 |Schur de|composit|
|00000060| 69 6f 6e 7d 0d 5c 62 65 | 67 69 6e 7b 72 61 69 6c |ion}.\be|gin{rail|
|00000070| 7d 0d 53 63 68 75 72 20 | 3a 20 27 73 63 68 75 72 |}.Schur |: 'schur|
|00000080| 27 20 27 28 27 20 4d 61 | 74 72 69 78 20 27 29 27 |' '(' Ma|trix ')'|
|00000090| 20 3b 0d 5c 65 6e 64 7b | 72 61 69 6c 7d 0d 0d 0d | ;.\end{|rail}...|
|000000a0| 5c 76 65 72 62 2b 73 63 | 68 75 72 2b 20 74 61 6b |\verb+sc|hur+ tak|
|000000b0| 65 73 20 61 20 73 71 75 | 61 72 65 20 6d 61 74 72 |es a squ|are matr|
|000000c0| 69 78 20 61 72 67 75 6d | 65 6e 74 2c 20 61 6e 64 |ix argum|ent, and|
|000000d0| 20 70 65 72 66 6f 72 6d | 73 20 53 63 68 75 72 20 | perform|s Schur |
|000000e0| 64 65 63 6f 6d 70 6f 73 | 69 74 69 6f 6e 0d 6f 6e |decompos|ition.on|
|000000f0| 20 69 74 2e 20 49 74 20 | 72 65 74 75 72 6e 73 20 | it. It |returns |
|00000100| 61 20 6c 69 73 74 20 63 | 6f 6e 74 61 69 6e 69 6e |a list c|ontainin|
|00000110| 67 20 65 6c 65 6d 65 6e | 74 73 20 5c 76 65 72 62 |g elemen|ts \verb|
|00000120| 2b 74 2b 20 61 6e 64 20 | 5c 76 65 72 62 2b 7a 2b |+t+ and |\verb+z+|
|00000130| 2c 0d 77 68 69 63 68 20 | 61 72 65 20 74 68 65 20 |,.which |are the |
|00000140| 24 54 24 20 61 6e 64 20 | 24 5a 24 20 6d 61 74 72 |$T$ and |$Z$ matr|
|00000150| 69 63 69 65 73 20 72 65 | 73 70 65 63 74 69 76 65 |icies re|spective|
|00000160| 6c 79 2c 20 73 75 63 68 | 20 74 68 61 74 3a 0d 0d |ly, such| that:..|
|00000170| 5c 5b 20 41 20 3d 20 20 | 5a 20 54 20 5a 5e 7b 54 |\[ A = |Z T Z^{T|
|00000180| 7d 20 5c 3b 20 20 5c 6d | 62 6f 78 7b 77 68 65 72 |} \; \m|box{wher|
|00000190| 65 20 24 41 24 20 69 73 | 20 74 68 65 20 61 72 67 |e $A$ is| the arg|
|000001a0| 75 6d 65 6e 74 20 6d 61 | 74 72 69 78 7d 20 5c 5d |ument ma|trix} \]|
|000001b0| 0d 0d 49 66 20 74 68 65 | 20 61 72 67 75 6d 65 6e |..If the| argumen|
|000001c0| 74 20 69 73 20 72 65 61 | 6c 2c 20 74 68 65 6e 20 |t is rea|l, then |
|000001d0| 24 54 24 20 69 73 20 69 | 6e 20 60 72 65 61 6c 2d |$T$ is i|n `real-|
|000001e0| 53 63 68 75 72 27 20 66 | 6f 72 6d 2e 0d 0d 0d 5c |Schur' f|orm....\|
|000001f0| 70 61 72 61 67 72 61 70 | 68 7b 45 78 61 6d 70 6c |paragrap|h{Exampl|
|00000200| 65 7d 0d 5c 69 6e 64 65 | 78 7b 65 78 61 6d 70 6c |e}.\inde|x{exampl|
|00000210| 65 20 6f 66 20 73 63 68 | 75 72 28 29 7d 0d 5c 69 |e of sch|ur()}.\i|
|00000220| 6e 64 65 78 7b 73 63 68 | 75 72 28 29 21 65 78 61 |ndex{sch|ur()!exa|
|00000230| 6d 70 6c 65 7d 0d 5c 62 | 65 67 69 6e 7b 76 65 72 |mple}.\b|egin{ver|
|00000240| 62 61 74 69 6d 7d 0d 3e | 20 72 6f 6f 74 32 6f 6e |batim}.>| root2on|
