 Welcome to a new module on linear systems and of first order differential equations and it is a particularly it is qualitative analysis. So far in our all earlier modules the topics we have completed the topics which are basically covered in a university syllabus. So now onwards the two of the main modules are one of the stability analysis basically what we call it qualitative analysis and qualitative analysis of linear system first we will do and then in another module we will learn the qualitative analysis of the non-linear system. So, you can view this module is a precursor to the module on non-linear systems and stability analysis the trajectory behavior around an equilibrium point etcetera. So, that is the main aim of this module it is not just the representation of the solutions which we can still obtain as a solo using the classical theory which we have already completed we can use the existence uniqueness theory of the system of the things which we have studied in the previous module you can apply in a similar fashion here, but the more important aspects about the study of the behavior nature of the equilibrium points behavior of the trajectories because the solutions of the trajectories of ODE systems are solutions. So, you can think it as a motion of certain particles. So, I want to understand that. So, first we will do it in this module about the linear systems and later in another module about the non-linear systems. So, you will also see the power of linear algebra and its diagonalization the Jordan decomposition is usefulness in analyzing these equations. So, what we have done in one of our earlier modules we have studied the first and second order linear systems. So, we can also consider the nth order linear system and what we will see quickly now every nth order linear system is a special case of general nth order system. So, how does a general nth order linear system. So, let us look at the nth order linear system linear equations. So, let me give before that is a linear system a title to this module linear systems and qualitative analysis. So, that is a title of this particular topic. So, how does a general nth order linear equation will look like a regular linear equation will have the following form. You will have the nth equation which a coefficient 1 d power 5 by d t power n plus let me use the correct one p 1 t into d power by n minus 1 y by d t power n minus 1 plus if you go here you will get the final term p n of t of y is equal to some g t. And if you are looking for an initial value problem you can define as you have seen will be seen rather for n equal to 2 for the second order equation not only the initial value problems you can also define the boundary value problem. And you will see few lectures on boundary value problems of the second order equations in a different module. So, for an initial value problem. So, you need n conditions say for example, y at 0 is equal to y naught y 1 y 1 at 0 means y prime at 0 is equal to y 1 etcetera up to y n minus 1 of 0 is equal to say n conditions you need it. So, that is a initial value problem if g t equal to 0 this called a homogeneous g t equal to 0 then it is a homogeneous linear nth order equation homogeneous linear nth order equation. What I am saying is that such a linear nth order equation can be converted into a first order system how do you do that 1 you put x 1 is equal to y 1 x 2 is equal to y 1 not y 1 x 1 is equal to y x 2 is equal to y prime that is nothing but x 2 is equal to y prime x 3 is equal to y second derivative etcetera up to x n minus 1 you will define to be x n minus x n minus 1 to be y n minus 2. And your last term x n is equal to the n minus 1th derivative that is equal to d power n minus 1 y by d t power n minus 1. So, if you use this one and we can using that equation here. So, if you look at that x 1 dot is equal to y dot. So, the derivative of x 1 the derivative of x 1 here is nothing but y 1. So, you can write down that way. So, if you look at that one x 1 dot is equal to that d x 1 by d t exactly will be y 1 I am using two notations dot in one both are for the derivatives. So, you can write down this is y 1 that is equal to x 2. So, if you write x 2 dot will be x 3. So, if you go like that you will get x n minus 1 is equal to dot x n minus 1 dot is equal to x n and x n dot has to be recovered from your equation x 1 at that if you look at here this is what your that will be if you take this to the right hand side you can write everything as let me write down that you will get minus p n into x 1 minus p n minus 1 into x 2 if you go like that the last part will be p 1 into x n and then you will also have your g t here. So, this is from the equation this first n minus 1 from the definition the last one will be from the equation. So, if you put this together here and if you define your x t now is equal to x 1 etcetera x n all evaluated at t and you can write down your equation if you combine all these we will write your equation as x dot of t is equal to you can write your matrix a t into your x t you see what is a t where a t is the matrix is a special form of the matrix you have a special 0 1 0 0 then 0 0 1 like that 0 0 the last one here here you will have minus p n minus p n minus 1 etcetera up to p 1 this is correct. So, if you multiply this one a t into x t a t into x t. So, you will have this kind of system. So, that gives you a linear system of equation with a t in this special form see a t is of this one this is your a t you have your a t like that. So, but this allows us to consider any general a t you do not have to consider that way. So, general linear system. So, thus a general system can be written as initial value