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Math for Economists - Lecture 8

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OK so last summer I wanna do there were put up this Agon value us situation again in order to expand little but there's so there's some interesting facts that come along with begging values that I didn't get a chance to to cover because I was trying to do a whole problem for you last last time we had a matrix a was the matrix and what we did was So we started looking matrices as functions and this takes the plane hard too Into the plane too but when you look at this But may Churchill really know exactly how it does this so the study of ideas values it basically finds that taxis that this operates on what we found was we found the hike in values to be I think it was those there's this was you this We already down to you was the egg in value 0 1 1 and that corresponded to an ID value land equals 3 and then we had a lame equals 1 and he was 1 negative What they really are that taxis on which this operates Get it chooses to view the plane like this and then if you will Choose to look at it that way to you can see what's going on it basically takes Anything along this axis and multiplies it by 3 the land equals Street and than this 1 just keep keeps multiplied by 1 of start-up looking at it as grid like this and then moves over and multiple is a song by 3 and stretches that then we also formed an orthogonal matrix cue from these vectors Analysts rate those down ahead 1 1 1 negative 1 of Hugh We had to divide by the likes of those with the likes of those guys if you like we can write he pulled with 1 or 2 out like that and then this is orthogonal we learned about orthogonal matrix was last time and what that meant was a few times transposed is equal to the identity and really what that meant was that the transfer was was equal to these numbers so that what we did was we multiplied AI neither side by QMQ Trans whom we will just I told you to do this this is what's called the process of diagonalization so we took our original matrix a multiplied on the left by Transco's which by the way is also a human errors then multiplied on the right and what we got was a diagonal matrix And the sky had the eigenvalues that's where we use this Capital landed to describe the matrix because inside of that nature to gut the little lamb those that all we can do a little bit more of this this is actually this is the part that the economy is actually just when we were when we're studying matrices rural we did what we start off saying OK welcome we Adam yet we learned that out of there we secular multiply and yet we learned that a multiply matrices what we didn't learn how to do it is by will you sort of learn Ataturk powers of matrices like a that would be a kinder times it
But when you do it again is much more complicated if you do it real math class the kind of focuses on this theory OK Some so of that target powers A couple of their Saks about I value
So the matrix form formed with that could be it so the reason use this this terminology matrix forms I wanna write this in terms of a product of a luncheon each fund that I have a matrix terrace where user capital
1st exposure to it over the whole and now and now we find these uh matrices from the quadratic form Start with an example here next ones where does for next 1 used blurs is a little bit more complicated once it it's like the 1 before but now I get this added they've already got a little excimer X X 1 X 2
Next to And I wanna right has
That It so this is an example Rowley the squared terms
