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COSC 4364 Numerical Methods - Lecture19_Eigenanalysis
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00:00 that's where you get. Yeah, , there is. It's like a
00:08 Probably just like a freshwater station. and this is the thing that you
00:17 only see crawfish is a lot of . Mm hmm. Some practice.
00:25 bigger versions. Yeah. You have get a little right, Okay,
00:35 let's see. Trying to get this to work. Mhm. Yeah.
00:43 continue on. I can't find you . Some of these details here.
00:56 , I'm trying to talk about So I guess I didn't comment
01:03 And there's a difference compared to most probably done and what we've talked about
01:12 but we talked about solving equations in with direct factors and other things that
01:19 matrices and factors mostly. Uh there's and when it comes to baby steps
01:29 fueling resources needed to solve problems. tends to be the thing, the
01:35 thing you need to know about and gives us an idea of memory space
01:40 and computing trips required to solve the . But it doesn't give you the
01:46 into how well things may work. it doesn't really take those things that
01:56 talked about And I think most they'd to see it outside of this
02:02 It doesn't really take into account other than size. What do you
02:09 So I can value an Eigen vector that we talked about also get insight
02:16 some behavior as a function of the in the matrix. So that's kind
02:23 the difference in someone's compared to what talks about this is what we talked
02:28 last time. But then I think the problem is that basically finding a
02:32 of vectors that when you apply the a to the vector is kind of
02:38 got the state that some of the back, so that's something to himself
02:44 the scaling factors and the argument and was my last time. And then
02:51 one way of looking at it is to solve this equation. Um and
02:57 have been valued this separation as a tradable solution in terms of acts and
03:05 the lander's then once and we were selling this expression to zero and then
03:12 if you under travel the determination to the polynomial and it's not just the
03:17 and then we went through a couple examples of the signal works And that
03:22 one small problem, but it doesn't , so this is something pretty much
03:32 , but it's time. So a things concepts that is still remember
03:42 people talk about american values and Eigen . These concepts comes in the
03:50 I think the first one here is one usually it is ideal vectors that
03:56 enormous in some sense because remember um there again, formulation, the effect
04:06 an Eigen vectors. So is in trailer factor of the exposed to get
04:11 order in chaos space they want um normalized themselves and we'll talk about what
04:18 means in case that's not familiar on next few sites. So especially to
04:23 to get some notional scale and we a common sense for scale among
04:32 Now the set of I am values is associated with the metrics is known
04:39 the spectrum and that's kind of related kind of frequency components in some single
04:50 some other structure and talked about the . So anyone who may have played
04:57 the audience so on your, they when you forget frequency components in it
05:02 that corresponds to, I didn't understand some models that will come back to
05:10 . Um, the maximum value of Iranian values have a particular significance.
05:19 come back to that. That was as a spectral radius. Um,
05:24 mom talks about the difficulties um arguing in that case of a particular
05:33 matrix had to be less than one that there's also a spectral radius or
05:41 . Um so in some sense magnitude somewhere it's less than one than it
05:48 be quite many times. Then it blow up suspected radius is a concept
05:53 is very important to remember as The spectrum is a useful thing and
05:59 we'll talk about that again. Space is as part of the space,
06:05 is SAm, there's someone would call down here all the i in vectors
06:13 with the set of Eigen values for and we'll come back to the illustrator
06:21 then there's also some something use vectors this space then then that to
06:29 So they stay in, that's what called in variables do not cross things
06:36 remember. And I'm sure I will normalization, mormons, spectrum and spectral
06:42 of various patients very well. The couple of leverage points. So
06:51 everyone immediately enormous. Okay, so just repeat quickly than what the properties
07:02 and which the most common ones are by no means there are no lots
07:07 norms being used depending on works for activists, what you want to
07:11 So this is what you're trying to propose a murder um, Vectors and
07:18 tackles the matrices. And the basic . Is that the norm of the
07:27 . I cannot and I got a . So the only case when it
07:32 be zero is this what you're trying measure is identical. Other things that
07:37 scale every component of the director of same, anything in practice and it's
07:41 same as failing take care. And there's this called triangle inequality that says
07:50 norm of Osama factors. It's the equal to the sum of the norms
07:55 thanks. Something you said. This the basic properties that the normal needs
08:01 have. And so so here is of those three basically normal in your
08:12 today. One is um and what place is basically sums up all the
08:22 that is the components of an It's not necessarily the most common norm
08:30 sometimes it's a desirable norm and the one which is to back to normal
08:39 very commonly used and us this from sum of the squares of the components
08:46 director and I think that's what we to sort of giving mean squared error
08:50 instance, that basically never to normal of the errors in that case.
