So, before we resume our usual journey about this eigenvalues and eigenvectors, I remember a couple of lectures back one of you asked me if the idea of eigenvalues and eigenvectors is predicated on the notion of determinants alone and I had clearly answered that well you know determinants is a very useful tool for establishing that you have these eigenvalues and if you have them how do you solve them. But of course I thought maybe I should give you a more crisp understanding of why eigenvalues will exist irrespective of whether you can manipulate some determinants and evaluate them there from just for a very fundamental understanding. So, I just quickly give you a sketch of why you must have an eigenvalue and eigenvector for an operator over a finite dimensional vector space ok. I know this is also a bit of a digression because, but I think we it is useful just to remember this. So, suppose you have V and you have this so again I will not distinguish between the matrix that captures the operator phi and so I will just use A. So, I will use this interchangeably for a matrix as well as for an operator over an n dimensional vector space. So, it is an abuse of notation I agree, but we will stick with that ok and as and when from the context we will say that yeah this is a matrix is an operator. So, for the moment you can treat this as either it really matters not, but let us just look at look at V A V A squared V and by the way V is F D V S with the dimension of V being N ok. So, we will take a look at this all the way until A to the N V. So, this is a set of vectors how many vectors are there right. So, what can you say about this set of vectors first thing that strikes you very fundamental concept of linear algebra sorry they cannot be linearly independent because the dimension is N and the cardinality of this set is N plus 1 vectors right N plus 1 yeah. So, let us call this set S. So, S must be linearly dependent field ok. Further we will suppose V is over F and the field that we have in mind is algebraically closed. I have described what an algebraically closed field is if you if you cook up a polynomial each of those coefficients are members of that field right then you will have roots of such a polynomial in that field itself right. So, if you are dealing with an algebraically closed field you will end up having all your N roots if it is an nth degree polynomial you will have all your N roots in that field. So, for example, if you are looking at S squared plus 1 is equal to 0 and if you think of this as a polynomial over the real field then it has no roots right. If you think that this comes from RS for instance then no roots, but if the same object is now viewed as some object in CS then of course it has roots yeah. So, in this case we have in mind a field which is algebraically closed like the complex field right. So, you will have all your N roots if you have an nth degree polynomial in that same field itself. What is the one thing that you can then say first from here there exist c 0 c 1 c 2 until c n not all 0 such that c 0 v plus c 1 a v plus c 2 a squared v plus dot dot dot until c n a to the n v is equal to 0 right. So, you might wonder why we digressed into that polynomial just hold on a second now consider p of x given by c 0 plus c 1 x plus c 2 x squared plus dot dot dot until c n x to the n belonging to f x. If it is algebraically closed I can factorize it into first order terms you see what I mean for example, if I wanted to write s squared plus 1 I cannot do. So, I cannot factorize this over the real field while maintaining that each of the first order factors are also from the r r x r s right, but if I am allowed to do this over complex field I can write this as s plus i into s minus i. So, anytime I am dealing with an algebraically closed field I can write this as some c n times the product of x minus lambda i i going from 1 through n. Because of the algebraic closed nature of the field where each of these individual objects. So, the coefficient 1 and the coefficient constant coefficient minus lambda i also members of that field these are first order polynomials. So, any nth degree polynomial I can split up into n first degree polynomials over an algebraically closed field such that each constituent also comes from that same f x same ring of polynomials as it were right. If this is true what can I do about this think about it can I not write this as c 0 i plus c 1 a plus c 2 a squared plus dot dot dot until c n a to the n acting on v is equal to 0 nothing fancy. So, far yesterday rewriting of that same expression as it for, but at this point I would make use of this fact about polynomials and I would write this is nothing but the same as saying c n times the product i going from 1 through n a minus lambda i i this whole object acting on v is equal to 0 c n is there in the factorization because I have to take a linear combination of these fellows know. So, yeah you might say that c n is any nonzero number and therefore I can divide by it and make it 1 if you are not taking c n as nonzero then that means you are taking a span of n linearly independent vectors which have no guarantee of being linearly dependent right they can be linearly independent n vectors in an n dimensional vector space can be linearly independent may not be, but it can also be the moment you have at least n plus 1 like so. So, of course this one has to be included to guarantee that you have at n plus 1 you are not ruling out the possibility. So, you could have divided by c n that is true in fact at this stage also you can remove the c n by arguing analogously that c n must be nonzero, but what is in fact we are going to do that if we proceed with this. The point that is so does it answer your question as to why c n is because this c n has got nothing to do with the characteristic polynomial do not confuse it with that. We are not claiming this as some characteristic polynomial or anything we are just saying we started with an arbitrary vector v and we kept acting on it using a n times until we had n plus 1 vectors and since we have n plus 1 vectors sitting in an n dimensional vector space they cannot be a linearly independent set since they cannot be a linearly independent set there must be a non trivial combination linear combination of those vectors which was vanish identically that is all that is there apart from that the paraphernalia we have built around this is this theory of polynomials not really theory very straight forward, but some result on polynomials on algebraically closed fields right. Now we just want to use that idea of polynomials not just over x, but over matrix polynomials and we are going to say of course there is nothing that prevents us from doing this no no no this polynomial is a different object this is completely a different object, but the point is that the way you can factorize this you can also factorize this because a commutes with itself and with any polynomial of thereof you cannot just do this with any arbitrary multiple like if you had a 1 and a 2 if you had x and y. So, if you had a polynomial in multiple variables you would be very careful the order of operations matters. So, x y squared may be equal to y squared x, but two matrices a 1 and a 2 you cannot always say a 1 a 2 squared is the same as a 2 squared a 1 particularly