And Kramer's role a X he's still the system of equations there's a expert on through equaled be year and then Kramer's role us to do is to just hone in on 1 variable while but said I just wanna know what Well X I was equal to The determinant of AI divided by the determinant today this being determined that minute to tell you with a AIA is a where I take less than foot in the I can't think what I did it before I use J. syllabus which affected GA represents columns truck with feed Want to Be a Millionaire and then leave everything else along than I would take the determinant of that appear to take chairman of all matrix and that tells you what that individual variables So on an exam I forgive you work for variable system and disable find me x 3 basket Final Four By disabled funny X 2 X 3 than you can just horn in on that 1 variable values Even those those ideas came from different parts of the book They were all part of the same thing or off solving systems of equations now 1 of the back in a while some the details like At you know what is the inverse of what is determinants of stuff because we had to learn most things along the way that's essentially what we were doing Time although through Chapter 10 Chapter 9 a lot of the stuff that we just studied with pure matrices sometimes is just notation you do you know a transpose by put it see there you know what to do that if I if I put a little negative 1 there you know what to do with a lot of his just notation but you have to know to do the basic operations operations petitioned a beat when you doing a plus B which matrices they have to be the same ordered a both that Begin by and you can't mix matrices but different when you're adding and then when you multiply the them you also you have to be even more careful I do a Times beat you can do in by Annan by Annan unless these little numbers match 5 of them twice and And die or are these must be the same and that the resulting matrix CD that's the outside numbers sources said Quick example for your notes by have 3 by 4 And multiplied by bike to I can expect that to be a 3 bags to hot Scalar multiplication let's see times matrix that means multiply everything in there that set this is an example of that I have 5 times 1 3 and 10 5 by the No don't forget is you're not just more buying 1 thing 1 row over a vault level we had the idea that transposed the notation was a to the seat power Now as say it may be that it's just notation that tells you switch the rows of the call Whatever the just a little pictures for that I have a matrix here and I had this road and this road in Israel in a trance turns on those guys In the column If Daiwa steady for the test I take these concepts and I go find looks to the notes its water that Useful facts that are associated it on doing this for something you could do the terms of the heyday of the chairman of a transpose we use this in 1 of the proofs of identified
Yourself spirit 9 . 1 if you like former members as a fact and we also had a chance to vote Said that does itself is useful facts about the transposed The removed our way to the concept of determinate peso there's the notation like absolute value of the Matrix If I write down Kazan and I ends you can even talk about determinant it's a square matrix to begin with straight that ain't as Annan and has got it 1 1 came on and a N 1 and that's the most general thing do and then what is determined it's quite a complicated thing to describe skull is little pieces too Ghostwriter the determinant is What's called cofactor picks Bianchi and prolonged any here Besides the need rope or call will Since we got this 1 here muzzle just do it must use the 1st so if by if I go along the 1st row here tomorrow I do with this cofactor expansion I take the elements of the Ichiro a 1 1 in the sense that it wants to know if and 1 and those of the elements in the 1st row and multiply them by their corresponding cofactor than I had I have to tell you what all these little Things a little cofactor so each of see Jay's that's equal to negative 1 I J Times em I J and now I have to tell you what am I doing this Wesco complicated formula them I J is the determinants Iowa my unit do a minute take A here's Thursday and taking the determinants us why have vertical bars here to indicate determinant and then I'm gonna removed Jake Columns and Let compact way of describing what to determine if you practice at a bunch of value to see somebody do it you'll really reminded all those things are but that's the whole world Mathematical formula for What does have a quick example go with make sense of all that notation example I like to users it's this 1 here and what I recommend we doing his duty to get really fast at this just picked