08:56 it's very common and it has very mathematical properties. So people use it
09:01 instance, can you least squares regression instance. And the basis for that
09:08 the usage out too. And the one when it's also not the kind
09:16 commonly used is infinity or max norm it's called the vegetables. Look at
09:24 maximum value of any component and they're useful depending on your materials. Most
09:31 the time in this course will be So where is this going to
09:40 Um back to normal? So In case yes some and use the L
09:48 norm, you know something of all components of that. So you might
09:56 a war it can't be enough travel problem or something else. But that
10:03 up all the the segments or whatever want to do. For instance if
10:09 have to norm in this case the component here and the norm is the
10:18 of the components of the in terms after this and then give some intuition
10:26 what The one and the pick out terms of the night. Now matrix
10:40 should have the same properties that the norm of matrix should Always be positive
10:48 long as the matrix is not zero scaling every element of the matrix
10:54 also scale the normally matrix and same with the triangle inequality that more about
11:00 summer matrices shouldn't be no more than , some of it, you
11:08 And um then so from here there's few more properties to products.
11:17 Um I have no secrets from the in this case a excessive actors and
11:24 was previously and there is that the more and the same thing actually to
11:31 a space and what's being used in book and is common is that months
11:39 the base and matrix norm on the of the world. So eight times
11:45 is a vector. So let's you what this is from the definition of
11:53 . But to get the matrix then one plays it on with it
11:56 bit. So in terms of this electric product you are allowed to just
12:02 it on use different sectors. And description here is that the chorus director
12:09 for X that as well refined should wrong. And within that measure of
12:17 norm there are many possible vectors Someone needs to try on all the
12:22 to sex with this property and then the one that maximizes every one of
12:27 major inspector product. So that's the behind this induced um matrix norms based
12:35 vector norms. Yeah. So um thing. So how does these things
12:44 up? So they used the one the L one norm. Yeah.
12:48 basically summing up all the components and we have an a check. So
12:52 not just the vector. So in case then one North Potter Matrix is
12:59 15 up the column for which the of absolute dryness in that column is
13:08 maximum the max corresponds to the And but it doesn't mean that the
13:16 , commonwealth something on there, max is also known as the the absolute
13:26 of it is another spectral norm. come back to this. And what
13:31 stigmas are, you should be used singular values of the matrix and we'll
13:38 about later. So come back to . But it's related to Yeah,
13:42 get mad. And also All How it's spreading and the internet the
13:50 coming instead of taking the maximum call son takes a maximum. Those are
13:58 the so depending on what you want do. I mean, I guess
14:02 relationship I promise to give you um , when one does the square is
14:11 turns out, but often one formulation up um looking at the product of
14:17 matrix with itself used on the new . Uh huh. Okay. So
14:23 that case as it was obvious that on the other side A N A
14:30 , they have the same Eigen So basically the Eigen value of this
14:35 is squared, I can tell you so And the typical one your son
14:45 sigma for singular values as I said the singular values corresponds to the I
14:53 values of A um and again the absolute value that is respectful ranges.
15:02 respectful radio system, maximum, Absolutely am ready. So and then one
15:12 I will come back to me not but that is very important. There's
15:18 so called condition first in the first , I introduced this concept of decomposition
15:30 and that is something that actually indifference our notion of using this conditioner which
15:41 yes, please tell you something of scale or the range our values and
15:51 relationship between the backs and the And here is some it's related into
15:57 north. So this is the very number is a very important quantity in
16:02 to understand italian american procedure for instance . Okay, so that was
16:15 Any questions on this norm. Thanks . Um No, but it's something
16:23 you should make sure you actually go the intuition or the memory of.
16:33 , it's like some of the matrix that could be used. Um so
16:38 will start, I took this from other so I start sites associated with
16:44 books with the book because didn't find all that much highlighted in the
16:53 that it's important for not only understanding one can expect from our own values
17:01 Eigen vectors of the matrix, but the procedures used to find them uh
17:09 already just the structural thing and try angular triangular, it's pretty obvious.
17:17 it's a form or something you may may not be familiar with, but
17:23 kind of the triangular matrix um plus diagonal adjacent to the main drag
17:34 So is that so it's an upper form. It means that It has
17:41 non 0 Bangle below the main Its upper because the rest is kind
17:46 a triangle and upper triangular matrix plus diagonal to mean that everyone only the
17:54 adjacent to the main diagonal, one arbitrary diagonal in the door trend
18:01 And the opposite is to just transpose opera to England. Yes, it
18:06 lower uh than orthogonal. The concept a thermal matrix is a matrix for
18:19 we take this transport and multiply it itself. Yeah, that's a
18:26 multiplication, left, multiplication. The of itself for the right multiplication of
18:32 transfer itself. If you do this , then you end up with that
18:38 . So it says that the best the columns, all the agents,
18:42 are Linearly Independent or 1000 onto each . Unitard a matrix is Kind of
18:52 extension of this concept of Orthogonal two . So information made it successful.