with multiple variables polynomials and multiple variables those things do not hold, but when you are dealing with single variable polynomials a commutes with any power of a. So, whether it is a matrix whether it is a linear operator that is represented by matrix the idea is still the same the point is that this follows exactly from this you can also think of it as an operator where by this squaring and this cubing and nth power is this basically composition of that operator multiple times it matters not actually. So, matrices is very elegant because it is just multiplication composition of matrices of matrix operations is just multiplication matrix multiplications right. So, this is true now if this is true what can you say then this object must have a non trivial kernel is it not that is what this equation is saying that it is v of course, you have to say that this v is not equal to 0 that is a given if you choose v is equal to 0 that is the point in composing that set yeah then that will be complete nonsense we do not want to do that. So, of course, that v is not equal to this. So, you have a non-zero vector such that this entire thing acting on that non-zero vector causes it to vanish. So, therefore, this v definitely belongs to the kernel and it is non-zero therefore, it has a non trivial kernel what can you then say about it is it possible that all these individual constituent blocks are non-singular and yet you have this situation is it possible because think about the way we write this we write this as a minus lambda 1 i a minus lambda 2 i until a minus lambda n i right. If each of them happen to be non-singular then compositions of non-singular operators can it have a non-trivial kernel. So, at least some fellow here must be defective in this whole product at least one of these fellows we do not know which one yet at least one of these fellows must be defective right. What does that mean? So, one of these has a non-trivial. One of these definitely has a if all of them so, assume the contrapositive that none of them have a non-trivial kernel. So, means they are all invertible. So, you just keep hitting them with their individual inverses and you will have to the conclusion that v is equal to 0 which is an absurdity. So, at least one of them must fail to have an inverse which means it has a non-trivial kernel and if any one of these fellows has a non-trivial kernel we have actually established the existence of an eigenvalue and correspondingly an eigenvector no determinants right. So, existence of an eigenvalue and eigenvector is not predicated on this construction of determinants. This is also another way of seeing this from the very basics from the very basic understanding of a dimension of a finite dimensional vector space and that is why finite dimension is very important. As I said if you are dealing with infinite dimensional vector spaces and operators therein no one can guarantee the existence of eigenvalues right. So, when you have algebraically closed field on which you have built up a finite dimensional vector space then the dimension of the vector space being n you will always have n eigenvalues. Of course, to see that the number is n it helps greatly to formulate it in terms of the determinant. So, we will not keep the determinant as a paraya or as something that we will not touch at all but we will not dig deeper into the intricacies of determinants because that will be again whole new you know set of results that we will have to derive. We will use it only as and when required but this just serves to illustrate that eigenvalues and eigenvectors will exist independent of your understanding of determinants. This is a proof of the you know in some sense the existence of eigenvalues and eigenvectors of finite dimensional vector spaces right. So, it is a it is a slight digression I agree but I think it was it was pretty instructive to see that this is also true ok. So, we were actually seeking this question answer to this question as to whether a particular operator can be diagonalized subject to a clever choice of basis and we already saw that if you had all your eigenvalues distinct then the set of eigenvectors provided exactly one such basis just sort of bringing you back maybe couple of lectures back yeah to jog your memory a bit or maybe you can just flip through the pages of your notebook and you can just figure out that is the result we discussed right. If the eigenvalues are distinct then the corresponding eigenvectors have to be linearly independent. So, if you have all your eigenvalues distinct then all your n eigenvectors are going to be linearly independent and therefore, that forms a basis for the ambient space n dimensional vector space V and subject to that choice of basis that operator turned out to be exactly the diagonal operator to the diagonal matrix completely decoupled right which is a form that we love and I will just again briefly talk about an application of that nice diagonal form assuming that we are able to diagonalize this is something we will be able to do what is it. So, we do have objects like e raised to the power a or sin of a or even say cosine of a and such funny looking objects where such matrices often applications they come from n cross n and you might think what is this these are not objects that exist in and of themselves we actually have to define these objects how do we define them we define them analogous to their scalar representatives through series. So, you know the exponential series is 1 plus x plus x squared by factorial 2 plus so on the infinite series. So, this is not by definition this you can derive using the Taylor series for any continuous function you know you have a Taylor series on the other hand this e to the a has to be defined this is very useful when you are solving for differential equations the matrix exponential allows you this is so called transition matrix. So, given that you know the state vectors remember that phase voltage we had drawn given that you know the state vectors at any time instant t naught what is going to be the value of the state at some instant t greater than t naught it turns out that it is just e to the a t minus t naught hitting x naught that gives you so x t naught maybe I should write clearly gives you x of t. So, this serves as like some transition matrix knowing the value of the states that some instant t naught you can get the value okay. So, the solution of the equation x dot is equal to a x with x t naught is equal to say x naught yeah then you can just replace this with x naught here yeah. So, this is the unique solution of such a differential equation alright where of course a belongs to r n cross n. So, I am not going into the details of that if you do a course on control theory basic undergraduate level course on state space methods you will encounter this every now and then alright. So, this is of course x we have given you some idea about this. So, x belongs to an n tuple. So, the ability to evaluate this e to the a is pivotal because if you have evaluated e to the a just t is just a scalar. So, what is this e to the a this object it is defined very analogous to a plus a squared by factorial 2 plus dot or dot until you have y factorial r and this goes add infinity. It is an infinite. So, that is how it is defined very important you cannot derive this okay you cannot derive this like the Taylor series expansion in the scalar case this is defined in this manner. And once you have defined it in this manner it can be shown that this is the solution this is the unique solution to a linear differential equation like so and initial value problem like so alright. So, it is very useful not just this you can encounter these sort of functions from different applications. So, functions of matrices are very useful is what I am trying to get at, but how do you evaluate this I mean forget about this this is like some n tuple numbers and their matrices are being squared and all sorts of crazy things are happening here how do you even begin to see the pattern you see this is an n squared entry points n squared data points in each matrix each individual term you have to see and you have to guess oh well this first 1 1 term looks like it is some cos 5 x plus sin 3 x plus something something some funny function.  