problem in this form x dot of t is equal to a t x t a t need not be in the above form it can be a a t will consist the entries of a t will consist of functions need not be the form given by the earlier thing x at 0 or you can put it at any point x at 0 is equal to x naught. So, this is a general linear system of course, in general we are not going to concentrate on system such as is this system when a depends on t a plus some g t will also come a non-linear term where you can have some non homogenous this gives you a non homogenous when a t depends on t is called non-autonomous is called non-autonomous. So, if a t is equal to a independent of t independent of t it is called an autonomous system it is called an autonomous system as you see even in our first order and second order equation for example, even when we are studying second order equation when the second order coefficients are independent of the functions you have seen the advantages and able to solve the equations completely. So, there is a larger advantage and many other even the qualitative analysis you see the difference when you study non-linear analysis part of this problem. So, for autonomous system is much more easier to study than non-autonomous system. So, most of these lectures though in this module we will concentrate only on the autonomous systems and we try to represent the solutions and the stability analysis of that system. So, our major aim if you look at case n equal to 1 that means a is equal to a then the solution is you know already then the solution is x t is equal to e power t a into x naught you see. So, our aim immediately we will see that we want to have we want to represent the solution for the system if possible of the form when it is a for an autonomous system. So, our what is our autonomous system autonomous system is x dot of t is equal to a of x t where a is independent of t that is more important thing and you will study the initial problem at any initial value is x naught which is also a vector. So, this system we are always going to refer it to as 1 the important point here is that system 1 is a is independent of t. So, the whole aim is to represent the solution aim represent the solution x t is of the form some exponential form exponential here is what are we have exponential form here what we want to recall the definition of an exponential of a matrix which we have studied already in the basic part of the module. So, if you have a matrix a if a is a matrix then e power a can be defined to be i plus a plus a square by 2 factorial plus etcetera where the series converges that is what a series converges in the set of all linear operators R n to R n. So, in this case given a matrix one can view it as a linear operator from the space R n to the space R n and this is also identified with a set of all n by n matrices that is what we are trying to say in this case. So, on this set of all linear operator from R n to R n we have the corresponding itself vector space which has a topology under that you will talk about the convergence of this infinite series and this e power a. So, note e power a is also a matrix note e power a is also a matrix that is the important part of it. So, you can define e power a in particular we can also define for any t belongs to R e power t a that is nothing, but identity again it is symmetric plus t a plus t square a square by 2 factorial and so on. It is again an infinite series and the series converges uniformly in this case converges uniformly because there is a t parameter. So, again e power t a is a particular is a matrix here which you want to do that. So, here is an exercise I want to give you an start with an exercise now which you can do it easily define x t is equal to e power t minus t naught of a x naught. Note that e power t minus t naught into a is a matrix. So, that matrix will act on the vector x naught and x t is defined that. So, the exercise is that show that x t is differentiable in t differentiable and x t satisfies one x t satisfies one the system one system one what is a system one this is a system one. So, this is the system one. So, you just compute x dot of t you can see that x dot of t is nothing, but a of x t and definitely x at t naught is easy to see that x at t naught is when t substituted for t naught the first term here will be identity. So, x at t naught will be x naught. So, that way we have the existence. So, this proves this proves this gives existence that is what I am saying. So, you have proved the existence directly by construction of course, as I said the existence and uniqueness can also be proved using the general existence uniqueness theory. But, so the existence which we have already have it. So, and we want to give you a uniqueness a quick uniqueness proof you can get the uniqueness using the general theory, but I want to give you a general easy thing. So, assume y be another solution y equal to y t be another solution. Then what you have to there to prove that you have to prove that y t is equal to x t that is what you have x t is defined already what is x t x t is nothing, but e power t minus t naught of a into x naught that is what you have to prove it. So, proving this is equivalent to that I can take this out e power minus of t minus t naught into a x y t I will take that to the left side is equal to x naught. So, what you want to prove is that this is what you have to prove it. So, this you have to prove it that means left side is a function of t right is just x naught which is a constant. So, you want to show that the left side is a constant and that constant is nothing, but x naught. So, define in that case just the left hand side of z t you define z t is