19:01 and congregate all the numbers and the is real. These two are the
19:08 but this is a complex matrix ever than this. What? Because compared
19:17 just transposing you also contradict that um welcome in here and things like that
19:27 the transport information. And by central major cities it's also that the complex
19:33 gates transport from that then you get same age, expect and normal is
19:41 that corresponds to the for complex. these properties are just go and don't
19:52 only in computing I. M. and I am records but also understand
19:57 properties. Eigen values definitely making sense property being forced quantified. My That's
20:20 . So it's just there's examples on and this transpose is everybody knows that
20:26 find the transpose is Did they take 1? And it transports values so
20:32 don't know about it. It's almost where they are. But the since
20:36 time on imaginary component. Sorry that plus effectively in other words these two
20:43 places transport and symmetric is pretty He transports this guy and then get
20:49 back and this symmetric matrix the commission that you get into transport and communication
21:00 moves here but is the victim's miners all this movement here. That's a
21:08 . This is not true but this because even the transports and navigation.
21:13 don't get. Oh man. Yes functional properties. It is because of
21:24 matrix. Bye bye itself should be identity matrix. And I guess I'm
21:33 to talk for all these. You see the effect. This matrix
21:39 It percentage fact 0101 then becomes the . So that's kind of the after
21:46 corner of their data in the Um Okay This one times this column
21:54 and he's in the virus. Find that they can benefit adventures for these
21:59 examples of majesties but it's not true this case. Um The unitary property
22:10 in this concern and initial matrix and I think it's uh transpose and congregated
22:18 this case transposed has more of the and plus and congregation changes that have
22:24 know. Mhm. Do this exercise that transports and congregation and multiply it
22:32 itself. Normal was something where it be 842. Um No, so
22:53 get a huge so far. It's a good example of what it was
22:57 normal. Uh Oh you should just the same sorry, this version of
23:03 identity matrix and there was unity. my confusion. So and then if
23:10 . Find both these motions. In generates this magic. Whereas this one
23:16 not true. So there's just a of making properties that are being used
23:22 both. Understanding as I said, inviting as well as computing competitors.
23:31 so here is not on the side shows a little bit of our Eigen
23:40 and Eigen breakfasts are related to some the properties on the previous stuff.
23:48 , but another one that was not the biggest side is I can values
23:55 the power of the many checks for someone to buy it by itself.
24:02 , the challenges of this lecture isn't union batteries, so the original matrix
24:09 raised to the power of K the of times is hard to find.
24:14 if you know the argument is okay you also know that I can values
24:21 K take for the car, Um, and conversely somehow find it
24:29 to get these ideas values, then can also derive what the values of
24:34 original nature and this is in fact in as well as the power
24:41 Eigen values and Eigen vectors, I get out of the full name of
24:47 lecture. Um then yes, that's , that's the annual values of the
24:56 is the inverse of the hanging So it's also fairly straightforward and some
25:03 the methods are actually using this property compute the Eigen values by instead computing
25:12 inverse of the environment because in some turned out to be simpler and then
25:17 just send the mercuries values Of the nature is the non inverted one other
25:27 that are sometimes useful, it is and all the Eigen values really.
25:39 it's complex and her mission and it's the case very real. So in
25:46 it is complex one should not necessarily them to be real but because of
25:52 symmetry and kind of being commission then comes out dying in violence.
25:59 If you have a procedure that generates know and on real uninvited in this
26:06 you should be careful watchful when you've something as well. Again, understanding
26:12 properties um of you should gaps as properly on the matrix. It is
26:20 and that will strengthen its not on commission but also positive definite and it's
26:26 all the members of now the one we're going to talk about. So
26:34 is testing so as was mentioned last I said, the one thing is
26:47 because you're lucky somebody gives you a matrix. And that's great because in
26:52 case it's already diagonal matrix. It's to see that all the animals elements
26:58 indeed the items like this underground. what this sincere what's known as a
27:06 transformation uh is if you generate the this time. Come on the next
27:15 that is reimposed multiply when major XP it's embarrassed then um yeah have certain
27:28 . I am values this matrix. the same as our new values
27:36 So if it turns out you find P here that makes this product diagonal
27:44 you actually have to take it off the onion. But I'm sorry.