different rose due to the determine along let column along the that road stood on different rows and columns the answer 0 So you can always check conceded just see people get past that The short cut further over this Part of the call factories just put the pluses and minuses of the upper left in quarters always a plus the scientist alternates from them so when I do this determinant like used the That is used this that road there so I see that that the plus-minus plus Like Star office 7 times a determinant of minus time determinant of something plus 9 times the Journal of something and then put those determinants of what I'm doing the 7 I get the 2 3 5 6 Where do the I get 1 3 4 6 When I do the 9th I get 1 2 4 Dodgers reduces down to a budget to buy 2 of the city 0 save us some time But that She should do to practice and then Andrew have to do a 4 by 4 of them yet I You 1 Divorce Now you should get yet what he's saying is if you have a matrix like this 1 2 3 0 4 0 5 0 6 and you see that that's triangular then the determinant is 1 times for 6 and even if he didn't you just use this column in you go 1 time determinant of 4 5 0 6 and then that's the one-time So you can just say that in your head go for
Don't forget these new role operations in order to modify the 1st you won't necessarily 1 that would certainly to buy 2 and 3 by 3 maybe that your preference I personally goal for the 3 trees and the other 4 by 4 and the time reduce the matrix 1st Florida terms
The inverse today would this guy represents like dividing by matrix here we write won over a week Israeli inverse that but it's like 1 over at the reciprocal of a matrix and the implication is that it may take gave you multiply it by inverse them with his vanity and vise versa you on that side that the idea the inverse is it's it's a reciprocal and identity behaves like the 1 in the number system the formula for a vs. gone however the determinant of that incident hinders doesn't exist unless the determinant is nonzero
At times the adjoint today in the of course and you tell you if adjoint is it's the cofactor matrix transport for every enough at the upper left-hand corner they won 1 you go you see 1 1 foot in the upper left-hand corner for every little entry through 1 of those ideas
9 factors that will make that matrix take the transpose would called the agile it's useful to just have the member said the formula for the 2 by 2 memorized foods have that right now
That's a little bit of a cumbersome formula would come too late use you can just memorize what it is like to a seat 1 2 3 4 a embarrassed and 2 by 2 case it's it's 1 over for minus 6 the determinant times well What we do we switch these 2 areas for 1 and make these 2 negative Take it do although the universe saying it's all those guys then a universe is equal to 1 over The determinant of Which is Latest PC times a minus be enough as long as A minus Germany is not If hoof
But That was all the Chapter 7 8 is that any ideas that Chapter 10 starts from the abstract things like vector spaces in the history of eigenvectors nite about lot stuff Chapter 10 Only chided narrow it down for you certain things you have to know about vector spaces so we had to talk a lot about it so I explain what it was but then there were just a few computations you have to worry about 1st of all what is the definition of a vector space
You and we pardon That But at these 2 guys and that means the some of them is and if I have any old in there that means a constant times the sector In the sectors so examples Artist unfamiliar setting the real numbers Always go 0 at the 4th 3 The vector spaces is being examples that will have to deal with it and then not vector spaces are portions of these you can have a portion of this India And be vector space so 1 of the examples of the exercises is the upper half point the reason this fails It is because if I have a vector you hear Sevey then minuses Daniel and minuses not in the vector space that's why fails to be elected not ending class I did the the 3rd quadrant of the 1st quarter of that if I'm questioning is that a vector space the way show that it's not a vector space as they take a victory