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COSC 4364 Numerical Methods - Lecture20_Eigenanalysis
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00:33 All right. You know, so with electric computation. Yes, the
00:44 um about what contracts, what are going to do it? So the
00:55 lecture or Hector's basically about the And today I will talk more about
01:03 procedures, algorithm yeah, actually computers environments and writing vectors. And there
01:11 kind of at least I mentioned last in terms of what effects, what
01:17 of algorithms would choose in different And that will show up in the
01:27 and want to talk about whether european to complete just one. I can
01:32 if they want their as well or they want that choose that affects the
01:40 of work we need to do. too much trouble and that's pretty much
01:50 . So thank you for I don't she works. Okay. Okay.
02:10 happened. Well. So let's see it's quick. It's supposed to work
02:21 the USB connector for. Well, think yes, it was. So
02:33 is a whole long list of methods the power method is covered in the
02:39 and but I think of variants of bar method and it's not the bad
02:47 , but it's not all that So what's down here is mostly water
02:54 one way or another is used for instincts. I wanted you to also
03:00 some insight into that those that are used in practice. More so.
03:05 the parliament. So the basics for bar method is this particular property that
03:17 values of the power of the matrix the power of the Oh, there
03:23 a long mistress. So what department that you will see when I talk
03:27 it is in fact generating a sequence implicitly of powers of a more it's
03:38 . And the corresponding thing is that method also works then by using this
03:44 the computer In Verse 11 values. the parliamentary in its original form is
03:54 Focusing on one Eigen value, particularly largest in magnitude Eigen values. So
03:59 means we can also find the smallest I'm in mind said working with the
04:06 limit. So these are the corresponding and then we can play some games
04:12 as usual, sometimes useful and using property that the onion values of kind
04:22 what college did make it so subjected matrix but sigma's in this case on
04:29 diagonal, then that matrix has shifted the end of this is something I
04:38 use in the work to find out department to find out the other Eigen
04:44 and the largest for this Parliament. how works? So um the
04:56 So it depends on that the largest magnitude Eigen value is strictly largest larger
05:04 the section in magnitude I can And the relationship between the rest of
05:11 doesn't really matter in terms of other order or kind of but it's one
05:19 value in magnitude has to dominate And you will see how why that
05:26 in terms of how the method works it's kind of a business.
05:32 But am I a new boss domestic amount your problem is and if the
05:43 values are distinct, that's a simple that is more or less underlying
05:49 So I hear it doesn't mean that and necessarily doesn't work if that's not
05:55 . But for now it's assumed that Eigen values are distinct and that means
06:02 standing I get values and corresponding to idea of breakfast corresponding to the Eigen
06:10 are not Colin yer. So there they can be made you something some
06:17 or 1000 of the challenge basically they be used to serve as and the
06:27 coordinate vectors coordinate access. Oh System. As I said, it's
06:34 basis for ceo so they that's kind the idea in space that has the
06:40 of the problem. And so it's environment matrix and investment and uh the
06:47 that are not for the multiple of other, they are independent. So
06:56 means then if as long as this up. But the item vectors are
07:04 they're independent. Not in line with other. So that the performing,
07:09 know, talking about basic shift that's on what? But that means that
07:13 vector and this and dimensional space and express is a linear combination of the
07:21 am vectors and that's that's sort of important property that underlines that this have
07:29 parliament is supposed to work for And so yes, in general.
07:36 a linear combination of the vectors of combination. But then to make the
07:41 easy said one doesn't insist that the is in some ways normalized. You
07:49 always scale them. It just changed Eigen vector As as well as the
07:56 version is still the 19 vector. we can use the president to make
08:01 simpler. And I'm just pretending that have to have the banks is just
08:06 linear combination of use sort of a is fair. And that just makes
08:12 finished our mission for the next few . Something for everyone. The power
08:18 is this, is it? Pretty And then I'll tell you why this
08:21 of works in the next couple But basically did you successive making spectrum
08:28 . So and if you do the then you will discover that in fact
08:35 you compute powers of the matrix. , that's not all that much magic
08:46 forward now and that's one of the fight again. But if you then
08:52 now what this operation does. Um the results and the declaration of this
09:00 that modification, then let's say Um The expression for the starting factor
09:10 terms of the idea that pops up system. So then we know that
09:19 the again value for the power of is the power of the I
09:26 So then it's a direct translation. for the first I am better than
09:32 product of a cage and it is the power of the corresponding I can't
09:40 . Um too for all the other vector, students get an expression on
09:46 linear combination of the I am factories the coefficients of the powers of the
09:52 . How do you think? And the commandos can manipulate this expression a
10:00 bit. You can pull out, onto the card. Okay. The
10:07 factor one for the first in vector to that value that we pulled
10:13 And then you have the ratios here all the other Eigen vectors. Now
10:18 point here was that the first For λ one was supposed to be the
10:25 in magnitude. I get value for matrix already. So that means all
10:32 are okay guys and multiplying the other Eigen vectors is less than one.