So, you cannot get a closed form expression in general through this at best you can give it to a fast computer and the computer can approximate and tell you oh it looks like this 1 1th entry or the 2 2th or 2 3rd entry looks like this function, but there is hardly any way that given any arbitrary structure of an A matrix you will be able to guess this. On the other hand if I give you a diagonal matrix yeah if I give you a diagonal matrix then what happens the game completely changes it is a game changer right now it is almost like we are back in the domain of scalars there is no coupling it is just the diagonal entries so each individual entry yeah it is easy to see right is it not. So, now you can just look at this each individual entry. So, if I know that D is a matrix like lambda 1 lambda 2 until lambda n what do you think is e to the d going to be like e to the d is going to be what just e to the lambda 1 e to the lambda 2 e to the lambda n beautiful is it not point is how do I get from here to here and what is the relation there of is there some relation there is lots that is at stake here that we have not clarified yet. So, here is the deal since I have already convinced you that e to the a is not just a unique function it is useful for our particular application of dynamical systems of course, but let us say there is any function f of a has a or rather is such that f of x is analytic. If it is analytic maybe I should just call it f of z then if it is analytic then it has a series expansion that is all that it guarantees and if it has a series expansion then I might go ahead and define this as that particular series in terms of the matrix I mean this is the scalar. So, this is guaranteed to have a series expansion because it is analytic if it is an analytic function then that same thing I invoke on the matrix A and I say that this also has a series expansion. Now, suppose a is diagonalizable by our experience with 2 by 2 matrices 2, 3 lectures back maybe what have we seen what do we mean by a is diagonalizable that means p inverse a t is this that is and this happens to be a diagonal matrix with lambda 1, lambda 2 until lambda n may not all be distinct. Then what can we say about the series? Well here is the deal each term in the series is some power of a is it not. So, a to the power r is what is equal to t the diagonal matrix t inverse t the diagonal matrix t inverse like this till t the diagonal matrix t inverse r times right part the funny thing is that I can merge these together and what am I then left with this is nothing but just one t and in between is sitting this diagonal matrix times r which is very easy to compute provided someone tells me what the diagonalization is and this is t inverse.  So, what is the short of all of this? This is difficult to evaluate your degree this is in general difficult to evaluate right because this does not have any structure in and of itself. But if this length itself to diagonalizability or diagonalization then this object is easy to evaluate because this is just like evaluating n series simple types of series right if it is a cos series then these are just cos lambda 1 cos lambda 2 so on if it is a sin series it is just the sin lambda 1 sin lambda 2 so on till sitting in the diagonals. Once you have that then it is just hitting them with a t and a t inverse on the left and right respectively and you have this evaluation. So, any continuous function or any sorry any analytic function whose matrix x you want to evaluate matrix function you want to evaluate through an analogous series expansion right if that matrix lends itself to diagonalization our job is considerably simplified in theory of course how you are going to find these lambdas what are the vectors that constitute t we will not get into that computational aspect neither in general for this course nor you know for a motivation towards I mean we shall not even aspire towards those things as in counting the flops and stuff okay that is not the goal of this course right. But at least in principle you would agree that any matrix A of arbitrary structure provided it is diagonalizable and e to the A just happens to be one such important function which is useful in solving differential equations for us right. So, this is definitely one way to evaluate the matrix exponential and who knows the matrix cosine the matrix sin any function essentially that has a series expansion right. So, I am just putting it out here for you so that you understand that whatever we are doing here there is a strong motivation behind diagonalization yeah and if you are not able to diagonalize we will at least try to massage it to a form closest to diagonalizability which is the so-called block diagonalizability but all that in good time okay. So, with this motivation out of the way let us now try and get a handle on some more crisp conditions on when such diagonalization is possible up until now we are assuming it is diagonalizable and we are able to do this but we would like to characterize by looking at a matrix and certain features thereof as to when it is diagonalizable and when it is not what are the telltale signs of diagonalizability or more importantly of lack thereof in order to do that we will define two very important properties of matrix one is the geometric and algebraic multiplicities okay. So, again we will not go into details of determinants but at least you know that if we are given a matrix A so suppose it is a mapping from V to itself again think of it as a matrix or an operator I do not care at this moment it has a characteristic polynomial right.  