in a week and thereby multiplied by negatively hits outside the vector space of fails the 2nd of the 2nd part of these Both are not vectors exist And then there's another example of the exercise Rice's shows the unit circle the vectors of filled it's not that means it's gonna fail 1 those 2 condition 0 A length of that debt The agent for that that double line so as not to terminate its double 1 the length of a vector if he is equal to Do you want to The new It is the square at me once where plus to where You just take the components to square foot the square is like a general version of the Pythagorean there 2 On the of again ask you to compute 1 of test Giuseppe recognize this Rotation that means take like that whatever you get in said the inner product that's another simple computations Hideo Nomo before July the United overdo it But you can be our end to transpose be that will you be safe if chance but point on Tsai become the horizontal Row vector se You want be 1 plus bodily damaging you if you 1 3 0 4 5 6 then in a project series here even have to ride out like this you just you just go in and that matching up the 1st CSU 1 and 1 and then you to indeed to matching those of multiplying eminent adding the results of 20 times for the last 2 times 5 plus 3 Times 6 which was 14 Pelosi if a
It'll probably be the case that I'll give you a couple of vectors and I'll say Are they were thought the piled sectors will say which ones are You'll see an example like that on the sample The key is EDG his where comes after the inner product theater product is tells you whether things areas are orthogonal not Inner product is 0 than their perpendicular are thought to have all you need so if you're To determine the orthogonality of 2 vectors it's basically like saying Pierre product and tell me whether I get 0 OK so now I've got that I want to put a little at a time into this linear independent but only that the two-year Vonnegut the Aggies values and make sure that we get another chance to practice is so I gave it recipe for a marketer repeat the recipe gave it to you last time as time so I wanna do it I want to be a hike in value from a sample views that as waiter review the number 1 just giving a matrix and ask you finally agony values and the eigenvectors physicists like Ghana hiding had a trick during So there's a matrix and the recipe is if I wanna find the hike in values what I'm gonna do it is I'm gonna solve this equation the determination of a minor slammed high school 0 that's what I As sulfur lamp that inning The ACGIH so what is that this is a the determinant of a major And what am I doing here on duty subtracting landed from because this Islam NYTimes I which just that land is in the bag also that's all I'm doing is subtracting 1 minus Lander 2 2 former slander and make regions said that he goes 0 and sulfur lamp Opec 1 I multiply that out but I determinant they get 1 miners land time for my missed land mines will ripple 0 and then multiply that up further Notice earlier for here a minus for it so aminic Atlanta squared minus 4 land mine slander mind 5 land equals So that landed final slammed mind 0 . 0 0 course It was like to argue that as the 1st step get gagging values that for each of those values a movie vector associated filmmaker down the process the 2nd step is to get hike sectors and we do that it's for each Linda salt a minor slammed high school 0 for the vector of Peso so 1 land equals 0 That means 0 here on market is distracting anything from a The dreaded after a minus 0 5 0 that gives me a year
That a marketer subjecting promises beginning at any time he wants to equal 0 0 at my goal is to find V-1 and V-2 the way we do this is weak put it it augmented matrix