10:39 that means something less than one to large. Vira means that the rest
10:43 it here it's going to drop and think that's kind of illustrated here on
10:52 bottom. But looking at some of usual one and saying that's something at
10:56 point the smaller contracts in this case first time. So this isn't quite
11:04 we wanted and we're supposed to compute one. Now this expression so we
11:09 to kind of figure out this ambiguous thing and one clear somewhere and that
11:18 guess continue on the next slide. but this just trying to illustrate that
11:26 successive modification kind of works in terms how they're different dr that major spectrum
11:37 quotation are they? In fact will to being aligned with A. I
11:46 back to corresponding to the largest flying , wow this thing here. So
11:53 is that expatriate increasingly in line with there are many electric correspondence to the
12:02 . I am not. So. now to try to figure out how
12:06 get their lamb does plan formed and vineyard use a vineyard functional of the
12:13 of ex. And the linear functional based at some form of expression that
12:19 it in your properties in terms of it says here so and Scale versions
12:27 two vectors X and Y. It's same aspects that scaling uh the function
12:35 to their the respected doctors. So from the directive condition. Some of
12:44 functions are still satisfied this criteria. within your functional is the very simple
12:52 . And just pulled out for one of the components of the
12:56 And that's used in some of the I'm going to show you. So
13:00 linear function, you know, doesn't to be very complicated. Exceedingly
13:07 So, so now that it's difficult apply this functional reinvented and whatever.
13:16 it's this one here, picking up of them. Um And look at
13:19 ratio between the function of the applied two successive integrations of the parliamentary and
13:26 we get something like this. Um just straightforward. So then your functional
13:32 remember what's two plus epsilon, What's ? So now it's the sum and
13:39 we realized that ε has gone to . The sked is sufficient energy.
13:45 this basically um it's wherever accurate expression blocked. And since the two different
13:55 iterations K plus one. And they okay so if we look at this
14:00 rent one escape was one divided by we take. So this ratio in
14:05 converted the largest and that's simply what parliament. It's doing it. Any
14:16 on the parliament will get examples Okay. So it's exceedingly simple network
14:26 expected multiplication and then informed this kind a ratio and want to have a
14:32 simple points are graduating for using Alright, so this is just simple
14:43 . I don't think it was any . It's very obvious. Major.
14:47 sure you didn't tell me what the was. Yes. At some point
14:54 converges and then you can be stopped particular area in terms of number of
14:58 and and then check the there is the animals. Yeah I decided on
15:07 an error criteria. Far off on . Ah yes compared to one.
15:16 here's a little tiny example. Show apartment that works here. Something.
15:22 . And take the starting vector like 01 and the police is successive iterations
15:29 matrix vector multiplication. So when we it zero here, 0 and
15:37 It's just coffee this thing in here this is the result of her doing
15:42 times this zero and at X0 and pretty obvious The Top 1. 1st
15:50 is not 0.5. The next parliament 1.5 so values and then you can
16:02 doing it. So you go down and that basically make inspector now the
16:06 one is many times this factor. it's not a detective remarkable. And
16:13 your former ratio and then they can how you select your fire and then
16:18 select your fires in this case the um components of interactive. So this
16:27 max component is on from five in new and the old one maximum is
16:32 . So the ratio is five And 2.5. one and 2/3 so and
16:42 it's um approaches to on which this to be the dominant diagnosed. Now
16:52 sort of the very basic version. to do that is used sometimes someone
17:01 to improvements. Um and I guess also sense ah that if Again the
17:13 vector zero does not have a component this direction of the corresponding I collect
17:22 correspondence to the value. It may in trouble. So it's important that
17:29 has some components in the direction and course you may not know their Eigen
17:34 and that's a bit of a guessing . But again, the method may
17:39 work if this will happen to choose next year that has no protection on
17:46 first time correspondent to the largest line the johnson issue, it's not
17:57 So if that's some of the publicity get ahead and of course if the
18:06 environment I can value is in fact complex on your value. And the
18:10 have started records real, they never into the imaginary space. That the
18:16 that comes to, you know, the time we respect. It doesn't
18:21 into conflict This one. The other is that the vectors the access as
18:34 hit rate maybe potentially get large and can cause numerical issues. And so
18:42 kind of getting very large and you this example, it was not very
18:47 , but you can notice stuff kind , the size keeps growing. So
18:55 order to kind of now that settlement one tends to use some normalization.