equal to my e power minus t minus t naught of a y t which I can define what I have to prove is that z is a constant in t and that constant is nothing, but x naught. So, to prove that constant here is an exercise again for you these are all simple exercises show that d z by d t is equal to 0 that is an easy exercise when show that that is a d z by d t is equal to 0 then this exercise will give you z is equal to t is a constant, but what is that constant constant, but that constant has to be z at t naught z t is a constant already proved. So, it has to have the same value as t naught, but z at t naught by the definition here put t equal to t naught here. So, you will get this is nothing, but y at t naught, but what is given to you y is a solution to your initial value problem and y at t naught is nothing, but your x naught. So, you have your uniqueness in a quick way without appealing to the general theory. So, we have proved the existence and the uniqueness existence by construction and uniqueness by direct application the linear system the autonomous linear system as a unique solution. So, we may wonder what more to be done here the issue is that show already proved the existence and uniqueness again if you go back to our second order linear equations just obtaining solutions are not enough just like here we have the representation of the solution x t in the form we have the representation of the solution x t in the form e power t minus t naught a x naught you have that one. So, the two of the difficult issues two remarks two difficulties here one difficulty the computation of exponential is very difficult computation though you have a representation but the computation is of exponential of a matrix is difficult exponential of a matrix is difficult in general that is one point. The second point is that as I remarked again x t you have to think it is a dynamical system is a solution to a dynamical system in other words x t you can think it as a motion of a particle. So, the important issue is that how this particles behaves especially near an equilibrium point which we are going to explain to you and you will also learn about the equilibrium points later in the non-linear theory. So, what we are more interested is that the behavior of x t especially near an equilibrium point especially in particular the stability analysis of the trajectory. So, this formula does not reveal anything about x t. So, the second difficulty is that this formula does not do not reveal about the behavior of the trajectories about the behavior of trajectories and equilibrium points which you are going to study trajectories. So, you see that is a whole thing. So, one is especially the computation of this thing and second one is the trajectory behavior and the whole module on this one in the non-linear theory is to understand this trajectory how does it behave what are the equilibrium points how the whether you have the stability near an equilibrium these are the questions from physics engineering point of view. But then there are some matrices where you can have a computation e c example if you want to see. Example suppose a is diagonal suppose a is diagonal that means I write a diagonal matrix in this form suppose it is a diagonal matrix. So, diagonal matrix I will read only the diagonal entries lambda 1 etcetera lambda n this is nothing but what I mean by that is nothing but lambda 1 lambda 2 only along this diagonal lambda n all the other elements are 0. So, I want you to give a it is a it is an exercise from again linear algebra which you would have done already. I will also give a feeling what happens to the trajectories of the solutions with this a as the matrix the thing is that you can immediately compute e power a e power a will be again diagonal. But then entries with e power lambda 1. So, this is a small exercise which you can immediately do it what you have to do is that you have to compute a square a cube etcetera. If you compute a square you will get the diagonal lambda 1 square lambda 2 square etcetera lambda n square and for any a power n it will be diagonal of lambda 1 power n etcetera lambda n power n. So, and then add it and you get the matrix e power a immediately. But what does this mean when a is what does this mean when a is diagonal as far as our linear system is concerned. If a is diagonal what does that diagonal system means to us diagonal then it just says that x 1 dot is equal to lambda 1 x 1 x 2 dot is equal to lambda 2 x 2. If you rewrite the system n x n dot is equal to lambda n x n u c. So, it is a decoupled system that is no connection between there is no interaction coupling between two variables. Any of the variables do not interact with other variable and each one is a simple equation x 1 dot is equal to lambda 1 x 1 and since x 2 etcetera x n are not coming into picture this immediately this is a decoupled system this is what it says decoupled. So, there is no problem in mainly this decoupling system you will give you the solution x 1 t is equal to e power lambda 1 t into the initial value at the first component that is all. If you go this way you get your x n t is equal to e power lambda n of t the n th component of your initial value. So, if you write this in the form. So, this is equivalent to saying that if you write this e power x t is of course, you have to change here if you if I look my initial conditions at t naught is equal to x naught. So, I have to change accordingly I will change here accordingly this will be e power t minus t naught into lambda 1. So, I will change here this is what I wrote was for the initial value of the origin into lambda n x naught this is just a multiplication. So, if I write the