Sommelier rid these by multiplying by Wrote to his replaced by negative to rolled 1 cluster of when you do in these Things you don't have to rent out these steps you know what you do when you can get back you can do is go this is crossed that offers you know that's what you're going to do the next day And then you just go right for this says that he wanted us to be 2 equals 0 so he wanted to be The school over here our Arab So let us take stock of what we have so far more interested in that sector we will believe sequel to leave 1 To what we found that We found out that you want this to too 5 gay thanks thanks negative Now backed up to and there's a ragged The Now you can't do this again with another Lambda that landed equals 5 for land equals 5 A minus 5 5 W Equals 0 the but a different sectors they're just so not having the same letters For that same as feared by subtract 511 negative 2 to 1 I'll make W W 2 0 0 this funk quickly for 2 to see have permit circuits theirs What is so when I am at this stage Series C I could get rid of this firms going across not just go right to the says that too W 1 minus W 2 equal 0 W 2 people to W 1 and the veteran interested in W W wash W 2 That's equal to W 1 2 Go to water I want to The issue The What we do it the eigenvectors now Willis's collective question Yet so that's a minus 5 I don't take these days there and subtracting 5 along with that of 1 minus 5 is negative for the former 1st lady would focus on that landed even 0 I guess we vehicle to put here on tho wears is 2 1 a negative In the end effort and 5 had that W. Gould but what Now I said this earlier than expected is Solar absorbers said the way this matrix is designed it's a it's a symmetric matrix and so What's gonna happen is a way for you to check with you get these vectors right these vector should be perpendicular they should be fervent so you can check with got right of their perpendicular fight to the death that In product alienated 2 plus 2 0 so that's like a check to make sure you get the matrix that I give you will make sure that the eigenvectors Perpendicular OK now on make this Q. accused the orthogonal matrix that puts the eigenvectors in there but the 1 thing I have to do is add to divide the likes of over the length of B & W over the length of W so length that he is equal to revive the lake W is also illegal to reach 5 so I get negative to over 2 5 1 over Route 5 1 on Route 5 and make their positive too That's Q and then I am left with the resulting diagonal matrix that comes from future and pose a few we get this right that you don't have to do this multiplication could you know what it is in advance if I put this Eigenvectors first-year than Italian value shows of 1st But put it in the 2nd eigenvectors hits by value shows There's the those of the ACGIH values in the back of the resulting debt Questions about that take those 2 example of their reproduced on you'll quick at this there's not a whole lot that I can do that any different than the 2 to by 2 there's not that many that I can give you the volcanic turnout saying so you practice these 2 Should be ready for the 1 on the using to their daily class this 1 the 1 on Monday gets yet so see these are the bag values It's a diagonal matrix I know these tours 0 but then these 2 values LPI values and specifically effect would be 1st year and a half with 0 1st by switch the order but this 1 here and that there the gets which do not reach economic that
Beobachtungsstudie
Ebene
Eigenwertproblem
Lineares Funktional
Matrizenrechnung
Oval
Prozess <Physik>
Matrizenring
Große Vereinheitlichung
Sterbeziffer
Natürliche Zahl
Orthogonale Funktionen
Kartesische Koordinaten
Wärmeübergang
Vektorraum
Stichprobenfehler
Diagonalform
Multiplikation
Rechter Winkel
Nichtunterscheidbarkeit
Mereologie
Ordnung <Mathematik>
Diagonale <Geometrie>
Leistung <Physik>
Matrizenrechnung
Prozess <Physik>
Natürliche Zahl
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Symmetrische Matrix
Richtung
Ausdruck <Logik>
Faktorzerlegung
Multiplikation
Symmetrie
Eigenwert
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Zusammenhängender Graph
Wurzel <Mathematik>
Leistung <Physik>
Zentrische Streckung
Matrizenring
Mathematik
Streuung
Orthogonale Funktionen
Vektorraum
Rechnen
Fokalpunkt
Teilbarkeit
Diagonalform
Flächeninhalt
Würfel
Tourenplanung
Mereologie
Körper <Physik>
Ordnung <Mathematik>
Matrizenrechnung
Turnier <Mathematik>
Kalkül
Punkt
Momentenproblem
Natürliche Zahl
Klasse <Mathematik>
Rechteck
Auflösung <Mathematik>
Element <Mathematik>
Physikalische Theorie
Weg <Topologie>
Multiplikation
Variable
Exakter Test
Reelle Zahl
