Birne Binegar: Okay, so first.
Birne Binegar: Logistical issues, let me.
Birne Binegar: see something real quick here.
Birne Binegar: Okay, so.
Birne Binegar: A couple things first of all.
Birne Binegar: I want to do course evaluations today and for me to do that I need a volunteer someone who's.
Birne Binegar: willing to collect the exams be the last one to leave effectively and take not the evaluations, to the math office in math building right next to whitehurst.
Birne Binegar: And is there anyone who traveling back East in it and could drop off Lee okay great.
Birne Binegar: So what i'm gonna do is i'm going to let your, for I guess 20 minutes or so, then i'll stop i'll pass out the evaluations then i'll lead and then, if you could just fill out the evaluations and put it back the envelope and deliver them to the math department that would be great.
Birne Binegar: Okay, so the plan for today is it's kind of minimal, what I want to do this.
Birne Binegar: quickly review what it means to diagnose matrices showing you the algorithm by which we diagnose matrices and.
Birne Binegar: The conditions that definitely guarantee that we can the algorithm can go to completion and we're able to actually diagnose major see i'll do a quick two by two examples So you see how that.
Lisa Fisher: goes.
Birne Binegar: And then not primary purpose of today's lecture is give you an application of.
Birne Binegar: Why would you want to.
Birne Binegar: diagnose the matrix To begin with, and i'll show you that diarization process allows us to solve an interesting family of ordinary differential equations without doing any.
Birne Binegar: differential equations per se will basically just be using linear algebra to solve.
Birne Binegar: differential equations involving functions.
Birne Binegar: Okay, so.
Birne Binegar: So what diarization means is, if you have a square matrix.
Birne Binegar: Say square matrix a we say that it's.
Birne Binegar: gonna get this thing centered here somehow we say the matrix is dying realizable if.
Birne Binegar: there's an auxiliary matrix see that when you do this matrix multiplication of you've given matrix I see you're actually able to convert it into a diagonal matrix and the application i'll show you what the end of lecture today.
Birne Binegar: actually shows you why not being able to die you like the matrix this is useful.
Birne Binegar: But now what we discovered this week is that.
Birne Binegar: A matrix can be diagonal eyes, but there's a condition for that to happen, basically, has to do with how we construct these two matrices what we need to get a matrix.
Birne Binegar: In order to satisfy this identity identity, it turns out that every column vector of C has to be an Eigen factor of the matrix a.
Birne Binegar: But not only do the columns of this matrix see have to be eigenvectors for a it turns out, we have to have enough of them.
Birne Binegar: So that we can actually invert the matrix see so we we actually need and linearly independent eigenvectors one for one linearly independent eigenvector for each column with the matrix see so that see can be inverted so that the matrix a can be diagnosed.
Birne Binegar: So this is the basic criteria, but what I showed you in wednesday's lectures there's some simpler criteria that you can impose Well, first of all.
Birne Binegar: yeah here's the construction, if you have a n by n matrix a and you have.
Birne Binegar: I can vectors a button that you have as many linearly independent eigenvector so I can vectors pointing in distinct directions, as you have columns of The.
Birne Binegar: matrix a, then what you have our end solutions to this eigenvector eigenvalue problem you use the eigenvectors that's called the matrix see.
Birne Binegar: You use the eigenvalues as a diagonal entries of this diagonal matrix D and then this identity is automatic so you will, as long as you have enough, I can vectors you can form this matrix see you'll also be able to inverted and you'll up effectively have diagonal matrix a.
Birne Binegar: Now.
Birne Binegar: there's a simpler way to see maybe earlier on, then actually.
Birne Binegar: Finding all and eigenvectors all and linearly independent eigenvectors of a sometimes you can see whether or not matrix diagonal I visible.
Birne Binegar: by looking at as characteristic polynomial so let me remind you, this characteristic polynomials determine a minus variable Lambda times the identity matrix.
Birne Binegar: You use this polynomial to actually find the um the eigenvalues that matrix and, in fact, what the eigenvalues are are exactly the roots of a.