27:48 there is a bunch of methods for either values that are based on finding
27:54 . So that this product yes Diana we'll talk about that. So
28:02 this is pretty much what they deserved , you know because anyone, it's
28:06 easy to see that the man is the matrix D. And one of
28:12 island values all the values on the . Mhm. But it turns out
28:18 also too they're the the majors is . So sometimes it's sufficient that you
28:25 find transformation that turns to matrix into triangular matrix people. Then again,
28:31 know that their values. Father wanted that I haven't. So um this
28:43 ah the stars in this particular case don't want to also do what the
28:49 am vectors. If it's the diagonal , let me know that I can
28:53 this threat from the matrix diagonal And it turns up also dun dun
29:02 Eigen vectors are the columns on the politics and that they are almost like
29:08 normalized because identity magic. It's just I'm the one that had. So
29:15 norm is one. But just one comments as well, that's for me
29:19 it's normally identical twins. And also it's not always possible to get to
29:28 final four. And in that case can potentially find something essenberg form or
29:34 support Jordan form that's like never before more general than you know as well
29:41 these bonds are after site down, has some work for um the lower
29:47 report potential to see that. And you can get closer to the dialogue
29:54 . Thank you. Yes. You what I'm for the number of these
29:58 that we'll talk about next time actually to find them. There's some that
30:05 correspondent to the United States and that's , we'll talk about this. So
30:14 my diagonal form form form of We want to they will find the
30:27 a couple of more concepts and it just mentioned last lecture. That is
30:34 . Again that if the matrix A not effective, that means that all
30:41 onion values are me and it is . Um, with this statement claims
30:50 um just about the deer in Yes. That you can then find
30:55 matrix. Thanks. Such as this . And it doesn't matter when you're
31:01 of cost. Spotify start to switch wonder from X- eight or X
31:06 Um, it doesn't matter. You still find that sometimes. All right
31:13 way and sometimes that was done on inside T. A. Friends the
31:22 . So what it says, it tell you anything about how you find
31:25 ex but it just tells you that a good hope. If you can
31:28 enough that you should be able to the next. That this is
31:32 And then do you have for your has only turned out also. And
31:44 fact that just the context and what is. The fact that wasn't mentioned
31:48 some some artificial sergeant Ron moore is of them. And this was the
31:54 of the multiplicity and this is just concept. It was no multiplicity example
32:03 values now. So this is similar transformations and a typical experimentation.
32:13 Postmarked rotation for the universe instead of . That's kind of almost so
32:25 So here is what they have. yes. Um one have done this
32:32 will have some tea and this that invulnerable to compute this product and they
32:39 finds their technique values for B. again be is diagonal is particularly simple
32:44 maybe it's still simple to find their values to be. They're not made
32:50 many of the algorithms dude. Um manipulation of multiplication is over with some
33:00 the matrix like this. So all got matrix B. That is easier
33:06 find the argument is for Okay, in that case so once you have
33:13 baby and you find an ideal value see and I'm going to go back
33:24 with the inverse of the years of gap. Okay. And the inverse
33:28 Y. And then on the right side to get under being the first
33:32 . So what. So what this that If Lambda is an Eigen value
33:37 this transform my 60 it is the argument value as for a. Because
33:43 here in verse volume it's not an that correspondent. That's cool. It
33:51 looks different but this is kind of ex context in some way. And
33:56 is the same for me. 14. So this is um so
34:02 I want to have potentially found american and vectors for this transformed problem working
34:09 DVD instead well directly and say I value also for a and we'll get
34:17 for a to the best thing this . So this is what they
34:28 Uh huh. No the triangle Here's that shows also that has the
34:35 characteristics. Calling on this is the for them because I didn't know the
34:42 of the characters from normal. It's little bit more of a rule
34:46 It's the same. This manipulation shows . Um Now so this is a
35:04 bit of human properties of a what profits on this? Except all Transformation
35:17 60 and correspondent profits of the similarity matrix. So so then again thank
35:34 say has a set of distinct target and some multiplicity then mhm. Yeah
35:41 is non singular. And if you to go ahead and thank you fix
35:50 property. So distinct talking about this can transform into a diagonal matrix because
35:57 are symmetric um He has additional properties the we columns of the matrix.
36:06 1000 also passed. Uh huh some those properties and comes from America and
36:13 the real and diagonal. And if have any permission then he has the
36:22 property that knows that. And then get transposed on playing with themselves.
36:32 or in this case for T then the advantage in managing but then again
36:39 one knows that B is real and . And also mm hmm billy
36:47 Other forms that you can get depending the properties of A. So that's
36:55 understanding the properties of a. Give ideas both about potentially define team and
37:03 takes on the properties of tea. more importantly what you can expect to
37:08 able to get that you'll find the . And I guess the first ones
37:13 easy because they know it. They what I think that was good times
37:18 unanimously Eigen vectors. Now we talked you and Jordan for my Children example
37:25 and get it and I want to about for sure and okay and it's
37:36 article concepts and profits be well but should be familiar with the concepts are
37:46 what they mean because um intervention is active in real life what type of
37:56 algorithm or software they would prefer to because that's related to the problems of
38:03 nature that you should know because that is giving you problems and distribute examples
38:15 the similarly the transformations. Ah yes, this is coming up,
38:27 trying and error. Yeah. Come with this matrix by whatever mechanism doesn't
38:34 the amount of find me. But you have this key ah wanting them
38:40 at the the universe is easy to in this case or something else.