Perspektive
Lineare Geometrie
Vorlesung/Konferenz
Leistung <Physik>
Matrizenring
Determinante
Mathematik
Reihe
Biprodukt
Arithmetisches Mittel
Flächeninhalt
Mereologie
Garbentheorie
Diagonale <Geometrie>
Matrizenrechnung
Subtraktion
Punkt
Hausdorff-Dimension
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Ungerichteter Graph
Bilinearform
Euler-Winkel
Term
Eins
Ausdruck <Logik>
Richtung
Variable
Algebraische Struktur
Reelle Zahl
Einheitskreis
Vorlesung/Konferenz
Warteschlange
Leistung <Physik>
Lineares Funktional
Kreisfläche
Graph
Vektorraum
Physikalisches System
Biprodukt
Linearisierung
Arithmetisches Mittel
Rechter Winkel
Offene Menge
Koeffizient
Mereologie
Ellipse
Parabel <Mathematik>
Garbentheorie
Matrizenrechnung
Subtraktion
Punkt
Kalkül
Hausdorff-Dimension
Zahlenbereich
Bilinearform
Euler-Winkel
Eins
Multiplikation
Variable
Zahlensystem
Nichtunterscheidbarkeit
Einheitskreis
Vorlesung/Konferenz
Warteschlange
Beobachtungsstudie
Matrizenring
Kreisfläche
Mathematik
Graph
Frequenz
Arithmetisches Mittel
Differenzkern
Rechter Winkel
Parabel <Mathematik>
Resultante
Matrizenrechnung
Multiplikationsoperator
Sterbeziffer
Extrempunkt
Hausdorff-Dimension
Klasse <Mathematik>
Gruppenoperation
Besprechung/Interview
Zahlenbereich
Gleichungssystem
Bilinearform
Ausdruck <Logik>
Eins
Multiplikation
Variable
Kugel
Vorlesung/Konferenz
Indexberechnung
Leistung <Physik>
Kreisfläche
Mathematik
Frequenz
Fokalpunkt
Rechter Winkel
Koeffizient
Mereologie
Dimension 3
Parabel <Mathematik>
Ordnung <Mathematik>
Matrizenrechnung
Matching <Graphentheorie>
Bilinearform
Biprodukt
Term
Division
Gerichteter Graph
Variable
Unterring
Heegaard-Zerlegung
Mereologie
Vorlesung/Konferenz
Garbentheorie
Indexberechnung
Streuungsdiagramm
Leistung <Physik>
Erweiterung
Matrizenring
Determinante
Natürliche Zahl
Klasse <Mathematik>
Rechteck
Zahlenbereich
Gleichungssystem
Lineare Gleichung
p-Block
Bilinearform
Physikalisches System
Ordnungsreduktion
Ausdruck <Logik>
Variable
Vorlesung/Konferenz
Tropfen
Grundraum
Nichtlinearer Operator
Matrizenrechnung
Matrizenring
Determinante
Wasserdampftafel
Inverse
Zahlenbereich
Gleichungssystem
Störungstheorie
Physikalisches System
Term
Gerichteter Graph
Skalarfeld
Übergang
Multiplikation
Variable
Zahlensystem
Exakter Test
Schwebung
Beweistheorie
Mereologie
Vorlesung/Konferenz
Leistung <Physik>
Matrizenrechnung
Subtraktion
Matrizenring
Determinante
Mathematik
Element <Mathematik>
Ausdruck <Logik>
Zahlensystem
Einheit <Mathematik>
Betrag <Mathematik>
Kompakter Raum
Mereologie
Vorlesung/Konferenz
Faktor <Algebra>
Wärmeausdehnung
Nichtlinearer Operator
Matrizenrechnung
Determinante
Nichtunterscheidbarkeit
Inverse
Zahlenbereich
Vorlesung/Konferenz
Ordnung <Mathematik>
Inzidenzalgebra
Term
Ausdruck <Logik>
Topologie
Matrizenrechnung
Determinante
Flächeninhalt
Vorlesung/Konferenz
Grundraum
Teilbarkeit
Ausdruck <Logik>
Resultante
Länge
Punkt
Klasse <Mathematik>
Reihe
Kartesische Koordinaten
Vektorraum
Drehung
Skalarproduktraum
Pythagoreisches Zahlentripel
Exakter Test
Eigenwert
Reelle Zahl
Konditionszahl
Mereologie
Einheitskreis
Vorlesung/Konferenz
Zusammenhängender Graph
Projektive Ebene
Matrizenrechnung
Prozess <Physik>
Physiker
Determinante
Orthogonale Funktionen
Zahlenbereich
Gleichungssystem
Vektorraum
Biprodukt
Skalarproduktraum
Eins
Flächeninhalt
Eigenwert
Stichprobenumfang
Vorlesung/Konferenz
Matrizenrechnung
Länge
Wasserdampftafel
Gruppenoperation
Klasse <Mathematik>
Reihe
Fortsetzung <Mathematik>
Vektorraum
Biprodukt
Symmetrische Matrix
Diagonalform
Multiplikation
Differenzkern
Eigenwert
Rechter Winkel
Tourenplanung
Vorlesung/Konferenz
Ordnung <Mathematik>
Resultante
Matrizenrechnung
Länge
Subtraktion
Abstimmung <Frequenz>
Gewichtete Summe
Hausdorff-Dimension
Natürliche Zahl
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Bilinearform
Zählen
Eins
Erwartungswert
Multiplikation
Exakter Test
Rangstatistik
Eigenwert
Stichprobenumfang
Minimum
Nichtunterscheidbarkeit
Lineare Geometrie
Vorlesung/Konferenz
Bruchrechnung
Matrizenring
Determinante
Inverse
Vektorraum
Teilbarkeit
Divergente Reihe
Ablöseblase
Ordnung <Mathematik>
Luftreibung
Explosion <Stochastik>
Mechanismus-Design-Theorie
Aggregatzustand
Vorlesung/Konferenz