19:03 again they can scale things from the and um absolute value. Any component
19:11 the results of the matrix vector that means this scale vector. Now
19:17 always less than one in 19 so these things are going to be well
19:23 in terms of not growing out of . And then there was an example
19:30 them applying this thing and then saying the normalized largest component and now you
19:38 see that the X values are kind nice. It doesn't change the
19:43 It's just make sure that the american to the game. The normalization is
19:51 an important aspect. You can choose chairman or move one but in the
19:57 things on them ah guaranteed to the symbol. I will give you this
20:02 up according to the north. Normally director has at most Size one in
20:09 measure or in the measure that you . So this is showing the extra
20:18 here and um no. Alright so next thing is somebody someone taken or
20:32 acceleration. You pick it up the instead of just using the racial that
20:41 whether from the functional for the current the previous illiterates. Um there is
20:49 particular formula that now involves three Or pass three difference. No special
21:01 Yeah. Value. And that supposedly a better job than picking up London
21:09 quickly. I should be able to fewer iterations by the time of this
21:17 instead of the architect. And the example uh started to illustrate that So
21:24 will find the matrix but the concerning in this case has been dysfunctional.
21:30 difficult. The second component of No idea why you would choose for
21:36 . It's dysfunctional but it's about it in that's fine And here is an
21:44 that provided and they both looked at numbers at first, it may not
21:49 sense unless you can figure out what the normalization um what they used in
21:56 credits going down here and then if use these ones and this in the
22:05 , that is the second component of . You discover that these numbers don't
22:10 episodes. It's not wrong. But turns out that these ratios here um
22:21 the earth functional for them or are was Um then X two divided and
22:30 successfully hit rate. So if you at these two ratios by no means
22:36 but it turns out if you use a normalized version on X one,
22:42 the end of the R0. So animal looks at his books and tried
22:48 help you and make sense of It's supported. Kind of added that
22:52 based on respect. Now in terms the exploration part um beginning here obviously
22:59 doesn't seem the first one. It give you much conduction worse than this
23:05 , but that's why I go down once I see that the essence it's
23:09 to the um Yes, the six this case 26 It approaches six faster
23:22 the odds. Not very much in particular example, but in principle.
23:29 this scheme should allow you to get merit programs within an editorial more quickly
23:37 just using District forward. So it's to be aware of the things ways
23:47 of kind of convergence to have fantastic using more than one on the information
23:56 . So that was the next one inverse situation and there's not much new
24:05 just to tell you that as I before, rarely do you want to
24:13 the inverse of the name? So want the what they have so mm
24:20 mm mm mm From the next time enough person since the I imagine the
24:29 is the universe of the question arises the original matrix state. Now for
24:36 for the part method to do this Well, they focused on the largest
24:41 magnitude. That means when we look work and with the inverse of the
24:49 then they want again Our um values the inverse of the matrix of the
24:56 one, which means it should be smallest the value of the original.
25:02 that's fine. Now one has a inequality about the smallest and magnitude I
25:09 , but it has to be strictly than and one of the other it
25:15 then gamers here is the largest in which makes so and here is the
25:25 that preserved. Well you don't really to form the inverse. So what
25:31 have this equation and this console, it says. Unfortunately in this case
25:37 ministers, it's not enough to do that come on the application and the
25:42 now you guys have to solve an systems so you can find experts solving
25:49 system of the question and of course way is to do this but that
25:53 shouldn't have to like. So that instance and your factory ization And since
26:02 matrix itself expects for all these so that means you do the characterization
26:11 and this is nothing more than a you're a matrix spectrum of application and
26:19 is the back substitution. That is the possibility. So the hard work
26:25 do once that is a fact possession this one is this operation is a
26:31 as well. So I'll just say I come from all this and it's
26:40 same age, same story and they form the inverse and this this guy
26:46 to do that but as they don't that. But I did anyway.
26:50 that's more than enough. Yeah, okay. And then we can follow
26:54 information here and I found that normalization using the largest magnitude reflected here.
27:01 the Lord is this one and then test convergence it's recently. Okay.
27:07 one is Than the smallest. So too bright inviting the 16 blocks.
27:15 this this But it just shows you principle of it's strange come up to
27:22 and arrested members. Wow the questions it's far more than numbers. So
27:34 there's additional versions too, basically gets find additional Eigen values. Not just
27:41 artist the smallest but if you want other Eigen values. You can also
27:45 the power and then the properties. hmm I get values of some of
27:53 shifted nitrous scenes. So here's the matrix. So to subtract the diagonal
27:59 with some value new on the diagonal the idea of values of this matrix
28:05 lambda minus new. And then then can make sure that if this quantity
28:14 significant and harder than other um, is basically the new maximizes the distance
28:25 the day and the distance between you all the other pagan badges. It's
28:31 . That means now you've actually take the value. Thank you the parliament
28:39 of this. Um, so that's idea and there is a correspondent for
28:46 university but it's an investment just to , so and sometimes. So this
28:54 the way they can figured out in sense, I imagine is closed or
28:58 closed. And you so all this for me inverse power method. Then
29:07 , here's now the correspondent thing, . And that's what I want to
29:11 the largest Mr calls is the new then the or the furthest away.