Now this one obviously again if the field is algebraically closed actually I might as well drop this you know at this point and just say I am dealing with the complex field which by default is algebraically closed so I will not have to write this again and again. So, whenever I write f now you can just treat it as you know the same object can be viewed as a real matrix or a complex matrix just go ahead and think of it as a complex matrix real field is a sub field of the complex field after all right. So, maybe from next time I will not write this you just assume that it is a complex field which means that just a while back we have done this we can write this as product of X minus lambda i to the di i going from 1 through k with lambda 1, lambda 2, lambda r oh sorry k lambda k being distinct and summation di i going from 1 through k is equal to n agreed. Yeah it is an nth degree polynomial some of those eigenvalues can be repeated let us say they are repeated the ith eigenvalues repeated di number of times the algebraic multiplicity is simply this it comes from the algebra nothing to do with vector spaces just hit it with the determinant operation get the polynomial you just did algebra. So, it is the algebraic multiplicity so di is the algebraic multiplicity okay of lambda i that is what this is right does a definition of the algebraic multiplicity of an eigenvalue right of course the sum of the algebraic multiplicities must be equal to n. The next object is of course although I have written it first I am now going to define it consider w i is equal to kernel of a minus lambda i i okay a minus lambda i i then the dimension of w i is equal to l i and this is in turn defined as the geometric multiplicity of lambda i. Is there any reason to suppose that the geometric multiplicity will be equal to the algebraic multiplicity I mean we have already seen an example of a matrix where we fail to diagonalize it again I will just reiterate that example maybe 2 1 0 2 yeah. So, if you take of course 2 is an eigenvalue we evaluated it so take a minus 2 i that is just going to be 0 1 0 0. So, kernel of a minus 2 i is equal to what the span of what is it going to be what vector I mean 1 0 2 0 i mean it says it is something in the span right it is a one dimensional vector space yeah. So, it is just and so this dimension is 1 whereas if you look at chi of a that is going to be x minus 2 whole squared. So, a m is equal to 2 and g m is equal to 1. So, obviously and that is exactly the problem as we shall see later with diagonalizability if you had this equality between geometric multiplicity and algebraic multiplicity you would have been able to diagonalize it that is actually the result that we will have in some time if we have time today that is. But this is the point see so this is a clear cut identification of the algebraic multiplicity and this is an identification of the geometric multiplicity yeah this is spanned by a single vector so its dimension is of course 1 agreed no doubts about this. So, you have seen what the definition is you have seen through an example how to get a handle over this the algebraic and geometric multiplicity. Now after all we are looking for a basis you see a suitable basis. So, we wonder if a choice of basis has any impact on the algebraic and geometric multiplicities. So, for example, and that would be very funny I mean it would be very uncomfortable if it did. So, our first guess would be it does not but if it does not then we should have to prove it why it would be uncomfortable is because then you know it depends on the basis. So, something sort of falls apart you choose a basis for the vector space your friend chooses another basis and they are capturing the same operator in terms of two different basis and they are coming up with oh this in under this basis it is going to have the same algebraic and geometric multiplicity under this basis it does not that would be very disconcerting and we would then not know how to even handle this problem, but clearly we do know. So, you might already guess that it would be basis independent, but we need to prove that right. So, that is going to be our next investigation that suppose you have an A under some basis and your friend has another A a matrix A under some other basis. So, what is the relation between these two matrices long time back again these are we are harkening back to the days of your when we spoke about basis transformations and all. So, if your friend comes up with a matrix representation of an operator after all the operator is the same your friend comes up with a matrix representation which is A you come up with a matrix representation that is A bar. So, what is the relation between A and A bar do you remember what captured the basis transformation is not it something like a T inverse A T a similarity transformation right you remember that the diagram we had drawn that lattice we had drawn that we go from this to this whether you go via the lower path or the upper path or via the direct transformation in terms of phi it matters not and then you get rid of the middle path that is phi and say that this is the path through A this is the path through A bar. So, how do you relate A and A bar it is basically padding through those matrices T and T inverse right. So, we are going to study how if at all such change of basis affects this algebraic and geometric multiplicities. So, suppose phi which is a mapping from V onto itself has two representations under different sets of ordered basis given by A and A bar. So, that A bar is equal to T inverse A T for some non singular T correct. See if you object to this if you object to this I would request you to go back to many lectures back when we spoke about the effect of basis transformation on the operators. First we studied how basis change affects points in a vector space that is objects in a vector space it is just T V V changes to T V under some basis change yeah like T captures this dictionary you remember we had spoken about it quite a few lectures back plenty of lectures back actually, but anyway this is all going to be very useful now. So, now your friend chooses A bar you choose A this basis change T connects the two right. So, your friend comes up with this and it is given by sorry sorry x which is determinant of again we will use some little tricks of determinants, but I hope that you will not object to it. So, this is x minus sorry rather x i minus A and you choose this as your determinant we have to check if these two characteristic polynomials turn out to be the same if they do then of course the algebraic multiplicities would turn out to be the same for every root. So, we have to show that these two are equal where your friend chooses determinant x i minus A bar yeah which is nothing, but determinant I am going to write it in a funny manner, but you have to bear with me T inverse T minus T inverse A T which in turn is going to be determinant up until this point we will not be using any tricks of determinants in the next step we will x minus sorry x i minus A times T. So, far so good I wanted to pull out the T inverse from the left and the T from the right which is why I just replace the identity with T inverse T okay. Now I am going to invoke a little trick here it is basically a generalization or rather a special case of a general result I will just name drop that result though I hate doing it it is called the Cauchy-Binet theorem and it has a very wide implication in the sense that when you want to evaluate the determinants of matrices that come up as a result of multiplication of rectangular matrices of course determinant only exists for square matrices, but suppose you have a square matrix that has come up as a result of multiplication of rectangular matrices even then you can do something using the Cauchy-Binet theorem for square matrices it devolves to the general case that determinant of A into determinant of B where A and B are both square is equal to determinant A into determinant B that is the only result I am going to use here without proof. So, applying it I