Birne Binegar: polynomial equation, he subscript a of Lambda is equal to zero this tall know meals only going to equal zero when Lambda is either eat for our one or two or.
Birne Binegar: up the arcade so these numbers are subscript in an index here are exactly your eigenvalues we call these powers here we call these algebraic multiplicity of the corresponding eigenvalue.
Birne Binegar: And one thing we saw last time is that if all these eigenvalues are distinct numbers so meaning the multiplicity, these are all equal to one, so, in fact.
Birne Binegar: Another way of saying this is that r1 through our case actually are one to our end and these numbers are all distinct.
Birne Binegar: Then, then you matrix is going to be diagonal usable because what we saw last time is that if you have.
Birne Binegar: To login vectors and have different eigenvalues they're always linearly independent of one another, if you have n distinct eigenvalues you're going to have in linearly independent eigenvectors and that's what you need to diagnose the matrix.
Birne Binegar: Right another criteria.
Birne Binegar: Is.
Birne Binegar: These powers of Lambda minus and eigenvalue those these are called the algebraic multiplicity that the corresponding like yeah now you.
Birne Binegar: there's another multiplicity associated with an eigenvalue and that's the dimension of the corresponding I can state.
Birne Binegar: The subspace of all vectors in our end but which the vector each vector is an eigenvector if you're given matrix with the given eigenvalue.
Birne Binegar: that's a subspace of our end its dimension is called the geometric multiplicity.
Birne Binegar: And what we saw last time is so long as the algebraic multiplicity is exactly equal to the geometric multiplicity then your matrix is going to be diagonal lines visible.
Birne Binegar: If any one of these numbers is not equal to the corresponding algebraic multiplicity your matrix is not going to be diagnosable.
Birne Binegar: And then have a further simple criterion for dying liability is whenever you have a symmetric matrix whenever the matrix is equal to X transpose than a is always diagnosable.
Birne Binegar: Alright, so these are.
Birne Binegar: Three sort of tests that you can check to see whether or not a given matrix is.
Birne Binegar: diagnosable that you can actually check before you actually compute all the eigenvectors have you given matrix.
Birne Binegar: But let's do a quick example here's a two by two matrix five minus one to two first thing i'll do is we'll see if it's dying unlikable.
Birne Binegar: So what I do.
Birne Binegar: Is i'm going to look at the characteristic polynomial first so here it is when you actually plug in.
Birne Binegar: For that matrix a this becomes the determinant of five minus Lambda negative one to two minus Lambda and then the determinant of that matrix.
Birne Binegar: can be had by taking the difference of the dive mental products, you get this polynomial which simplifies to this and then factor eyes it's like this, and so what you see is you have for this matrix you have two eigenvalues three and for each occurring with algebraic multiplicity one.
Birne Binegar: In particular notice since three is not equal to four, we have two distinct eigenvalues we also have a two by two matrix and on the previous slide.
Birne Binegar: This first condition.
Birne Binegar: Since we have a two by two matrix and all the eigenvalues are distinct that two by two matrix is going to be diagnosable so I haven't even computer the eigenvectors.
Birne Binegar: But I know from results of Wednesday that I will have enough literally independent eigenvectors to athlete diagnosis matrix.
Birne Binegar: So what i'll do next.
Birne Binegar: i've answered this question, it is dying advisable, because it has two distinct eigenvalues but let's go ahead and diagonal ISIS matrix.
Birne Binegar: To do that, I have to solve the eigenvector eigenvalue problem for this matrix I need to find this eigenvalues which i've done now, I have to find the corresponding eigenvectors.
Birne Binegar: So for each eigenvalue i'll calculate the corresponding I can stay, so this would be.
Birne Binegar: substation our to consisting of.
Birne Binegar: Apollo he to its sorry three that's going to be the subspace of our to such that every vector in it is an Eigen factor of a with eigenvalue three that's what it is.
Birne Binegar: But the way we calculate it is we simply subtract from our original matrix the eigenvalue along its main diagonal and then we.
Birne Binegar: We think of this as a coefficient matrix for linear system and we try to find a basis for that solution set well to do that well first we calculate this matrix split was blissfully then.