29:19 that's the way you think. But , um, in this case it
29:27 no, not on this. So the matrix are still fixed. It's
29:33 example again, but it's not. , so this is fixed to the
29:38 once and then the same thing. there is a way and it's that
29:45 form the part that I think it required working well the artists are going
29:54 the indian market is distinct and then fixed it up. Uh and then
30:00 also picks up the smallest denomination value there. The matrix is the universe
30:06 instead of or universe partner. Ah then you can go for other again
30:13 this point shifting the electrics then there kind of a twist. Another twist
30:25 from the second thing that is more than is known as surveillance coaching and
30:34 is still forming. That's the discussion we have a normal I can't
30:43 I can actually questions so if access I don't have to conform columbus it's
30:49 and I'm not pre multiplication or the of exodus external X. Transport exorcism
30:58 number of scalar value. And on right hand side we get another
31:02 Mm hmm. So I can look this ratio then which always can see
31:09 So what the Israeli unfortunate method is constant computer. The right hand side
31:15 for the 8th race for that. this example here chose um Same in
31:28 . Same starting vector. Uh I think this ratio is successful,
31:34 Bit rates from this make Okay, is formed as a matrix spectrum of
31:44 then what's normalization and stuff. And we can look at this is now
31:49 thing that's supposed to convert trance lambda well. Um It's not a huge
31:56 but I plugged in the table Just compare one use normalization without this remote
32:03 Tonya uh you know the eight literate from the coast to but this is
32:11 closer in six and it's not just from an example but it shows that
32:16 actually does help and picking up but it's not expensive computation make expected
32:29 Mm Remember each iteration in the environmentalism spectrum of applications. And this is
32:37 sending a product and sex is a where the new U. S.
32:42 the power method. So this is inner product and another inner products that
32:50 . Um and um somebody else and that produced this estimate of the I
33:02 value them then if you look at shifted onion value and if uh the
33:12 here is really close two the value predicted the largest Eigen value and that's
33:19 designed to capture then the Eigen value this is very small. So this
33:24 kind of amusing universe step of universe this as an estimate and then and
33:33 you can normalize it. And it out that this converges very quickly and
33:39 have been stable unfortunately now and it's what I said on the phone,
33:44 signed it now this matrix is not . So they need to re factor
33:51 song with situation and it stepped up do something question now the factory ization
33:58 we need to do and that's a that pays off competition. Aly but
34:04 was it does cost something is what inducing the trailer culture and perpetuation that
34:12 this tiny matrix again. But it's exactly right. And even one step
34:18 was very close to the thing. a very powerful way of. Mm
34:24 sort of basic using them. Kind the basic idea of the farmers.
34:30 guess we have 31 day now just moment. Mm hmm. Okay.
34:44 right. Thank you. Next. spices and then it's move away from
34:49 parliamentary and talked about other methods. anyway, this is the basic idea
34:54 unification on this ratio and then like the tricks and make the quickly or
35:03 for the nurse. That's fine as as value. And we're shifting ideas
35:14 doing that. Step forward to find I am values for it's in the
35:20 in several years. 20,000. So was about the bar method. Any
35:28 on the bar method. Very simple form essentially matrix doctor modification that plus
35:37 out this racial functions. But the that is quite different dimensions. Um
35:51 best producing similar to transformations. So um also convinced when I am down
36:07 at the time of the fire method it's just a couple of different procedure
36:15 many times 19 shift. Um That's more accurate. So if you
36:25 this picture and then affected where the factor submerges. I can vector corresponding
36:36 the August nine years. I am new. Then the difference between the
36:44 of breakfast and successive generations increasingly small large in the number of information and
36:50 on. So that because self convergence our netflix, this method is not
37:00 to convict the problem. So So remember that properties um um matrices
37:17 to each other. So if you a matrix and to this particular transformation
37:25 this matrix here the similar to the and this particular mentions here then has
37:35 same Eigen values as A. And the basis for this dissipation method and
37:41 can also remember the properties of similarity that if one has the I am
37:47 to this matrix then I can find immediately Iranian doctors of the major
37:57 it's in those times the conductor for subjects. So that is the basis
38:04 similarity transformations somehow. The question is what kind of how do you issue
38:09 aged? An interesting and useful method for the decoration? What one does
38:18 one chooses age. Such that the things matrix for their similarity transformation as
38:29 particular structure. So To pick out value that is on top of the
38:37 column and the rest of the column zero and then the rest of the
38:45 and and that is written in the . The roll back to them.
38:50 for the first bill And then there an end -1 by N - Winemaking
38:56 . So how this definition that works first to pick out disk. I
39:03 back to chain corresponding I am I correspondent, I am vector and we're
39:11 to continue to work on the that not smaller than the original matrix.
39:16 work on successively smaller agencies in the . If you want more than
39:23 Yeah. All right. So how you pick your age and age is
39:34 in this part of the world? strange. Right? And that's you
39:39 , as a householder transformation and I I mentioned it and you should do
39:48 I talked about solving a system of questions. The book doesn't talk much
39:56 it and talk about the elimination but said method depending upon the metrics that
40:04 to be doing that data dependent procedure in the kitchen fine what the electoral
40:14 carbon is used. So whereas the stated independent and it's still that's the
40:25 that this can be preserved. So don't need to worry and look for
40:31 . So the control phone is probably the government of values. Mhm.