can say this is determinant of T inverse times determinant of this times determinant of T right so that is determinant of T inverse times determinant. So, this determinant is what it is nothing but the characteristic polynomial of A times determinant T this is the only step I have used without proving. So, the general theorem is the general theorem is Cauchy-Binet theorem ok, but of course you do not need to go as far as Cauchy-Binet theorem to prove it for the square case for the square case you can just try it out and you will see it does work out all right, but the general theorem if you are interested you can I mean absolutely not part of this course ok absolutely not, but if you are interested you can just read up it is a very elegant way of representing multiplications of rectangular matrices and then seeing how the determinant resulting determinant can be elegantly represented yeah it turns out you choose certain special sub matrices and the sub matrix that you choose from the premultiplied is the same sub matrix you choose from the post multiplier and just combine them pretty much like some inner product in some sense and you choose all possible sub matrices of the requisite size yeah if you read it it is very interesting result, but anyway so I will just erase it because this is not something definitely we are not going to cover in this course ah so this is ok right I mean if you believe me just try take two square matrices multiply them first and then take the determinants or you know take the determinants first and multiply those two scalars it is the same thing now if that is true then of course the next step I can combine this T inverse and T within the determinant back again that is just determinant of identity so this is going to be nothing but determinant rather this determinant and determinant these are inverses of one another in the scalar field so this is nothing but chi A x I started with chi A bar x so indeed the algebraic multiplicity does not change under the similarity transformation yeah so am is invariant to choice of basis that is the deal this is clear ok now this is going to take slightly more work little more work than this to show that indeed the geometric multiplicity does not alter either because you see for geometric multiplicity you cannot take them all at one go you have to take for individual eigenvectors so consider wi is equal to kernel of A minus lambda I I know how can they change you ended up with the same characteristic polynomial so the same characteristic polynomial will have identical roots right so eigenvalues obviously cannot change so and neither can they multiple because the whole characteristic polynomial remains invariant and the algebraic multiplicity is determined solely from the characteristic polynomial so it does not change now for this we have to show we will be interested in the degree of wi all right so you have chosen A and your friend has chosen wi bar which is kernel of A bar minus lambda I I which is kernel of T inverse A T minus lambda I I right I will have to check so the question is is this equal to wi bar rather where this so this is the question at least the question it is very important half the job is often done if you have posed the question right so you agree that this is the question if we answer this in the affirmative it would mean that under the basis transformation under a change of basis the geometric multiplicity doesn't change here the question has been posed right that is important to check so suppose v1 v2 till vk is a basis so here's the proof is a basis for wi the claim would be that T inverse v1 T inverse v2 T inverse vk is a basis for wi bar now the question has been changed a bit if we can prove this claim will be done right because this entails that the dimension of wi is k and if I am now able to prove the claim that it would show that for wi having dimension k wi bar k sorry wi bar must also have dimension k so what do I need to show that fact that this is a spanning set or a generating set for wi bar and this is this is linearly independent so now the entire proof rests on my ability to show the two things that this set that I have written down in the claim here is linearly independent and that it spans or generates the entirety of wi bar all right okay so let me erase the question now because we can continue writing here so suppose summation alpha i T inverse vi i going from one through k is equal to 0 this implies I can pull out the T inverse because of the linearity of the operation and then I have summation alpha i vi is equal to 0 of course T inverse is invertible so this must have a trivial kernel yeah so this means that summation alpha i vi is equal to 0 but look the vi is already linearly independent by my very consideration here right so this is true then I must have alpha i is equal to 0 for all i since v1 v2 vk are linearly independent right so therefore this set is linearly independent indeed but now I have to show that this is a spanning set yeah for what for wi bar so consider v bar belonging to wi bar then what what does vi satisfy it means that a minus sorry a bar minus lambda ii acting on v bar results in 0 right so what is a bar T inverse at T inverse at V bar or rather let us keep it that way minus T inverse T acting on V bar is equal to 0 so what can I then say can I not get rid of this T inverse again from the left because T is after all non-singular so I can act on this on both sides with T without any trouble whatsoever so this essentially means at minus T yeah at minus T into V bar is equal to 0 yeah sorry I forgot the lambda yeah lambda i thank you thank you yeah yeah what does that mean what do you think I can write this now as this is T here so let us pull this T outside shall we and we have a minus lambda ii acting on TV bar is equal to 0 what does TV bar come from this means TV bar belongs to wi not wi but wi right therefore by my very choice of basis TV bar can be written as some summation beta ii going from 1 through k vi's right because after all this is a basis for wi remember this is a basis for wi the moment I have that TV TV bar belongs to wi it can be represented as a linear combination of the fellows in the basis of wi which basically means that V bar is now going to be equal to summation beta i T inverse vi which belongs to the span of TV i I am writing in shorthand notation you have a want of space but essentially what I have shown is that this is a spanning set I have already shown it is a linearly independent set now it is a spanning set for the kernel of wi bar so of course the overall object has changed the kernel of I mean sorry the subspace wi is vastly different from the subspace wi bar but the dimension of wi continues to remain the same as the dimension of wi bar so the kernel of a minus lambda ii is quite different from the kernel of a bar minus lambda ii of course we have seen that the eigenvectors themselves change in fact these are just nothing but eigenvectors you will see right so therefore the subspaces themselves have changed but what remains invariant is the geometric multiplicity is invariant under choices of under different choices of basis of basis so both the algebraic and geometric multiplicities they do not depend on what basis you or your friend have chosen they will remain invariant and this is great this is what will allow us in the next lecture to concretize the result and state upfront what is the equivalence of diagonalizability in terms of these algebraic and geometric multiplicities okay thank you