Birne Binegar: We reduced to a row I fell on form or actually probably want to go to reduce show Echelon form.
Birne Binegar: So in reduce rational on form this would be the north face, I have to make that pivot one, so one minus one half 00.
Birne Binegar: Once I get the matrix in reduce show Echelon form, I go back to equations first one is telling me that X one.
Birne Binegar: minus.
Birne Binegar: One half X two has two equals 07 world just says zero is equal to zero, we look at this matrix the second column doesn't have a pivot That means, this is the free parameter.
Birne Binegar: So I use this equation to express X one in terms of the free parameter X two, then I write down a solution vector X one X two substituting where I can.
Birne Binegar: One half X two X to substituting where I can put the three parameters and then I pull out the common factor.
Birne Binegar: to reveal the basis factor.
Birne Binegar: it's a basis factor for this knowledge base, which is the basis it's basis factor for this Eigen space and so that's our eigenvector.
Birne Binegar: For the eigenvalue landing equals three alright, so what i've found my original matrix was I guess.
Birne Binegar: That on the previous slide.
Birne Binegar: Five minus one.
Birne Binegar: minus one to two what I found.
Birne Binegar: eigenvalues.
Birne Binegar: Then they were three and four and the corresponding eigenvector.
Birne Binegar: In this case it worked out to be the spectre one half one.
Birne Binegar: One half one, but we have another we have another eigenvalue for so i'll do the similar calculation.
Birne Binegar: For the Lambda equals for Eigen space.
Birne Binegar: And that will proceed similarly.
Birne Binegar: We subtract the eigenvalue from the diagonal elements of a and then we try to figure out the know states of that matrix well that's done by reducing to reduce your rational on form converting to equations and then.
Birne Binegar: Big Bang a basis for this solutions that in this case it's going to the equation says that X one minus X two X equal to zero, or simply the X one is equal to X two and back every solution of the homogeneous linear system with this coefficient matrix it's going to be.
Birne Binegar: A scale of multiple this factor so here's your basis vector and in fact you now interpret that as the eigenvector of your original matrix a corresponding to the eigenvalue Lambda equal to four.
Birne Binegar: All right, well, let me put that matrix down as well.
Birne Binegar: So this solves.
Birne Binegar: The eigenvector eigenvalue problem.
Birne Binegar: For a.
Birne Binegar: Once you've done that, and you've verified that you have enough literally independent eigenvectors, as you have the size of the original matrix.
Birne Binegar: Then you can proceed with a diarization and you have all the information you need to write down the matrix see that will die analyze a as well as the matrix D what you do is for each eigenvector.
Birne Binegar: use that to form columns for the matrix see so I can put one half one for that first column corresponds to that eigenvector second column, will be the second I can vector this will be the matrix see that it's going to be, can be used to convert a into a diagonal matrix.
Birne Binegar: that's not going to happen to see the minus first time a time see is going to be diagonal but I don't have to use this formula to compute D.
Birne Binegar: In fact D, I already know what it's gonna be it's gonna be a diagonal matrix it's entries are going to be eigenvalues have a.
Birne Binegar: The only question is this for become before three or just three to come before for well we've already chosen and older and.
Birne Binegar: We chose this to be the first eigenvector so we'll use the first eigenvalue as the first diagonal entry chuck D and then corresponding to the second column well is this, I can vector for so i'll use that as the diagonal entry in the second call and now i've got.
Birne Binegar: Just from this eigenvector eigenvalue calculation i've got the matrix see that converts a to a diamond on matrix and i've also got the diagonal matrix D.
Birne Binegar: Alright, so that's.
Birne Binegar: that's how diagnostician works that's how you are able to convert non diagonal matrices into diagonal matrix and now I want to show you an application of that.
Birne Binegar: Alright, so let's just reviewing we found the two conductors corresponding eigenvalues and we can immediately write down those nature CC and D okay so here's an application that first glance doesn't look like has anything to do with linear algebra suppose I have.
Birne Binegar: Two functions of a variable key and together they satisfy differential equation of this form the derivative of the first function.