40:39 . Alright. So d here. this is you have you know you
40:45 it say that's what you start with than so. So the basic thing
40:56 come back up here A one is first column with. OA so it's
41:02 hey column corresponding to the first column the transform nature. So we know
41:09 factor and what they want. And we by this age to the first
41:16 . My official gap is best for same first vector in the correspondent continuation
41:27 . That's nice for this point and the first axis of this space.
41:32 then it has a scaling factor because not the manufacturer but it's the
41:37 They're basically rotate this vector A one by the first column into a vector
41:44 is aligned with a vector on the system. And then from a once
41:52 know A you inform your vector That is the one you need to
41:58 out in order to get H. b. They're fine. So it's
42:02 to take director that comes from the this case the first column of
42:08 And then you subtract wherever it ends and the alpha is not known.
42:16 has to be chosen. It can be chosen for instance as using the
42:23 norm for the length of the. this is the scaling that preserves that
42:28 is what's known as unitary or orthogonal normal depending upon motel. But it
42:38 whether it's complex career. But it aah times H. D. Or
42:44 . Times X. Permission. Um matrix is a unit. Hey
42:52 So that's why skin preserving and by D. In this particular form then
43:01 have an age that has a reasonable . And so you can both get
43:08 from the original input structure on the aims and you get three also then
43:15 that's all. Um I'm really scared . And the someone here is I
43:27 I guess in terms of that this an item vector. That's what I've
43:33 here. Just to show that as was done for the, when I
43:40 about similarity transformations that it's easy to the I am vector. Oh the
43:48 matrix E. For the similarity transform and it's just take inverse times whatever
44:00 idea back today is something okay. the one is an ideal record to
44:07 similarity transform matrix and there is what result of this transformation is this matrix
44:15 and then the number 121. So is to relate an idea director of
44:20 transform matrix correspondent to be striking. then there's the same thing also for
44:28 up and correspondent to the London was electric. So this method gives us
44:36 first um for one I get value the matrix and then you also corresponded
44:47 in vector and it's having easy So I guess I did this depending
44:55 adopted. So I'm not going to to it ah wants the problem from
45:01 inside itself which is kind of an that it does for. And then
45:07 have an example here forming age and the first colony of some nature.
45:13 was no 21 2 And we want produce the first axis. That's 100
45:24 going in the normal. Okay, to 94 before that's one that's nine
45:33 square off the last stage. So what you get to north for.
45:36 squared square root of the sum of . And then we must forget what
45:43 he is. And then We have judge which was 1 - up here
45:50 times really transport the members of This is a condom and this is
45:56 row vector. So this is actually because this is the number on the
46:02 . So that's the are you some ? The translating transport is the matrix
46:10 to be divided by this number is factor of the elements of this
46:15 And now you never form that's another of this also. And that they
46:21 performed this explicit matrix. You apply vector do whatever it ages, operating
46:30 , so that becomes director transmitted application and then it just becomes shaving.
46:41 here is what's happening here. The age is the identity matrix -.
46:48 two things here. So, another in this case, as I said
46:53 apply 18 to Director A So this the main Director in this case is
47:02 common vector. So this is simple products and this is an inner product
47:09 this reactor is still director that matches for this in terms of service and
47:14 else. Okay so this is basically scale factor for the and doing the
47:22 consistency in cox this simple exercise for way of doing it, defining the
47:28 and the house of the transformation. exactly what I was supposed.
47:38 So if you're done with that and want to find some more and I
47:42 started to work on that smaller and you can repeat the process and remember
47:49 of the properties of the similarity Yeah I can values of this transform
48:01 are the same as arguing values of original. That means the rest of
48:06 Eigen values. So it takes all exclusive this one are actually fighting values
48:13 deep so you can get the remainder the item values now by operating on
48:18 then smaller. That's a simple So it's so again because of the
48:25 transformations and the property that you can additional. I invited southern correspondent Ryan
48:33 by working. Yeah, nothing dr this here and again in a
48:44 Tell the surgical science force Lawrence from book office but reflection method is a
48:53 method and it's used for frequently the methods not working as well. So
49:11 here is mr population assume that To hear them Is formed this way
49:18 Y two is an Eigen vector corresponding the next item of this european things
49:24 be now so you do the same and I got after corner of
49:29 It's Amanda too and the rest of first corner Moby is zero. So
49:34 then you get the correspondent, I , they know that I am doctors
49:45 right and then reform new vector and choose to make alpha this particular volume
49:56 . That looks so strange, show why that makes sense. And then
50:01 pretend that for saying that this I to hear is an item that for
50:10 four, sorry. So here is procedure right? That this is a
50:19 transformation and we said that business and um or this similarity transform nature.