38:43 can validate that the universe and speed fact gives you the identity matrix and
38:49 that at least these two guys are . Now it should then carry out
38:55 multiplication and is this very moment? in this case started with the symmetric
39:07 matrix and this is also in our majesty. See that it is that
39:15 electors columns are prominent in each And then as promised by the previous
39:23 he becomes diagonals and arguing captain values the obviously but remember that these are
39:32 the Eigen values. Okay, so and we had so we had one
39:41 . It was 1 1 admit that is four and yeah, I didn't
39:48 you how to find that this is . But they should I decided to
39:53 this. Maybe the corresponding on your mm hmm. To the argument to
40:01 can see that she takes eight times x one what we got on top
40:08 basically the value to 12 and three -1 times one. So send something
40:14 um and if you look the same here right, one day minus one
40:20 three is another two. So the is affected too too, which is
40:25 on this fact escaped by the number similar with the so you can easily
40:31 that these are italian directors correspondence and none of this told you can have
40:38 find tea and how to find What's the point about? Been told
40:44 one? Doesn't reflect about it. verifying the property but it doesn't tell
40:48 about. So this is fun. . And we'll get to examine the
41:02 and mom can also live yes, if one wants to do that.
41:07 you wanna step work polynomial to find onion values different machinery. And you
41:17 . But that's my right. I'm sure again came up for approved
41:29 for what type of matrices you can good transformation matrices. And so that's
41:37 mentioned himself matrix operations particular argument. in this case is similar.
41:49 30 assists, genitally matrix you such . Um And I'm gonna turn will
42:04 this product its identity. So now is then sure. Rule that every
42:14 matrix his unit terribly similar to the . So if you have a square
42:20 , it doesn't matter whether it's complex or symmetric. That's just it
42:27 the square form, then it should possible to find these units are the
42:35 that terms um this can't be or the triangle. Well that's again,
42:46 properties it doesn't require much of the accepting. Right. And then there
42:54 hope to find something that trumps a the diagonal line. And I'm sorry
43:03 . Yes. An additional it's It's also that's a subset to august
43:08 square matrix. Then you also get matrix. That is another space
43:15 Mhm. And here is one of . No problem. So it's it's
43:26 top of it information. That means can you get transpose symmetric for submission
43:39 ? You attacked diagonal? It's it's just a regular matric. So basically
43:45 there Squire metrics symmetry then you can the diagonal matrix. And of course
43:53 real matrix is a subset in some of her mission matrix. That's also
43:58 but that's some of the cemetery Yes sir. The dying animal making
44:04 not just the scoring matrix. So is a clear shows that um yeah
44:13 making so many things. In fact matrix is simple. two minutes that
44:21 transposed. But since the Triangle Matrix . Non triangle or something. And
44:28 other triangle if you transpose it um it's not she reflects to them
44:38 I don't know. So And this one hour almost at all.
44:54 So okay so now showing a little of you. Yeah Aryan values and
45:03 vectors for do you think they're hiding for a triangular matrix from the formulation
45:10 the determinant. It should be fairly that the union values of the triangular
45:18 are in fact the violence on the don't know. So now to find
45:29 corresponding onion vectors, countries that are your answer because there's the picking lambda
45:36 correspond to one of the diagonal. is something one of them Diagonal and
45:41 x equals zero. But this is kind of triangular matrix where that is
45:49 the africa and the matrix I should do you have an anger? So
45:54 is a bunch of columns and And then there is 1/5 column and
45:59 particular role I picked out that corresponds the diagonal value to zero, that's
46:08 . And then there is the rest the columns. And but of course
46:12 the rules and the rest of the remaining roles that gives you the triangular
46:19 here. And then six months, nobody. And they kind of current
46:30 the corresponding vector X. And it that this is in fact. And
46:36 vector by the components corresponds ability, set of rules, 15 -1 for
46:44 So now we'll pick up here, , down zero for the rest of
46:50 roles, you know? And then it looks like the product here,
46:54 we get is 11 times one Um You -1 finds you And this
47:03 is considered out and the rest of is zero. So then um as
47:11 as this non singular matrix and we a solution basically finding a way the
47:19 is known the matrix. Yeah, you're finding. And we have a
47:27 I am back. Do this Mm hmm. And then something was
47:36 of like a concrete example. Finding in ah this is not trying the
47:43 . But so yes, yes has through with that's before. Again,
47:54 doesn't tell you how to find this investors. But using the obvious stuff
48:01 have these equations than the republic These are also having Breakfast correspondent.
48:16 . So um, so that must been a lot down there. Knowing
48:23 about the nature of a what type transformation makes this should be possible to
48:31 intentional. And then there's an example some solid find my brothers and I
48:39 care just for him. Sorry. agents. Some always this is not
48:46 believing like I am very nervous and am an actress is a standard procedure
48:51 it's being used and and some of other and something to talk about.