Birne Binegar: is proportional to the first function plus ski a multiple the second function, and the second function of basis similar equation it's time to read it, it depends on both X one X two.
Birne Binegar: All right now, these kind of systems are pretty common.
Birne Binegar: You get an equation like this in chemistry when you're looking at reaction rate so maybe the rate at which say you're looking at up chemical reaction of hydrogen and oxygen.
Birne Binegar: The rate at which the concentration of hydrogen is disappearing it's going to be proportional to the concentration of hydrogen but it's also going to be proportional to how much oxygen that are is to burn with it, and so you might have a relationship like this that the way.
Birne Binegar: That.
Birne Binegar: The rate at which hydrogen is consumed is proportional to how much ideas you have, and also how much option to have and, similarly, the rate with the oxygen is concerned.
Birne Binegar: consumed is going to be proportional to how much hydrogen, you have and how much oxygen you have, and you get a system like this.
Birne Binegar: You also get equations like this, when in biology, or more precisely, maybe ecology, or you have predator prey systems, maybe you have a an ecosystem, or you have a bunch of boxes those would be the predators, and you have a bunch of rabbits now.
Birne Binegar: The growth rate of those populations well it's going to be for each species it's going to depend on certain that are natural.
Birne Binegar: birth rate manage death rate that will not the difference between the birth rate and the death rate would give it's not sort of rate of growth that population but because they're predators.
Birne Binegar: The foxes can eat the rabbits and they can contribute to the death rate, so I would expect the coefficient here that's negative because the foxes are going to eat those guys and.
Birne Binegar: The more foxes you have.
Birne Binegar: The more it's going to change the rabbit population in a negative way, so you have without the options there the gross with the rabbit population would just be the comparison between how fast they're reproducing.
Birne Binegar: versus how fast they're they're dying off, naturally, but with the foxes in the picture you have an additional term that impacts, the how the rabbit population grows and, similarly, the fox population grows it's going to depend on its natural.
Birne Binegar: Its natural birth minus death rate, but if there's a big food supply, you can actually increase fox population, and so you get a pair of coupled differential equations like this.
Birne Binegar: Another example in physics, if you have a couple oscillators.
Birne Binegar: You can write those down as this system of linear equations and they'll be of this form and I guess the last example.
Birne Binegar: yeah you would also see this, and I can move this.
Here it goes.
Birne Binegar: In electrical engineering, if you have simple passive circuits, with an inductor a resistor and a capacitor and you put those in the closed loop, there are certain circuit equations basically.
Birne Binegar: Look at how the voltage changes as you transverse the circuit and what you get a differential equations that basically have this form.
Birne Binegar: So the upshot of this is that these sort of systems of differential equations occur, a lot in many applications and what i'm going to do is show you how you can use just linear algebra to solve them.
Birne Binegar: So what we do, first of all, is we make the differential equation look like more familiar linear out algebraic objects, so we take the two functions of tea.
Birne Binegar: And, rather than thinking them as individual functions, we think of them as components of a vector value function function of tea that has two components.
Birne Binegar: And then, what i'll do is i'll grab those coefficients here and it's kind of natural this kind of looks like an a matrix times a vector with two components that both terms do in fact.
Birne Binegar: The original differential equation component by component can be written as a matrix differential equation.
Birne Binegar: This column vector variables once I differentiate it the derivative is going to go right through the brackets and attack each component individually and then, if i'm when i'm matrix multiply.
Birne Binegar: This column vector by this matrix what i'm going to get is exactly.
Birne Binegar: A one one X one plus a one two X two a two one X one plus a two two X two and, on this side we have.
Birne Binegar: DDT X one and DDT.
Birne Binegar: X two and so component by component i'm reproducing the original pair of differential equations but now I got it written as a single sort of matrix differential equation.
Birne Binegar: Right so that's The first step is I.
Birne Binegar: made my system, a differential equation I transported it somewhat into the context as linear algebra but now it's still kind of mix it's both the differential equation and there's matrix multiplication involved as well.
Birne Binegar: Alright, so.