50:35 I'm just plugging things in here. this was making appear this um and
50:42 we applied this sector has been constructed the indian director for B plus this
50:51 plug in your report and outside that the if you just carry out
50:57 make you expect them up tradition and this thing here. Mr at the
51:02 is obvious .2 was American values for to. So this is clear um
51:09 just doing the matric investigating your product , you've got the top row and
51:14 plug in your alpha and it may , it's kind of one step but
51:19 just simplifying but this um entity is the new york London alpha, You
51:27 see the λ one. The insisting is divided by Number two money from
51:34 one because this thing here. So you make a comment the nominator then
51:40 results in space program that really changed and believe which was in fact so
51:46 that case now we're seeing that this is now and I get infected
51:53 The similarity transform matrix corresponding two. then they got the high director for
52:03 original matrix A. We're going through usual exercise justifying the chambers perform the
52:13 transformed. So any questions on installation . Alright. So the next step
52:29 left this here dealing with professionals anymore one area time corresponding vectors. So
52:40 is what financing so and principal and step forward so one what this so
52:54 simultaneous iteration is that instead of working a single doctor and in situation to
53:03 effectively with the collection of put this of actors into another matrix. And
53:11 do you want to this matrix has columns and then one morning three columns
53:17 such as. And the number of here depends on how many I again
53:22 question trying to Yes. And the is not formed up like this.
53:34 that um remember the basics of the . But if you just you know
53:41 one story director that was the basic and eventually converges to being aligned with
53:52 I. N. Vector corresponding to value. But that's independent of the
54:00 . So that means in principle this does carried out simultaneously for several doctors
54:09 the end yourself. Is that then K. Increases all the columns of
54:17 matrix and to be aligned with the in magnitude correspondent. Right? So
54:28 this matrix becomes quite in condition because all pretty much so that's the problem
54:38 trying to clever and yeah and there ways around that and I'll talk to
54:44 next but just saying straightforwardly it looks know better but it does issues together
54:52 all the properties how big insurance the spectrum of rotation it works suffering.
55:07 that's what I'm trying to do them . Here's the picture. Right?
55:12 well I'm trying to make sure that each integration instead of working with a
55:22 results of a times X. More now. And Matrix one trying to
55:32 use another set of directors divide out this nature. So the fund balance
55:40 known as through our characterization this is matrix, it turns out the columns
55:46 this cube they are criminals in each and they can also be normalized.
55:54 in that case on this but things them to you one but for systems
56:02 upcoming $4 and then I rescheduled So even with this is one of
56:10 may have just a small components in other direction from boosted by normalizing.
56:15 I'm trying to always keep the whole correspondent to more than one I get
56:22 to. So expanded in the But that's the intuition behind what's going
56:28 here now. So how did you the Q and R. And again
56:35 going to say my favorite method but very possible that it is Q.
56:42 family farm. He was a householder so it's produced and I'll show that
56:52 some strikes coming up. So basically all the transformation is available workforce for
57:00 linear systems for legislation that the for and made disease. That our because
57:08 is X is not the square matrix it has whatever number of columns to
57:13 or corresponding to how many alien values . But the number of rolls is
57:20 to the size of the nature. it's kind of very tangled religions.
57:24 hmm. And then the iteration or iteration dollar. I'll show you Based
57:32 that one too is then using this of vectors and during the next major
57:39 of protection steps, some of these around the day they're not with each
57:47 . So keep on three or Thorgan ah directors who are going on every
57:55 and this same opinion situation mistake on street form of ocean. And that
58:05 is enough. Okay, now so you are interational sign and how about
58:16 household the transformation works then the member the discussion of inflation. You found
58:22 transformation of zero, everything over diana then they get another summary things that
58:28 capital B on the operations side and you will find a new household of
58:34 and zero or whatever it's been out and they keep going potentially because they
58:42 the triangular matrix stuff. So now um, so this contract gives you
58:58 the sequence of assholes transformation and what's as you can see it's enough to
59:07 any language. So, and that on this side here. Um,
59:14 excess rectangular zor today and forever 100 10 or something. R. Is
59:25 , 10 by 10 the triangle and not full. So that's kind of
59:33 insured from this side here as always rudeness square matrix. But if there
59:39 highly rectangular that were way deep here the, after a few steps there's
59:46 more left on this because the rest it to the right results there.
59:50 it's not working. And that's, kind of the same form of just
59:57 four in terms of harder than great successive transformations and so the set of
60:06 the that they use for H1 also breakfast that are the provenance of
60:17 So yes, yes, this was to secure our confrontation of an
60:25 X. A. Um then being is what's going on, what is
60:33 that this production romantic system which it then it preserves the norman the
60:42 Mm hmm just terrorist tunnel now back work we're supposed to talk about this
60:48 separation that is slightly different from No transformation and the difference is
60:56 And this method they don't work with collection of starting back to X.