solutions you have your x t is nothing, but your diagonal of e power t minus t naught lambda 1 with n raise this one that is equal to e power t minus t naught of lambda n this is a matrix this is your matrix operated on x naught. So, that is nothing, but your e t minus t naught a into x naught you see. So, you have your solution and you have your complete solution if a is a diagonal matrix and another interesting property with this which will be useful for our analysis we are eventually want to do more analysis. So, if the given matrix is diagonal the corresponding ODE is a decoupled system since it is a decoupled system is solved in each equation each n of the equations independently because it is a first order equation in one variable each one can be solved separately and then you can write down that solution. So, as long as the matrix is diagonal you have no problem of solving it another property which I would like to recall here the property is that in general for exponential map e power a plus b naught equal to e power a into e power b. So, this property of real numbers is not true for general matrices, but if a will be commute a b commute that is a b equal to b a then e power a plus b this will be useful in our analysis later e power a into e power b this you can verify if a and b commute you can prove that result. So, this exponential one of the fundamental property of an exponential function in one variable is not true for exponential functions of the matrix exponentials of the matrices, but if the there is a commutation and that is not contradicting for real numbers this is always true a b is equal to b a for real numbers and that is the another important thing. So, I will recall today some more interesting things which is necessary for here another property which we want to call a property. Suppose b is of the form p inverse of a p such matrices are called similar matrices as we know from linear algebra linear similar matrices represent the same linear transformation it is only according to a different basis you will have different matrices, but it will represent essentially the same linear transformation. Suppose b is equal to p inverse a p then what is e power b I want to know. So, this is a simple result again I will leave it as a small exercise for you if you have not done these things in the linear algebra this may be the correct time to do that one this will be p inverse of e power a p you see. So, you have a nice representation you can do this other way also. So, this is equal to this implies of course, a is equal to p b p inverse and that is also implies e power a. So, you can write down because p is an invertible matrix p e power b p inverse this is one important property which now we are going to use it under this linear transformation this helps. So, you have a linear system x dot equal to a x, but then this allows you to solve for a solving for x dot equal to a x that means namely the system x dot equal to a x can be obtained by solving another system corresponding to b. So, the system one can be solved system one let me recall system one system one means a x equal to x dot equal to a x with x at t naught is equal to x naught this is my system one this is my system one can be solved by solving wide system two namely corresponding to b. So, I introduce a system b equal to y is y at t naught equal to something whatever it is y at t naught is equal to some y naught which we will solve how do you achieve this one. So, recall. So, what is one one gives x dot is equal to a x, but what is x is nothing but p b p inverse p b p inverse of x. So, now put yeah. So, this if you call it here you will get a p inverse of x dot is equal to b p inverse of x not that p is a invertible matrix with constant coefficients. So, p is independent of t. So, you can take inside x. So, you put y is equal to p inverse of x that implies y dot is equal to b y. So, you see and what is y at t naught y at t naught is nothing but p inverse at x at t naught that is x naught and this is your y this you call it to be y naught. So, you can solve your system. So, if you have two matrices are the solution of the system corresponding to one matrix can be obtained by solving the system corresponding to another equivalent. So, such two matrices are called the linearly equivalent. So, this leads to the concept of the leads to the concept of linear equivalence concept of linear equivalence. So, you have a definition for this one two systems the systems one the system one that is x dot equal to a x is said to be linearly equivalent to said to be linearly equivalent to a system to the system two that if there exists an invertible matrix invertible matrix p such that b is equal to p inverse of a p. That is the point. So, whenever two systems are linearly equivalent you can go from one system to another system. What is so important about it? The important thing which you are going to see soon in next lecture or coming lectures one important property the one important thing is that under linear equivalence the nature of the equilibrium points or the stability of the trajectories do not change. For example, if a particular point is equilibrium point there will be a corresponding equilibrium point to the other system and this equilibrium point is stable you will see what are the kind of stability and stability are available soon and then the other point will also be thing and the behavior of how the trajectories will even nature of the behavior will also remain the same. So, the nature the first important property right now you may not know it we as we have not defined any sort of any the concept of stability, but the important point which you are going