What's up guys, this is Theron and tonight we're taking another dive into the para-watch wiki Tonight's thread was started by user Yero William on June 21st 2015. It's called whitetail Let's begin There's a place in the forest by my town where the deer don't die Time and time again hunters talk about it eyes full of darting half-lidded suspicion a Stretch of roadside south the town around the Appalachians, but not quite in them where the hunters don't go You'll hear stories about it if you listen long enough Rambling tales of deer that keep on living when shot through the heart the head splattered on the windshield of a car shrugging it off and darting back into the forest Trappers never seem to catch their quarry there and the number of times I've seen a car viewer off the road in the last 20 years Faced with something that didn't seem to catch the headlights quite right. I can't count on one hand. I Never believed it though Particularly rough patch of forest where the animals are healthier than the weak Stick-legged things that walk along wherever humans have gone by Then black ice on the road reflecting starlight to go where it shouldn't They'll blame it on the Indians, but these folk blame them for all sorts of things never actually go and look That's where my uncle differed He was a savvy hunter and a taxidermist too Never really believed in those things His garage was full of trophies some of them half completed and whenever someone said he couldn't do something He took it as a challenge He caught one of the stories and one day made it a goal of his to get one of those deer and bring back Its head in the bed of his pickup We drove there one night and his old beat-up of a dodged Dakota. I was the spotter in the back Hit his drink told him where to stop It felt strange something by a roadside and most people just drive past Out of staters usually took this route eyes on the road in front of them On the ground below me were pebbles of all sizes Tire worn dusted with the black coating of rubber that you see on these older stretches of road Forest was thick but didn't look unusual to my eyes There was a smell though Barely noticeable through the doors of the car.   I Didn't know what to look for. I could carry a gun, but I couldn't track Soon though clear as day. I saw them My uncle's flashlight darted across the forest floor Illuminating depressions in the soft earth cut deep enough to lay a finger in pulled with water from the recent rain They leaned along and skinny close together, but far from front to hind and something had dragged along in the dirt to the side of them Leaving a wavy trail The spore was wrong in a way. I couldn't pinpoint Uncle identified it though Hoops weren't splayed the deer was lighter on its feet than any this size ought to be It moved quickly to given the density of the underbrush Starving he said it was probably too sick to move properly. He didn't buy it. I Grip my rifle more tightly as we continued on The track stayed prominent hours sinking much deeper into the mud The sense was hard to ignore now although my uncle was indisposed to mention it It had gained a tinge of sweetness somehow like burning honey mixed with sewage water dripped Our stakeout spot was unwieldy in the lower branches of a stout white oak My uncle's breath heaved hot on my neck as we watched A sick deer couldn't travel far he reasoned and the head in bragging rights would be worth more than any meat it lacked Long minutes passed the hidden moon failing to give any indication of the time I was about to try and persuade him to call it quits when something moved My uncle holding the rifle didn't hesitate to shoot A low dark shot rang out and whatever had been moving stopped Quickly uncle dismounted the branch and started walking Wherever it was it was distant and the flashlight beam didn't penetrate the low branches The stench was overpowering now It seemed to come from nowhere in particular, but the air was warm and still Holding my breath. I continued after him In moments the flashlight hit something and uncle stopped A dark figure lain over in the ground cover as if injured sat at the edges of the flashlight's beam It somehow was still shown in silhouette as if the flashlight hadn't touched it One eye gleamed the other shot through with a bullet And the point where its antlers stopped and the branches began wasn't clear It breathed slowly steadily In a careful practice almost deliberate fashion The thing that was not a deer got up Turned backward and started walking away My uncle was silent quick to follow At that moment the forest seemed to melt around me as I ran Sharp thorns and branches blocked my way where uncle steps were swift The flashlight in his hands waving wildly I gave chase now acutely aware of my lack of gun And managed to follow him into a small clearing A place devoid of trees where a creek must have once flowed His flashlight tilted up and I stopped in my tracks The deer stood before us gazing intently The right half of its face was caved in and from the wound antlers grew Extending up and into the branches above it They billowed thick and gnarled like frozen smoke Around it from either side heads of deer hung like trophies And crusted down to the eyes in the same dark material Not quite catching the light I stood delirious with the smell under a canopy covered in the stuff Hard, finely twisted antlers gnarled together into great burls Half forming the shapes of heads or hooves Uncle turned my way looking back to me smiling You look sick What's wrong? It's only a deer I bolted wild through the night And woke up sometime later by the side of the road surrounded by paramedics My memories for the next few weeks were fuzzy Full of hospital rooms and doctors and tubes Suicide, they said Evidence was found in a nearby section of the forest Chugs of brain matter splattered by gunshot on nearby trees A flashlight and some clothes A couple scattered bones As for his body, they said the scavengers must have come after it My name was clear quickly enough as it went As they learned what was inside of my head Some kind of poison, biological in nature Neurotoxic Said I was lucky I came out with only minor brain damage Only thing that saved me was that I was found unconscious on the roadside Days before his estimated time of death I sometimes hallucinate now Sing antlers, trees, hearing the clicks of hooves They blame it on the brain damage But these folks blame that for all sorts of things Never bothering to go and look On warmer days, through the humidity and the still, heavy air A smell sometimes wafts into my yard from the south A sweet smell, like honey Like taxidermy in a rot That still, in that direction My uncle awaits Among the crown of horns I take up the scent deeply now As repulsive as it once was And know, quietly in my heart That my uncle did not die Shit, guys, that's something Do you think that's real? Or is just this guy's brain damage? Leave your answer in the comments below Subscribe and hit the bell And I'll see you next time