61:02 we also organizing the doctors are using every step. In this case you
61:07 directly with the matrix A and after matrix stage. And then you're doing
61:14 in the way that is described So again, if you are factory
61:19 of the matrix and starting first with matrix given to you. Then you
61:26 the next history by giving the reverse application the characterizations and responsibility. And
61:36 may take some triangular ah analysis based say first it's a zero. Then
61:43 get a one based on things do factory ization and using reverse kind of
61:51 of the nature and stuff. You to get the next picture. So
61:55 is. That's the ideas is kind the same in some ways of using
62:00 characterization. But you don't start with guest of his collection. Initial Starting
62:07 . Let's start with the Nature And you form the interests quite differently.
62:13 then it turns out that then this going to be a good thing.
62:16 hope the next time. Yeah. you can also look at the reception
62:23 sequence of operations on it doesn't look that different except kind of the nature
62:30 all this red light. This type product. Yeah. So yes,
62:43 depending on the properties of the nature say that to start with They get
62:48 All right. Is it dying on lots of the matrix and symmetry is
62:56 in a bunch of problems. So cure iteration. Yeah, it is
63:02 problem. one of the lost um methods for dance major cities.
63:12 So they don't cut various libraries sanctuary find and I think that used something
63:23 was one of them. That's right . So as an example and will
63:36 that this individual um the tribute to characterization of this matrix. Now as
63:44 can see in this case this is or matrix and it's up the triangular
63:49 about the first to enter here and there's the correspondent this matrix make its
63:56 the question to get there. The our margins and then you form the
64:02 iteration here we're doing the reverse Able to get some. Um so
64:10 this case of American, sorry, and so you can keep doing this
64:20 that and powerful method. That is reason for even though it implies specialization
64:28 each step. So it's another worker this film. Um I think I'll
64:38 you his shifts um instead of fact say the current interests, the fact
64:46 shifted person and then recovering figuring the value, then you get don't convergence
64:55 here I am call me fortune. other thing. The most clever form
65:02 to use um the combination of uh far the rest more power lessons And
65:15 inverse artist because it's used the outcome one as some estimates and they rejected
65:25 in this case um similar to the of not just work on the matrix
65:33 then a shifted version and if you're and figuring out the shift you get
65:41 very and great convergence. Same Um how your picture of sigma one
65:52 not necessarily given by your recipe, this is the um four. Um
66:04 then what we've got here and now guess the best thing here and now
66:09 something that was really good With the was there was a young age and
66:16 . So this time example will concede shifted person using this. Yes,
66:22 was into that. And this uh value is here on the dialogue are
66:30 to the lisa. So okay, now that's um so any questions this
66:46 . I wanted to make sure the look at the size and very important
67:00 and it will come back and I talked about composition and that this particular
67:15 comments here about the two Americans are . So remember him. Thanks.
67:22 all right. Let's make sure about system. Mark checks that is the
67:30 matrix Plus one Somebody. So that's the animal. If it's an upper
67:37 matrix, it's up a triangle bus Bag on immediate it at all the
67:45 time and it is lower. It's opposite the white triangle, it's
67:54 The upper triangle is zero except for first superman history forms. And it
68:00 out those are very useful and finding values and we'll talk more about it
68:12 find similar values and it turns out finding I can values then s number
68:25 matrices. The same place of finding is in order to make so many
68:35 used to first transform the nature's community into historical form and then you're trying
68:41 find the ideal values from the person . And it turns off if you
68:48 with symmetric nature. Sneeze things are simple because since it's symmetric consider between
68:56 is preserved in transforming the matrix. don't know where you're in champion.
69:01 things as they operate in the world . Well triangular, upper and
69:07 Heisenberg is partying the pride they had it. That is very important.
69:15 made to stay after driving on and transform it, then it becomes no
69:21 has some birth. And as they saying And the only thing that's something
69:25 is the elements wow the manual and and he did after the word.
69:37 , so let's see what the next are. So this is not about
69:42 actually. So I for proliferation is . And in principle again, all
69:52 I am values you have to be least end iterations. So that means
69:59 mentioned earlier that was into the 4th . But normally you don't need all
70:05 idea of badges and it's if it turns out the convergence And I
70:10 this little tiny two x two examples one step that wasn't too bad.
70:15 that means even if you want all valued surveillance have to be carrying things
70:24 . Thanks. That's when the steps size of this system. Okay.
70:31 this is best for the transformation the have several forms on the responsibility.
70:37 happens if and then the strangler and terms of the lawyer finished and it's
70:47 . So in that sense, you , now potentially kind of cheat a
70:54 bit of things off the that you to enter the disease are small
71:02 So there's kind of symmetric values And I got confidential trying to travel
71:13 quickly and mm hmm. So for medium that it's the first question I
71:22 for this thing. I know it's . So let me do you want
71:25 know this recording method? Um, will do it great. I'm happy
71:31 use that instead. Anyways. So is the common method? Common method
71:38 before symmetric fantasies. It's very simple . It's very easy to program and
71:48 . Different processes, processors and processes that matter. Then you can use
71:55 so what does it mean? It's on Yeah, that's rotations.