48:57 that's why I brought it up. how many have are familiar with
49:02 The same here judgment of phenomenal Okay one. So I'll talk about
49:12 and hopefully come somewhere completely. Especially a set of bacterias that are not
49:21 for abdominal. And ministry is not the access over there. The coordinate
49:27 X, Y. Z. Right X. And Y. Or government
49:30 each other in the three dimension These are probably not the both the
49:34 form 90° in three different dimensions. with the number of the 4th toe
49:41 persons to make them. That's clearly having projections on each other. And
49:47 normalization is to make the unit. that's what this Graham Schmidt of the
49:54 process does. And in the book actually not in this chapter, but
49:58 being used in some other discussions in chapter. And a lot of sequence
50:04 far as I'm concerned. So here kind of the basic idea. As
50:12 said, they were having two vectors be one and 2 that are not
50:19 linear because if they're cool linear, only cover one dimension. So they
50:24 be calling your so there ah have direction other than submitted to have some
50:34 or something that is cannot be projected the wall. So in that case
50:40 can protect someone component of the two another axis. That is then orthogonal
50:46 the one that is called linear the . And it's based on this
50:53 there is step that you take um been in the product to find the
51:02 or director on two. Another And in this case We take 31
51:10 a starting point and make the direction Iran kind of first union directors.
51:15 they don't just scale me and my every component of the for the normal
51:22 direct means, wow. To take normal the scales better than that becomes
51:28 union. So that's the first Then you take the projection of me
51:35 enter this thing And then there's the . So you find this other vector
51:41 is orthogonal to the direction of the this report. Um this thing was
51:49 . Here you take me to um this. So again, some practice
51:56 you get this one and that becomes youtube that is done as far enough
52:04 the one or so. There's no that we want. And then you
52:09 it firing and divided by the thankful said yeah and you're next and then
52:18 and visualize things in higher dimensions. then it keeps going as many dimensions
52:23 you want and but the rest of day sometimes to take a new vector
52:32 projected whatever subspace earlier of actors have and what they found the remainder is
52:40 that is that is orthogonal to the slide projectors, all of them.
52:45 then the scale of getting into what length of that relax the ground's swinging
52:54 . And I think that's giving yourself example here in this place, I'm
53:01 anyone. Uh I don't know and kind of normalized against what they become
53:13 potential of this. So in this one of these factors are not Ortho
53:20 the use um so let's go do you want You two is not
53:28 and new ones in this direction. obviously we're talking about um yeah,
53:37 used here. And then the normalization the square root of five because this
53:41 vector is mm hmm. To be is the sum of the squares and
53:46 square root four square this and square is one. So one plus four
53:52 five. And the square root mean root five. So now they're scaling
53:57 was square root of five. This and that vector than our unit length
54:03 the north, that is used So, and this is vector 11
54:10 two and make it the common vector doing the transport and you'll find it
54:15 . That's one of the columns and paychecks And the other one was this
54:23 . Okay, he was not. , so this is what's going wrong
54:30 . So sorry about that. I discovered it actually this one that corresponds
54:34 this one. And then you can out that these two problems are talking
54:40 each other and they're also getting So that To see that this matrix
54:46 majoritarian matrix to process product for So, um, so this part
54:56 the show's these particular vectors. They be found first too the process here
55:05 make them um, they used orthogonal normalized and in this case the first
55:13 taken apart. And then somehow the happened in this space, but it's
55:19 fine. And then it then shows the shoes terms correct for this particular
55:26 but again They didn't directed 10ure how find these two factors. Oh okay
55:37 . So that was again this process is going to be used when they
55:41 to this matter down here. Any questions on interview yourself of the
55:53 . So the next stop against this or I get values. So it's
56:00 very simple procedure. In fact that known as discs. So this is
56:09 doesn't tell you exactly where the annual are. It can give you english
56:18 space. There are new brothers are certainly. Um and the detector is
56:28 to figure out just like guaranteed not be zero because there's somebody here by
56:36 zero that is an aged six So it's a very simple procedure to
56:41 whether the metrics is guaranteed not giving definitive answer if these risks don't contain
56:53 original department system. That means this that has come to the table.
57:04 it's um I'm giving examples but Have different kinds of versions of it.