Birne Binegar: you're transported the problem into a linear algebraic setting now what i'm gonna do is i'm going to start applying some of our linear algebra techniques.
Birne Binegar: So let's start with a simple case where the the matrix a.
Birne Binegar: that's being specified here is actually diagonal.
Birne Binegar: In which case what's going to happen.
Birne Binegar: These off diagonal entries are going to be equal to zero, so, in fact.
Birne Binegar: When I write down the equation.
Birne Binegar: DDT.
Birne Binegar: excess of vector is equal to a times X T and this matrix is diagonal what happens is on the right hand side of the Lambda one times X.
Birne Binegar: that's right it Lambda 100 Lambda to what i'm writing it's a times X one X two what's going to happen is you're going to get net first component Lambda times X one plus zero where you're just going to get this equation, and this.
Birne Binegar: The second differential equation is going to come from.
Birne Binegar: Taking the derivative of X two and then looking at the second component of this product, a times X and that's going to be zero times X one plus land, the two times X to.
Birne Binegar: Go inside one the matrix is diagonal is both.
Birne Binegar: The differential equations effectively decouple the first differential equation depends only on X one, and the second differential equation depend only on X two, since they decoupled I can solve them independently.
Birne Binegar: But these differential equations are really simple they're the equations of exponential functions were saying is the rate of change of X one is proportional to X one itself and the solutions to those are well now.
Birne Binegar: Any solution of this differential equation is going to be an exponential function, where this is exactly.
Birne Binegar: The constant that appears multiplies the variable inside the exponential function.
Birne Binegar: Let me I can just simply verified the satisfy the differential equation, if you take DDT see one he to land, the one T the differentiation is going to ignore the constant.
Birne Binegar: And then, when it hits this exponential function will bring down that factor of Lambda and then reproduce the same differential option and now what you see is that the derivative of your solution is exactly equal to that.
Birne Binegar: Lambda one types of solutions so that's Lambda one times see one ease Lambda one T so.
Birne Binegar: yeah so effectively.
Birne Binegar: This function see what he Lambda one key always satisfied to differential equation, and similarly for.
Birne Binegar: The second differential equation its general solution is just going to be a constant times the exponential function with the corresponding.
Birne Binegar: Factor Lambda Lambda to.
Birne Binegar: A, by the way, I should point something out, if you remember when i'm talking about general linear algebra.
Birne Binegar: I said that functions can be thought of vectors and that differential operators can be thought of as linear transformations or equivalent Lee matrices in some sense.
Birne Binegar: And notice what this is saying is that that linear transformation corresponding differentiation this.
Birne Binegar: function X one thought isn't vector is actually an eigenvector of that linear transformation because it's derivative.
Birne Binegar: At this is linear transformation apply to the vector is scale or multiple of that vector so there's some linear algebra even hidden in the formulation of these two differential equations.
Birne Binegar: But that's not.
Birne Binegar: The.
Birne Binegar: point of this, though, rather, what I want to do is, I want to solve the more complicated.
Birne Binegar: linear systems systems of differential equations where this is a matrix that's not die.
Birne Binegar: And i'll show you.
Birne Binegar: How the process of diarization.
Birne Binegar: allows us to solve differential equation dx dt is equal this matrix times X T how it allows us to solve it directly.
Birne Binegar: Alright, so.
Birne Binegar: let's now consider the case when a is not diagonal but at least you can diagonal is it.
Birne Binegar: So, in the case of the two by two matrix what i'm going to assume is, I have to linearly independent eigenvectors for the matrix a.
Birne Binegar: And I have to corresponding eigenvalues so that when I take the eigenvectors use them as columns I get a matrix see when I take the eigenvalues like use them as a diagonal entries of a diet little matrix D, then this.
Birne Binegar: This diagonal ization of the coefficient matrix is achieved.
Birne Binegar: All right now, of course, the find these vectors V1 and V2 and indeed to find the eigenvalues Lambda one and Lambda to you would do that explicitly, you would have to solve the eigenvector eigenvalue problem for a.