Hey, what's good y'all should boy down for the down showing friends and we back for another kpop reaction now check this out We have team and want dance practice. Oh, yeah, listen. I love how y'all was talking in the comments I was like, yo, yeah Prince that Michael Jackson feel all of that. I love that. So we definitely gonna be doing a music video So yes, look out for that. That's on the way. Um, and let's continue on this journey All right, so please hit the like button share ring that bell and also Thank you for being on this page showing love for me to do these dance review reactions So I appreciate this journey a lot. So let's get into it. Let's stop on it. Okay. Let's get into the dancing All right, here we go y'all and three two one Mmm. He already coming in that field dramatic like he He's tapped in. Okay. See what he working with. I'll see what we got y'all. Mmm Okay Yeah, so that's let me know how the music video is going to be I'm not really looking at the dance right now. He didn't really do nothing yet I'm kind of looking at how the camera is working Just from a dancer's perspective Very artistic. I see what he's on for the music video. I see what he's on Okay, a lot of wave movements isolations second shot Okay. All right. So Again, when you watch these data practices, I love how y'all talk in the comments, too Y'all talk about how y'all also notices when there's something for a tour of music video or a war show I love that that y'all got that I and you know if you're about to become a dancer keep that same energy because you're gonna really make it out here if you know the different perspectives of how to Establish what you're gonna put on the stage or a video this right here automatically screens to meet music video So I can't look at this the same way I will look at something that's just on mainly on one stage with one shot So let's keep watching But I like how it feels because Taemin has a feel You can't look for the trick moves. You gotta look at the feel that he's on notice. He's isolating a lot You notice that notice how he's keep isolating this like this feel So he's looking for that money shot It's this feel right here. Okay, so with this And then you start to really create with the shoulders with this Yeah, I understand and then also look at the dancers look how they setting he want them being on some cool Shit, you know I'm saying being cool. I see it Yeah, yeah, yeah, yeah, yeah. Yeah. Yeah Yes, yeah Uh-huh, what do we say? He's really focused on the feel of it more than having explosive Choreography at this moment of the music You could tell and he's really making shorter shots See, I haven't really never seen any of his music videos or nothing And that's why I want to definitely watch him because this feeling that he has tourists his work really got my attention Okay, so, you know him and lay and Somebody else told me to look at two but him and lay are really two of my favorites like real talk Like I've been looking at more stuff and lay outside of just doing these reviews for y'all and Taman It's about to be one of them as well Yeah, even these moves. There's variations to hit this, right? But he's on something else. It's very internal how he's dancing. Okay, here come the ladies Yeah, see yeah, I see the shots. I see what he's going with this Yeah, yeah, yeah, yeah, yeah, yeah, yeah, so I know it's so damn funny that I feel like there's another dance studio next door Because they already getting any and moving around. I'm like I'm trying to see what they doing over there. All right, let's continue. Yeah, this really got that Mike I saw that I saw that Hey, hey, hey, okay Okay, ah You can just see the inspiration and feel of Mike. I love that. I love that shout out to Michael Jackson, man Like for real for real. I like this choreo though. I like this. Oh We can't just I gotta see that again to set up with that I Need to see the setup Seek-seek-seek-seek-doop-doop-da-da. Good. Okay. Okay Dun-dun, what was that? What was that? You hear that? Tun-tun and watch the show to work Right here. Uh-uh So for people that's trying to get into the dance world, you can look this in your body You can look this in your body. So Boom boom boom you can look that okay So keep that in mind when you're dancing that you could actually look that way dancing just some tips I want to give out for people that's trying to dance and they love doing choreography from their favorite artist You can put you can loop that. Okay. Let's keep watching Yep Hit it down on that beat. Uh-huh. Da-da He hears it he hears it he made the music I Like this it's the feel for me y'all. It's it's the it's the feel that he got. Yep Yep, oh I want to see this in the music video how they set that up. I want to see I set that up He really you see how his texture set up like he's like he's hunched got this hunch in his body and got this Got this feeling right here. You know, I like that. I like that I think right lately right now for me as a dancer. I've been really paying attention to more feel and The storytelling did anything So I'm I'm kind of I'm really enjoying these videos that this texture that some of these artists have on their dancing so Go so now let's see what we do right here with this Let's go. Let's go. I hope you two don't block me So I'm gonna try to chop this up So I wonder how the camera shots is gonna be with this one Because y'all know I'm gonna have to do Tamey one music video That's another music video I'm gonna have to do because I'm curious how he is coming right here Notice how he just kind of post up So I wonder when it get right here. Do they do a lot of shots that focuses on her Because look how she's posing you have to get multiple different shots of that so many angles you could do with that So I wonder if this is his moment because I'm just watching how Taman is and Let's see when he activates really So this is probably the part of the music video probably and I'm assuming I noted music videos probably out already But this is fun for me to kind of assume what's gonna happen and then I find out I was right or wrong So now she's still moving He's definitely in the shot pride for here. Okay, okay Mmm, so maybe not Maybe it's a maybe a couple shots. Maybe be saw shots, but look we activates womb and Then they come in. Okay Ta ta ta ta ta ta ta ta ta.  Yes Ah So notice my nose is itching. That's one thing. I know this Notice at the beginning with the guys how it's more simpler that they did it and look how more explosive it was I don't know why my nose into y'all yikes Yeah Whoo, go back. We'll go back Here you I like that That up there he goes again Just so to work. That's all it is Yeah, you can look that I'll finish the video. Yep, I will finish the video with that. Yeah Nice nice We are definitely watching the music video we're definitely watching music video y'all with it Yeah, let's definitely do the music video Let's definitely do the music video. I'm with it. Y'all with it. Let's do it Perfect, let's do it. Let's do it. Let's do it. Let's do it. Let's do the music video So we have to we have to 10 main music videos and then a thousand more videos that we have to watch So let's see y'all. Thank y'all so much. I appreciate y'all. We're on the road to 20k once we hit 20k I'm going live and the new playlists are coming out choreography playlists battle playlists and then my second channel I will be doing gaming and vlogging and Also, my dance work will be on here too on this page The other channel is still growing slowly and that channel will have reviews on like Things I'm in too far as art and music and so I'm gonna have that on there But other than that y'all that's about it more videos on the way Who knows you might you might see like three videos drop at the same time or maybe just two we'll see alright y'all. See y'all soon. Peace