|00000250| 32 20 3d 20 0d 20 72 6f | 6f 74 32 6f 6e 32 20 3d |2 = . ro|ot2on2 =|
|00000260| 0d 20 20 20 20 30 2e 37 | 30 37 0d 3e 20 4d 20 3d |. 0.7|07.> M =|
|00000270| 20 5b 20 73 71 72 74 28 | 32 29 2f 32 2c 20 73 71 | [ sqrt(|2)/2, sq|
|00000280| 72 74 28 32 29 2f 32 2c | 20 30 3b 20 73 71 72 74 |rt(2)/2,| 0; sqrt|
|00000290| 28 32 29 2f 32 2c 20 73 | 71 72 74 28 32 29 2f 32 |(2)/2, s|qrt(2)/2|
|000002a0| 2c 20 30 3b 20 30 2c 30 | 2c 31 5d 0d 20 4d 20 3d |, 0; 0,0|,1]. M =|
|000002b0| 0d 20 20 20 20 30 2e 37 | 30 37 20 20 20 20 20 20 |. 0.7|07 |
|000002c0| 30 2e 37 30 37 20 20 20 | 20 20 20 20 20 20 20 30 |0.707 | 0|
|000002d0| 20 20 0d 20 20 20 20 30 | 2e 37 30 37 20 20 20 20 | . 0|.707 |
|000002e0| 20 20 30 2e 37 30 37 20 | 20 20 20 20 20 20 20 20 | 0.707 | |
|000002f0| 20 30 20 20 0d 20 20 20 | 20 20 20 20 20 30 20 20 | 0 . | 0 |
|00000300| 20 20 20 20 20 20 20 20 | 30 20 20 20 20 20 20 20 | |0 |
|00000310| 20 20 20 31 20 20 0d 3e | 20 52 20 3d 20 73 63 68 | 1 .>| R = sch|
|00000320| 75 72 28 4d 29 0d 20 52 | 20 3d 0d 20 20 20 74 20 |ur(M). R| =. t |
|00000330| 20 20 20 20 20 20 20 20 | 20 20 20 7a 20 20 20 20 | | z |
|00000340| 20 20 20 20 20 20 20 20 | 0d 3e 20 52 2e 74 0d 20 | |.> R.t. |
|00000350| 20 20 20 20 31 2e 34 31 | 20 20 20 20 20 20 20 20 | 1.41| |
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|00000380| 31 65 2d 31 36 20 20 20 | 20 20 20 20 20 20 20 30 |1e-16 | 0|
|00000390| 20 20 0d 20 20 20 20 20 | 20 20 20 30 20 20 20 20 | . | 0 |
|000003a0| 20 20 20 20 20 20 30 20 | 20 20 20 20 20 20 20 20 | 0 | |
|000003b0| 20 31 20 20 0d 3e 20 52 | 2e 7a 0d 20 20 20 20 30 | 1 .> R|.z. 0|
|000003c0| 2e 37 30 37 20 20 20 20 | 20 2d 30 2e 37 30 37 20 |.707 | -0.707 |
|000003d0| 20 20 20 20 20 20 20 20 | 20 30 20 20 0d 20 20 20 | | 0 . |
|000003e0| 20 30 2e 37 30 37 20 20 | 20 20 20 20 30 2e 37 30 | 0.707 | 0.70|
|000003f0| 37 20 20 20 20 20 20 20 | 20 20 20 30 20 20 0d 20 |7 | 0 . |
|00000400| 20 20 20 20 20 20 20 30 | 20 20 20 20 20 20 20 20 | 0| |
|00000410| 20 20 30 20 20 20 20 20 | 20 20 20 20 20 31 20 20 | 0 | 1 |
|00000420| 0d 3e 20 52 2e 7a 2a 52 | 2e 74 2a 52 2e 7a 27 0d |.> R.z*R|.t*R.z'.|
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|00000440| 2e 37 30 37 20 20 20 20 | 20 20 20 20 20 20 30 20 |.707 | 0 |
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|00000460| 20 30 2e 37 30 37 20 20 | 20 20 20 20 20 20 20 20 | 0.707 | |
|00000470| 30 20 20 0d 20 20 20 20 | 20 20 20 20 30 20 20 20 |0 . | 0 |
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|00000490| 20 20 31 20 20 0d 3e 20 | 43 6f 6d 70 20 3d 20 31 | 1 .> |Comp = 1|