Birne Binegar: What i'm assuming here just for speed is I already solved the eigenvector eigenvalue problem for a I found two linearly independent eigenvectors to eigenvalues and now using that data i've constructed.
Birne Binegar: The matrix see that converts a to a diagonal matrix and i've got the diagonal matrix itself.
Birne Binegar: Alright So what do I do with that knowledge.
Birne Binegar: well.
Birne Binegar: First of all, i'm gonna be original.
Birne Binegar: The original differential equation that i'm trying to solve is D amp T T is equal to a exit T where this is a two component vector.
Birne Binegar: And this is what i'm.
Birne Binegar: The goal is to solve that equation what i'm going to show you is that the well we've already seen the diagonal matrix these.
Birne Binegar: There are associated differential equations are easy to solve because they're diagonal.
Birne Binegar: When I write down the similar differential equation, using the diagonal ization of a random a itself it's readily salt, all I have to know is what the diagonal entries are and then soon as I know what those dangle entries are i've got the corresponding general solution to this.
Birne Binegar: System of differential equations where the matrix is diagonal.
Birne Binegar: So.
Birne Binegar: So that's This is my goal, and what I have is not jewelry result.
Birne Binegar: I said, if.
Birne Binegar: Lambda.
Birne Binegar: Lambda two are.
Birne Binegar: The eigenvalues.
Birne Binegar: A there's an associated.
Birne Binegar: You form the matrix D, the diagnosis of a by using those eigenvalues diagonal entries and there's an associate differential equation.
Birne Binegar: di.
Birne Binegar: et is equal to D.
Birne Binegar: y of T and because.
Birne Binegar: In this situation, the matrix defining differential equation this diagram is possible, we know that the solution to this.
Birne Binegar: is just going to be some constant eat At first I can value tea and some constant need to Lambda to T, because in a diagonal case the differential equations.
Birne Binegar: The couple and effectively be why one is going to equal to Lambda times, why one which is going to tell you that why one.
Birne Binegar: A T has to be a constant time he Lambda one T and the second component is its derivatives that equal to Lambda two times y te that's telling me that my second solution is some other constant times eat him to to T.
Birne Binegar: Like till the diagonal systems are easy to solve, you can write down the solution as soon as you know what those diagonal entries are.
Birne Binegar: The more complicated situation I haven't yet presented the solution but i'm about to what i'm gonna do is i'm going to solve the diagonal case first.
Birne Binegar: And then.
Birne Binegar: What i'm gonna do is i'm going to take this solution to the diagonal problem and i'm going to multiply it by that matrix see the matrix see is constructed well D was constructed from the eigenvalues and the matrix see when we dive analyze the matrix a is going to consist of eigenvectors.
Birne Binegar: I conductors of the matrix a this first eigenvectors going to have eigenvalue Lambda one second I can vectors can have eigenvalue land, the two, so it makes your C C and D are coming from the diagonal ization of a.
Birne Binegar: And what what I want to show you is that, if I simply set X equal to that matrix time the easy solution to the easy differential equation what's going to happen, yes, well, first of all of us to my proposed solution and for xrp.
Birne Binegar: This matrix is going to be a constant matrix it only consists of constant eigenvectors so the differentiation respect to time will pass right through that guy he will.
Birne Binegar: treat it as effectively as a constant and then I have dydx it, but you will it T is should be equal to that diagonal matrix times y.
Birne Binegar: So I substitute in the left hand side or the right hand side of this differential equation, and now what i'll do is, let me write this a C.
Birne Binegar: d times, why what i'm gonna do is i'm going to insert the identity matrix in between the and why, but the way i'm going to insert it is i'm going to write this identity matrix and see the miners first times see.
Birne Binegar: The reason for doing that is that see times, why that's what i'm calling X and see the see the miners first.
Birne Binegar: well.
Birne Binegar: See the minus first times a time see is D, the matrices.
Birne Binegar: C and D were constructed precisely feel that the diagonal is a and from this identity, if I multiply.
Birne Binegar: From the left by see and from the right by see the minus first.
Birne Binegar: I got that right.
Birne Binegar: yeah from this identity.