57:13 On tape the diagonal values of the and then on some such the absolute
57:25 In one case is the other elements the matrix in the role. Oh
57:35 and energy. And the other version guess this one in some of the
57:41 value ah McConnell. So let's see it is. So the view is
57:50 for the sun radio here, radius the Yes it's the sum of the
57:56 values of the elements in low So this instruction on this line is
58:04 take a disc center, latin diagonal and that's the radius equal to the
58:09 of the absolute values all the elements that. No, so you get
58:16 set of discs one for each diagonal and they may or may not be
58:21 now and that's the thing and the one is best Vanessa said in that
58:30 to some in this case the radius the disk is now the some former
58:36 of all the elements off the back of this one disk seven formed by
58:45 sums of absolute drivers excluding the And the other one is this formed
58:52 summing up absolute values and the color correspondent for their values in absolute contexts
59:00 excluding For those who have two sets disks, that's kind of victoria looks
59:06 this. So See whether one is From the annual values from this case
59:13 three clean by these natures. So East Bangor Elgin you get the different
59:20 for the discs received. Um and the radius depends on the values of
59:26 diagonal and and then Miguel said this this case, that's when that happens
59:34 all of this, you know cover in that case there's no guarantee that
59:40 Aryan brothers although the matrix has not but if it had not been concluded
59:47 any one of the disks That would been guaranteed definition just 1/2. That
59:55 tell you where the Eigen values but best values are within the union
60:00 all these cases. Okay. Services ourselves. Mm hmm. So here's
60:20 little three x 3 matrix. So a complex matrix. So it has
60:26 real and imaginary axis in this Um, and so we have previous
60:33 centers and the centrist almost independent of but then unless you go through the
60:43 sons and then bring disks here among of the diagonals. So we have
60:49 other things for uh, center of disk and the low some absolute
60:58 absolute value of I is one. 2.13 of the radius of the disk
61:04 here. Next time is to some these again Give us one absolute
61:10 This is also free And the last and this is three disks And correspondingly
61:18 go same. That I'm not doing anymore. There's some of the common
61:22 students and three. So that gives um, and that's pretty much this
61:32 that in Australian waters it is soft that means so it doesn't give you
61:42 annual bonus. But it's sent to notion of where they are. I
61:45 yes, can only be used for . So there's nothing but it it's
61:57 gathered for complex. I would say if these were realized it doesn't
62:02 it will be just be on the and then you get intervals on the
62:08 for so um, and so This Kansas Mississippi, I can pick
62:18 up and maybe there is a picture , plug it in, right?
62:22 you can see it founders. We sorry, somewhere, this is my
62:32 . So somewhere here from things off central years. All right, thanks
62:46 that. Um, the use of to know is very simple to compute
62:53 on also in terms of exams and to have something that simple as something
63:04 . I said you stopping to remember be localized because if you can find
63:09 again that matrix just can't seem not be singular. Then the difference that
63:13 tools you can use as from the of the natives. Alright, so
63:22 it's a little bit um, and coming back to properties on the
63:29 you know, remember the video from collapsed so trying to have to relate
63:36 precisely than um the relationship between dynamic and arguments. So, so here
63:52 one of these very simple, all leads that we talked about dr solve
63:57 and in this case distributed on an solution. But this past green and
64:03 became find a solution but a us critical part and show that there is
64:11 is related to Eigen values and that's a subsequent related them to resonance frequencies
64:18 structure and that's why there is a with the values um and personal properties
64:30 . So no, that's more than a single equation. But then and
64:35 in simple equations of the remote system . two equations still in your system
64:44 . So formally in terms of this we'll come back to this system in
64:51 concrete and the example later on in case you also have now in the
64:59 form similar solution that X equals uh financial declaring version version of the use
65:12 in the matrix and vector formulation. they've done here was there are the
65:17 value see here, zero mm It generalizes two the venue system of
65:27 form and we'll play with this a bit and runway it was the positive
65:34 dissolution. I found out I want figure out what this means. Ah
65:40 that with an aging seven the circumvent this exercise here by this by postulating
65:47 director solution is similar to what they single place that is exponential decay for
65:55 or an increasing and diversion of initial . Think so just just find it
66:06 here and then one does it just something that each of them.