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|000004b0| 64 28 33 2c 33 29 2b 31 | 6a 29 0d 20 43 6f 6d 70 |d(3,3)+1|j). Comp|
|000004c0| 20 3d 0d 20 6d 61 74 72 | 69 78 20 63 6f 6c 75 6d | =. matr|ix colum|
|000004d0| 6e 73 20 31 20 74 68 72 | 75 20 33 0d 20 20 20 20 |ns 1 thr|u 3. |
|000004e0| 20 20 20 20 20 20 20 20 | 20 30 2e 39 33 35 20 2b | | 0.935 +|
|000004f0| 20 31 30 69 20 20 20 20 | 20 20 20 20 20 20 20 20 | 10i | |
|00000500| 20 20 34 2e 38 38 20 2b | 20 31 30 69 20 20 20 20 | 4.88 +| 10i |
|00000510| 20 20 20 20 20 20 20 20 | 20 20 33 2e 38 33 20 2b | | 3.83 +|
|00000520| 20 31 30 69 0d 20 20 20 | 20 20 20 20 20 20 20 20 | 10i. | |
|00000530| 20 20 20 31 31 2e 35 20 | 2b 20 31 30 69 20 20 20 | 11.5 |+ 10i |
|00000540| 20 20 20 20 20 20 20 20 | 20 20 20 31 33 2e 34 20 | | 13.4 |
|00000550| 2b 20 31 30 69 20 20 20 | 20 20 20 20 20 20 20 20 |+ 10i | |
|00000560| 20 20 20 31 39 2e 35 20 | 2b 20 31 30 69 0d 20 20 | 19.5 |+ 10i. |
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|00000590| 20 20 20 20 35 2e 32 35 | 20 2b 20 31 30 69 20 20 | 5.25| + 10i |
|000005a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 38 2e 34 31 | | 8.41|
|000005b0| 20 2b 20 31 30 69 0d 0d | 3e 20 53 63 68 52 65 73 | + 10i..|> SchRes|
|000005c0| 20 3d 20 73 63 68 75 72 | 28 43 6f 6d 70 29 0d 20 | = schur|(Comp). |
|000005d0| 53 63 68 52 65 73 20 3d | 0d 20 20 20 74 20 20 20 |SchRes =|. t |
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|000005f0| 20 20 20 20 20 20 0d 3e | 20 43 6f 6d 70 20 2d 20 | .>| Comp - |
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|00000610| 74 2a 53 63 68 52 65 73 | 2e 7a 27 0d 20 6d 61 74 |t*SchRes|.z'. mat|
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|000006d0| 2d 31 35 20 2d 20 31 2e | 30 37 65 2d 31 34 69 20 |-15 - 1.|07e-14i |
|000006e0| 20 20 2d 36 2e 32 32 65 | 2d 31 35 20 2b 20 35 2e | -6.22e|-15 + 5.|
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|00000700| 2d 31 35 20 2d 20 31 2e | 30 37 65 2d 31 34 69 0d |-15 - 1.|07e-14i.|
|00000710| 5c 65 6e 64 7b 76 65 72 | 62 61 74 69 6d 7d 0d 0d |\end{ver|batim}..|
|00000720| 53 65 65 20 61 6c 73 6f | 3a 20 50 61 67 65 7e 5c |See also|: Page~\|
|00000730| 70 61 67 65 72 65 66 7b | 66 6e 3a 73 63 68 6f 72 |pageref{|fn:schor|
|00000740| 64 7d 20 5c 76 65 72 62 | 2b 73 63 68 6f 72 64 28 |d} \verb|+schord(|
|00000750| 29 2b 2e 0d 0d 5c 6e 65 | 77 70 61 67 65 0d 0d 0d |)+...\ne|wpage...|
|00000760| 0d | |. | |
+--------+-------------------------+-------------------------+--------+--------+