to see the nature for example, stability or instability of equilibrium points will tell what is equilibrium points equilibrium points do not change do not change under the equilibrium point. Do not change under linear equivalence. So, what is this equilibrium points? So, recall so what is equilibrium points? You will see again as I said more on non-linear systems about it correct definitions again equilibrium points. Equilibrium points are actually solutions of the linear system equilibrium points of x dot equal to A x. So, equilibrium points so though we call it point it is a equilibrium it is a solutions steady state solution. Suppose A x equal to 0 suppose x b such that. So, let me call calculate up not to confuse with initial values x naught b such that A x naught is equal to 0 that means it satisfies the high tensile vanishes at that point. This implies if I define x t is equal to the constant x naught constant will be a solution. That means if you have an equilibrium point and if at t naught if the solution is at that point that the solution will remain at that point all the time it will not move from there. So, that is why it is a steady state solution. So, whenever you have an equilibrium point and the solution starting from there it will not change that. So, for example, for linear system x equal x naught equal to 0 x naught is always an equilibrium point always an equilibrium point. And when A is invertible and this is the case when we are going to consider when A is invertible 0 is the only equilibrium point because if A x naught equal to 0 if A is invertible x naught will be 0. So, if A is invertible 0 is the only equilibrium point. You will see more examples in the next class I want to set a kind of thing therefore, if you start a solution at the origin at time t naught the solution will remain there and here is where your concept of stability and the gravestone. In applications you can never start from a particular point you will always going to make error. So, the question is that if you start a solution at the origin at an equilibrium point then the solution will remain there. Suppose you make an error and the solution is not starting at the origin, but then a solution is starting say at a point initial point x naught which is close to the origin. So, the question is that whether trajectory will deviate from there very far away causing instability because this is important in the point of view of applications you will never be able to start at a particular point you are always going to make error. And this kind of concepts also you have to know that when a solution starting near an equilibrium point will remain there will go to the equilibrium point where it will move away from that leading to various definitions various types of stability when especially when x t is a vector it will have different components certain components will go to the origin certain components will have the stability certain components will not have the stability and all that can happen. And that is what basically in the qualitative analysis we going to see that. The second point so this is about the equilibrium point. So, we are going to initially we will give few examples even when a is not invertible and then you can see the various equilibrium point and how the solution behave. Our more interest is that when a is invertible in that case we have only one equilibrium point near 0 and we are going to study the stability analysis near the point near the equilibrium point. This is the first point second point of interest. The second point is about the change of variable. So, basically we are making a change if you go back to our earlier analysis the previous analysis. See if you go back to this kind of previous transformation. So, you see this is the transformation we made to convert the system x dot equal to a x to a system equal to y. So, there are two linear transformations basically. So, you have the transformation. So, you have transformations x going to y is equal to p x that is one transformation and the other one y is equal to p x that y going to x is equal to p which one is the correct one both are fine y is equal to p inverse of x y equal to p inverse of x does not matter you can replace that x equal to p y are linear transformations are this is important these are linear transformation not some arbitrary transformations why it is linear because a p is a matrix because it is a linear transformation you can view this as a that is a important one can view it as a coordinate change view it as a coordinate change. So, basically when you are considering coordinate change. So, when you have a some coordinates you will be basically looking at two coordinates change either x going to y or y going to x. So, you can see that if this is y 1 y 2 you will have a coordinate change x 1 x 2. So, effectively when we are considering under linear equivalence x dot equal to a x and y dot equal to p y we are making a coordinate change under this coordinate change the when you for example, if I is invertible and x naught is an equilibrium point and you will have a y naught is equal to p inverse of x naught will be an equilibrium point that are the only equilibrium point you can get that equilibrium point in the stability system x naught is stable which we will see y naught will also be stable that is what I say it does not change there may be a difference in movement of the trajectory for the stability analysis you cannot convert a stable equilibrium point to an unstable equilibrium point under linear equivalence. So, the fundamentally is a is a coordinate change that is what under this linear transformation one has to understand and that is not