72:02 So everyone heard about dinners locations and just rotating breakfast. Nothing matters.
72:13 and so the whole idea and try That's this 195 explain why particular symbols
72:24 cns. My best of this is in some ways. 1st.
72:30 So still so much that it's saying is on a diagonal and the off
72:36 elements and the two metrics is that something minus this. So anyway,
72:41 doing matrix vector multiplication. And the is that I want to choose
72:47 So when you do this, operate this vector A one A 2 then
72:53 get so it's not It's a little x 2 examples. Not too different
72:59 the householder was also deputy because there things the household is zero about the
73:04 lot of stuff. Um explained Just The result, one control moment
73:12 I'll give them more expensive example. this really is pregnancy. The
73:17 they're doing this thing now since what don't know C. And A one
73:22 A two is given frightening just reformulate things and put the A's in the
73:28 and that C. And S. . Unknowns and pencil and now we
73:31 solve the system of equations. Um since they are known um how far
73:39 can figure out how to get in case. We can of course.
73:44 There's someone squares for these guys could one and that's the reason to cns
73:50 said remember it's fine and co sign sum of squares of finding So we
73:57 what out for his for now make on this thing. Mm hmm.
74:04 huh. To find out what this be and what they should be.
74:08 in fact this is a sign and sign inside. So this is just
74:14 from this requirements. That's they want to be true and it's based in
74:23 as well. Mm hmm. Now is just gonna go for a guest
74:29 yet during the sexual computer in the . Yes. And so I'm applying
74:33 to zero but I promise to be little bit more serious example where,
74:40 know, produccion s into bigger environments could be whatever size the ones.
74:46 we choose a particular row and second column, second goal and 4th
74:53 & 4th. So this is where a lot of things in. And
74:57 your clients are going to see about component, as I mentioned and they
75:01 hear that. So now you're going pick C and S. Such as
75:07 outfront. zero is too, but actually maps back to two years.
75:14 what this, um, you can it but I'll skip gears and get
75:19 to, you know, at the . You really want to finish
75:22 So corporate duration then applies this given on the front and on the
75:31 So if the similarity transformations of the again. And this is how you
75:38 the matrices day and then today it's this guy here, let's see it
75:44 funding all the proper places to figure out. Um Okay, sort of
75:49 example. Again we'll make sure that you seeing us and I'm here,
75:54 think it should be. And you know, several things up and
76:00 again the finance trivial example symmetric That's the first thing C N S
76:05 this has probably been in this but that's not how it works in
76:14 . Your work like this. The by step to choose whichever elements in
76:20 symmetric they want to figure out. there's a reinforced multiplication. You get
76:27 and if you use the successive transformations this rotation matrices as well. Eventually
76:33 guy is going to converge to Yeah, here is an example.
76:41 then I'll put the Yeah, it's zero. Find the rotation matrix here
76:50 this case here, on the attention wanted to find to do this
76:57 You've got something like this to These particular values after I choose
77:04 So at least they started and while have to see her already but then
77:10 continue to the next step to this here. So now we're going to
77:15 on these four elements. So did notice this pattern here is the obvious
77:21 about something, the next step So the problem is solved are
77:29 This makes sense. And it turns it eventually does make sense because if
77:35 compared to um even the first matrix , If one looks at the off
77:42 elements. So it was student So it looks at what's off the
77:47 here, there are smaller then when started. So even if it seems
77:53 it doesn't do much progress, what actually does is suppresses the elements even
78:00 It turns zero elements into non zero in the next step. And here
78:06 kind of just following up and eventually see that now quite a lot
78:13 You have 2 3 steps and we'll see the potential of this
78:20 That is going is terrible. Uh I think it's cute. So oh
78:28 one of the guys relax worked. my God, we're gonna work that
78:36 from the bank commissions. So remember is supposed to be eventually turn into
78:42 matrix. And you know in the of the videos you can still see
78:47 eventually transitioned to and this was the shot was in random matrix the second
78:56 that mhm. Except a structure about random values And in the non-0
79:05 Yeah, Things get zero without for dying. So at least they chose
79:13 dynamics of this. I am saying that System one. So yeah,
79:22 I think my time is pretty much . Um So okay, these matters
79:31 not to touch about because I included . Anyway, that was fun.
79:40 these methods are essential. Matrix vector based methods. And need to expect
79:49 multiplication based methods are particularly useful. has forced legacies because they employed a
79:59 less work then that means that works matrices. So these methods are particularly
80:06 when you're as far as my So and maybe I'll come back to
80:13 one of these in the next lecture comment on the birthday. Otherwise we'll
80:17 about simply by either the composition. is a special version in my
80:22 Well, I never Okay I think the time is up. So questions
80:33 that you're all of this. You that's a lot of sensible about
80:41 It's turning through Western metrics. Such position. These are a little bit
80:46 lot more. You should be aware it. Okay. Thanks.
80:56 Yeah. Mhm.
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