Birne Binegar: By multiply from the right by see I get see be a SCI.
Birne Binegar: fi multiply this equation from the left hand side bites from the right bye see ya so the sea is going to hit the C minus first and i'll just get a factor of one and on the other side, see who's going to hit D and then I can get rid of this guy by.
Birne Binegar: multiplying from this side by see the minus first.
Birne Binegar: And I had seen the minus first there now get a is equal to see the see the minus first.
Birne Binegar: Alright, so what i'm trying to show you is that this matrix here.
Birne Binegar: can be a dinner yes D is equal to see the minus first AC that equation can be solved for a and what it tells you is that a is equal to see time D time see the minus first, so this is a, and this is X.
Birne Binegar: And so what happened here i've constructed a solution of the original differential equation using my dying licensing matrix and my diagonal matrix D.
Birne Binegar: This this.
Birne Binegar: This differential equation, the diagonal one is readily solved all you have to do is use the diagonal entries to create the exponential functions that will be solutions for each component of the solution.
Birne Binegar: vector function.
Birne Binegar: But then what I can do is I can use the corresponding diagonal I think matrix see to convert this solution of a different differential equation.
Birne Binegar: This differential equation, I can use that matrix see to take the solutions of this which were easily found, and I can use it to find.
Birne Binegar: A solution of the original differential equation, the X dt is a non diagonal matrix times X itself, so that this function here what I get by.
Birne Binegar: multiplying this particular solution that the the couple differential equation when I multiply it by this matrix I actually get a solution of the more general highly coupled system of differential equations.
Birne Binegar: And in fact yeah shows your solution will work out to be that, and in fact this is a general procedure that works, not just for two by two matrices.
Birne Binegar: But it works anytime you have a large system well potentially large anytime you have a system of linear differential equations.
Birne Binegar: where you can have multiple functions, whose derivatives are a possible in terms of your unknown functions anytime you have a system of couple ordinary differential equation, this form, you can solve it by first.
Birne Binegar: diagonal icing the coefficient matrix that's implicit in this in this form.
Birne Binegar: You diagonal is that matrix and then you solve the corresponding decoupled system which you can do component by component.
Birne Binegar: All you'll need to get the solution to the decoupled system is you just need those eigenvalues and use those to create exponential solutions.
Birne Binegar: put them in a better and then the corresponding eigenvectors those are you going to be used as well, what they do is they use those to take the.
Birne Binegar: solution that the diagonal system and convert it into a solution of the original system and that's done simply by doing matrix multiplication so in.
Birne Binegar: More precisely, you take the coefficient matrix and you diagonal is it here you're going to get the eigenvalues that you'll use to create a deacon a fully decoupled system that is readily salt.
Birne Binegar: you'll use the diagnostic matrix see to create this auxiliary matrix see the utility of this in the context of differential equations.
Birne Binegar: it's going to convert the solutions of the easy differential equation into the solution of the more difficult differential equation that you're trying to solve.
Birne Binegar: All you have to do is use the eigenvalues to construct the solution of the diet be coupled system, and then you multiply by the diagnostic matrix to get the solution of the non diagonal.
Birne Binegar: Non highly coupled system of ordinary differential equations okay so actually talk 10 minutes longer than I hoped i'm going to stop here today, I want you to.
Birne Binegar: fill out the.
Birne Binegar: survey is instructions for this course, when you.
Birne Binegar: Put the course information up on the blackboard and said once you fill these out if you could.
Birne Binegar: insert your responses into that yellow envelope.
Birne Binegar: And what you'll be done today.
Birne Binegar: Okay, let me pass out and i'll respond to your question.
Birne Binegar: directly in front of the elevators on the fourth floor it just crossed over and then there'll be a staff person, you can pass the amplitude yes alright thanks so much Okay, so in your question, yes, so.
If.
Birne Binegar: Does this mean that the solution to any system linear system of differential equations is.
Birne Binegar: A linear combination basically of exponential.
Birne Binegar: yeah well yeah yeah that's what's gonna it's ah yeah it's a linear combination of exponential functions, but what you don't know.