surprising because this linear equivalence is between two similar matrices as you see that the similar matrices represent the same linear transformations hence you should be able to do the same thing. So, what is our aim now with these are general remarks about these are basically what is our aim we should not forget about aim what are we trying to do this one see what we have seen is that you are interested in solving a linear system x dot equal to x and that you have a representation you have a solution x at t naught is equal to say x naught you have a linear system and what we have so far discussed is that if there is a linear equivalence linear equivalence that means you have a b is equal to p inverse of a p and you have the corresponding system y dot equal to b y and y at t naught is equal to some p inverse of x naught that is your y dot. And here the solution as I said that the one of the major difficulty is the computation of the solution even though you have a solution x t is equal to let me recall again once again because this is the important point x naught is equal to a into x naught the computation of this is difficult suppose you are able to transform this under linear equivalence you look for a linear transformation. So, that the computation of e power t minus t naught into b into y naught this is called y naught. So, this may be easy this may be difficult may be difficult probably easy you see this is a situation we are looking at it. So, the question is that is it possible to find a linear equivalence. So, that the solution to y power dot is equal to b y is easy to compute when I say that solution to y dot equal to b y is easy to compute this is easily computable. And we have seen one situation where this can be computed easily when b is diagonalizable. So, this leads the question of diagonalizability of a matrix. So, that is where we are looking at it. So, that is where we are looking. So, the so far we are discussed this is a linear equivalence. So, that we are looking for a matrix b corresponding to a using the linear equivalence. So, that b is diagonal you see suppose b is diagonal that is a question suppose b is diagonal lambda 1 etcetera lambda a. So, then what is solution of y t the solution y t is immediately you can write it y t is equal to the diagonal of that one. So, diagonal e power t minus t naught lambda 1 etcetera e power t minus t naught lambda n of this is a matrix acting at y naught what is y naught p inverse of x naught. So, this is what your y t and what is your x t x t is nothing, but p of y t that is what our transformation. So, you have your solution. So, you have your solution immediately p diagonal e power t minus t naught lambda 1 etcetera e power t minus t naught lambda n of p inverse of x naught. So, you see you have your complete solution here you can write down if you can question. So, that is why this is the question of diagonalizability immediately. So, the diagonalizability the importance of you would have seen the diagonalizability importance not only here when you are studying the linear algebra whether if you can solve your linear system if the matrix is diagonal because there again is a decoupling system exactly what you are showing that. So, the question is that given a matrix given a matrix a thus there exist an invertible matrix invertible matrix p such that b is equal to p inverse of a p is diagonal. If so our problem is solved our computational problem is solved because now the computation of e power b is easy because b is diagonal, but unfortunately in general the matrix need not be diagonal the matrix need not be diagonal. Example even when n equal to 2 that is the diagonal need not be diagonalizable need not be diagonalizable that is what I said not diagonal when n equal to 2 if such a matrix exist is called diagonalizable. Example you take lambda 1 0 lambda is not diagonalizable how do you prove that what is the kind of diagonalizable. You see even such a matrix simple simple looking matrix simple looking matrix is not diagonalizable. So, again let me recall from linear algebra diagonalizability is equivalent to the existence of n independent Eigen vectors. So, remark so recall diagonalizability is equivalent to the equivalent to the diagonalizability is to the existence of n independent Eigen vectors. This you are already seen in linear algebra we also recalled some of these things in our basics independent Eigen vectors. So, here is an exercise for you for the problem for the above matrix for this matrix for show that it has only one Eigen vector show that it has only one Eigen vector one independent Eigen vector independent Eigen vector you see that. So, when you are studying the linear system our main aim is that given a matrix can we have a diagonalizability property. The moment your given matrix is diagonalizable that means you can find a matrix B. So, that is a corresponding B you are completely the system is solved if the system is not diagonalizable what can we do about it. This is where the linear algebra and the Jordan decomposition comes into help and we will use that to understand the how much we can do it if the matrix is not diagonalizable. What we will do in the next lecture we will do a complete analysis in a 2 by 2 system because 2 by 2 system the possibility what are all the possibilities is easy to classify. Even in higher dimension the Jordan decomposition tells you the all the best possible way you can do it and but we will spend a more detailed study in 2 by 2 systems and in the process we introduce the concept of phase plane and phase portrait and then the various stability. So, we will have a complete analysis in the next lecture. Thank you.