66:12 Components that the derivative of this postulated , respectful number in France since we
66:19 coming the initial value. So then ? Yes. To control them that
66:27 response to kind of X and this under X. So in this case
66:32 experiment here that was in the solution in fact an Eigen value um to
66:40 mentioned so that's kind of important part relating ah the I get batteries feeling
66:55 of properties of the matrix. And I think there is a more complicated
67:01 well. I think this was the example that we use in the book
67:10 sort of vibrating glasses separated by spring two. Thanks a lot of our
67:18 um separations of motion and not not . I'll say the coordinates of the
67:26 mass is related to the supreme How much are full of the force
67:32 the present um that spring and then also the other side that is how
67:38 difference location between these two masses and are you then effects for the forces
67:46 place during the spring constants on this and of this type some of that
67:53 the next map. And you get similar questions. So best to look
67:57 this company to um equation system to the location of the screen. There's
68:07 one and 2 and then this kind a matrix. So there's a function
68:13 the methods and the cost and then was done. It's in the as
68:22 as I remember some reason really changed But anyway, someone gets this matrix
68:27 is all the spring constants are the and all the few matters are the
68:33 . So it's simplified nations here that taken also notices symmetric and someone formally
68:44 back to the or something premieres side terms of trading and constance then like
68:53 then the prostrate solution the way we it. And that was formally questioned
69:01 . Primary percent context. So I to try something of this phone and
69:08 plug it in. Um so we the derivative. We need the second
69:13 in here. So take the relative this thing twice And then what they
69:20 something of this form where again this the vector X. And then we
69:27 rest for them you know that make matrix pants from this sector experience that
69:36 it falls to this was the right side. Double prime business on that
69:42 . So now Well they had eight x equals something times X. So
69:53 finest moment to spare is an argument to this structure. So oh my
70:00 ! The Aryan values here are obviously to several of them from the
70:08 Sorry, playing around with distance and alphas and betas and then you know
70:16 is related to the matrix violence. then so they all of us here
70:24 they are respecting resting on frequencies of structure. So then when I got
70:34 here, Paula that's dissolved for launching to mm hmm values are just all
70:47 all. Um the options that's different solution to this depending on on this
70:59 we get this solution and that So we missed her in this
71:07 Part of this, the four different . Um because this solution was in
71:14 of the square. Right? So is this sort of plus and
71:18 I want to take the rules That for the right hand side.
71:23 that's why they basically get always and a solution to mm hmm equations of
71:31 for these two mass systems and it out the values of the matrix and
71:39 . Um this bickering solution will just in the normal frequency of somebody that's
71:46 me as a way of trying to the relationship and the onion violence from
71:53 nature cities and dynamical properties of the that are making the same models.
72:01 not covered by Sullivan union system impressions ah And so this gives a totally
72:11 aspects of our system that is dynamically study for test just manipulating the
72:23 So okay, just a little bit preamble. But at home start for
72:32 next time. So so farmers will up more or less um not quite
72:42 machine er but telling indication of properties are important and what's going to happen
72:51 . This is one method and there's whole lot of more methods um that
72:57 then do so all these different features find. I didn't have breakfast in
73:05 own life. So so here's things you're actually trying to solve something and
73:15 is not much discussed in the Um so computing, arguing values are
73:25 and I remember so yeah, the expensive things kind of talked about so
73:32 was constant elimination to have an N N matrix. It was included
73:41 Um any ideas, values, computations more And to the 4th or
73:48 So they're calm potentially quite demanding. the one thing when these fast oneself
73:56 one doesn't need all the items. if you want to need a few
74:03 are procedures that that doesn't go employed whole machinery to that follow. That's
74:10 simplified versions like guarantees that I think for instance the largest to the smallest
74:17 as I said, the largest time , respect of radios and sometimes critical
74:23 convergence rates so long as they're going announce in some cases ah getting the
74:30 and the darkest, Remember this condition of observations between the two. So
74:36 jurors about the condition number because that convergence and America properties. Maybe you
74:43 need to, so there is a lot of things deciding. Um but
74:49 you are, and while they're arguing this at all getting examples, they
74:56 necessarily care for the onion vectors just the onion violence may be good
75:01 And then the other side when I to talk about the argument is just
75:07 , this example of data structure those methods tends to work with Just
75:14 one Eigen value and one Eigen vector the rest of it. That's
75:20 Other things are really complex and other that affects how do things is but
75:25 major cities have been exposed so there's whole lot of considerations before you choose
75:31 method. The continuing Okay, that not so lucky. There's the
75:42 What methods are going to do is of them use this similarity transformation to
75:48 get to a diagonal matrix or triangular or is one of four matrix and
75:55 doesn't do that. The parliament that is this property that the I give
76:02 that values the power of the matrix the power of the Vatican the majors
76:12 10 years. This particular property that we have on the for investment
76:24 Other things you just said, mm notion dot fit in. I can
76:31 of the inverse of the nation since values so for instance department and in
76:37 identified an artist. Thank you. that's one way of doing. But
76:43 you're going to have the smallest item they could work for them or
76:46 several nature. The current effort is these two methods and then it turns
76:54 you can also use this property that the problem. So instead of having
77:00 matrix A He used an hx A was subtracted the constant for all the
77:09 that sometimes it's that's and stop with line also of yes friends algorithms that
77:21 being used. Or I'll get out your computation and I will say this
77:30 iteration to iteration and these methods down from the bottom are the ones that
77:38 in practice most common. And I'll a little bit about this function
77:46 Next time I'll start to talk about district former instead of covered in the
77:50 . Many of these are not covered them. I think my time is
77:57 . So thank you for today. you ever do we do you ever
78:06 a daily and a computer for this I mean this is Hayley. Erin
78:11 fearing useful. I think it is to try to remember. Okay,
78:26 thank you looking up and I'll answer better next time. Well thank
78:32 No, I was an exam for
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