Birne Binegar: yeah cuz that's all that's going to do when this when this solution gets hit by the this constant matrix.
Birne Binegar: is just going to get linear combinations of these entries, so that they are going to be linear combinations of exponential functions, but they're going to get end up getting mixed up in general, so a solution for the X one component might involve both of these X X one X two.
Birne Binegar: that's right that's right, so let me, let me step back the one tab yet.
Birne Binegar: It basically you have differential equations are determined by this coefficient matrix.
Birne Binegar: What makes this work is these entries here don't depend on the on the parameter T and that's that's why, when I.
Birne Binegar: To show that actually satisfied differential equation, I have to move that derivative through this matrix see so I don't want so.
Birne Binegar: yeah it should be constant but as long as their content and that that work.
Maybe be easier.
Birne Binegar: It was this example example with.
Birne Binegar: Right so maybe what I did is like I took that eigenvector with one half and I multiplied it by two just to get rid of that too and.
Birne Binegar: I was just worried, I was just asking.
Birne Binegar: So we said that as soon as we found that the eigenvalues we said there's two eigenvalues is a two by two matrix therefore design document right.
Birne Binegar: yeah so like what you need or two linearly independent eigenvectors eigenvectors that point in different directions and because I have to eigenvectors.
Birne Binegar: That we first talked to you I in values yeah and we found two is enough to conclude that.
Birne Binegar: right if I only found one, then, then I wouldn't be able to direct line but luckily I had to eigenvalues in each eigenvalue has its own eigenvector So my question is.
Birne Binegar: And then we find that we found that I can vectors and we found that they're not a trivial like the no space is not a trivial solution.
Birne Binegar: Is that.
Birne Binegar: That works out because as because we know like we expect that for a fact or that just like yeah so that's that that condition that the determinant of a minus Lambda is equal to zero death guaranteeing that your solution set that no space it's, not a single point it's a it's a it has.
Birne Binegar: subspace that happened because these two rows cancelled out But what if they didn't cancel out well then, you made a mistake, see what you're doing here is um.
Birne Binegar: If.
Birne Binegar: A minus Lambda I.
Birne Binegar: Times X, so this is basically.
Birne Binegar: Having this banish on this telling you that X is an argument vector of eigenvalue Lambda now what you want is a non zero solution of this.
Birne Binegar: And so, if you only have one solution, then you're going to have a unique solution and that's going to mean that if you only have one solution that would have to be the zero vector because you're like you're satisfied that would mean that.
Birne Binegar: A times a non zero vector produces zero it means the inverse of this matrix yeah so.
Birne Binegar: If the only solution is the zero vector That means that this matrix Ada minus first exists.
Birne Binegar: Or it means that the.
Birne Binegar: It yes, well, it means that the determinant of this matrix does not equal to zero.
Birne Binegar: But because you specify you because you found Lambda by demanding that this determinant is equal to zero, then you know that you have to have solutions other than the zero vector so it's got to be at least one dimensional so you're going to have at least one basis factor.
Birne Binegar: So, so when you're when you found the eigenvalues and now you're looking for the eigenvectors what should always happen is that when you start reducing it matrix you get a zero with the bottom.
Birne Binegar: Because he got an A non zero at the bottom, then you would be able to.
Birne Binegar: You would be able to you would have a unique solution.
Birne Binegar: You would be able to invert the matrix and get a unique solution, but the unique solution if it exists it's always got to include the zero vector so when that determinant is not equal to zero.
Birne Binegar: that's when you only get the zero solution when the determinant of a minus Lambda is equal zero you have to have more than one solution.
Birne Binegar: And that's going to mean that that coefficient matrix when you reduce it to one form cannot be the identity matrix.
Birne Binegar: And, if not the identity matrix which gotta have a zero row it at the bottom, yes, because you're you're.
Birne Binegar: filling out the attention that you go 11111 you get to that last row if it's a one there, then it reduces to the identity matrix if there's a zero in in that last row then it's not reducible to the identity matrix.
Birne Binegar: yeah so there's there's lots of moving years but they all connect together.