Why do narcissists always argue with you whenever there is an argument with a narcissist? It will always have something to do with you. It always has to be your fault They always have to put the focus on you So they can avoid responsibility They have no interest in understanding you Or coordinating with you Because they're very selfish and entitled Everything has to be done their way. They have to believe that they are more important than you are Because they have very fragile egos Deep down they feel completely worthless and insignificant So they constantly have to do things to make themselves feel like they are something To make themselves feel like they exist They have very fragile egos, so they cannot take accountability for their actions It's too much for them to deal with They have to blame and accuse you It's a coping mechanism They have difficulty managing stress They have difficulty coping with life Life is hard for them Which is why they're like ticking time bombs Anything could set them off They lack stability They have very low self-esteem Which results in them having no care or concern for you Because everything has to revolve around their fragile egos They have to defend and protect it That's why they're so controlling That's why they have to take away your free will Because they fear your sense of self They fear your differences and individuality They have to control you to regulate their emotions And manage their low self-esteem Which is why they act so arrogant and entitled Which is why they're so defensive They cannot be open with you They cannot be vulnerable Instead they create an alternate reality They make stuff up Whatever works for them Whatever makes them look good They will manipulate you to get what they want They don't care about fairness or equality All they care about is getting their needs met They lack empathy So there's no mutual understanding Whenever there's a difference of opinion They become highly argumentative Which should reveal to you that you are dealing with a very weak fragile person They want to display strength, power and dominance But underneath that facade they are very weak and afraid Which is why they always have to be in opposition to you They cannot listen to you They cannot agree with you on anything Because in their minds that means that you have the power Which means that they lose control When everything they do Is designed to gain control It's designed to provide them with a sense of security and stability Because that is something that they lack Which is why they're so unhappy That's why they always seem so agitated and annoyed Because underneath their facade is a lot of pain resulting from that unresolved issues Which is what they bring to the disagreement Narcissists are shame-based people And they're doing everything they can to avoid reflecting on their shame At some point in their lives they learn that when there is an argument or disagreement Someone is going to experience shame And they don't want to be the one who has to go through that So instead They devise a strategy Where they could be the ones to assign that shame to you Because if they can assign that shame to you Then they're no longer in difficulty or trouble When they argue with you They're trying to take away the focus away from their shame Which they deeply fear They have very low emotional intelligence They don't know how to cope with life Which is why there's no mutual understanding Instead their strategy is just belittle you To dismiss you as something that is unimportant They want to put all of the blame on you So that they can avoid accountability Because to them Accountability is a bad thing It's not something that they want to deal with The narcissist will argue with you And while some people may choose to fight fire with fire That is not something that I would advise Do not reciprocate Do not respond in kind Do not take the bait If there's one thing you can learn from their actions It's that you shouldn't want to be anything like them You shouldn't want to join their lack of responsibility You shouldn't want to join their dysfunction by doing what they do Let them do that Let them break the rules Let them run themselves through the rules Through the mud While you take a different path By taking accountability Which will lead you to a more functional way of life Accountability is the key to success It's easy to blame other people and take revenge But that's not going to benefit you in any way It's just going to make you more like them You're just going to be acting out of the fear and shame They have projected on to you Which is not going to result in anything good As Albert Einstein once said You can never solve a problem on the level on which it was created You have to take the higher road You have to upgrade your level of thinking And your level of consciousness And you can do that by taking your focus off of the narcissist Take accountability for where you find yourself And then take a different path Go in the opposite direction Because you will never find peace or joy in the direction that they're going in That level of thinking and processing Will never result in anything good It will only make them more miserable and dissatisfied with life in the end Thank you for watching I hope this video resonated with you Please like Comment Share And subscribe If you would like to donate My paypal link is in the video description Coaching inquiries You can email me at coaching.narcsurviver.uk Thank you for watching and I'll talk to you soon


This study examined the relationship between parity number and multimorbidity, millimetre, polypharmacy, chronic disease, and abdominal obesity among women in the Azokohort population. The results showed that as the number of live births increased, the prevalence of millimetre, polypharmacy, hypertension, cardiovascular disease, fatty liver disease, stroke, rheumatoid diseases, chronic obstructive pulmonary disease, and cancers also increased. Additionally, the odds of having abdominal obesity waste to hip ratio greater than or equal to 0.85 and waste to height ratio greater than or equal to 0.5 were significantly higher in women who had a high parity number greater than or equal to 5. These findings suggest that parity number may be associated with increased risk of developing multiple chronic conditions and abdominal obesity in this population.


Hi and welcome to Learn WordPress. In this session, we are going to take a closer look at caching and how you can use it to your advantage. Caching is a technique used to improve the performance of websites. It involves storing frequently accessed data in a cache, which can be quickly retrieved when needed, rather than having to retrieve it from the server every time a user visits the website. Caching is like having a personal bookshelf for frequently used files and data. When a user visits a cached website, the browser checks the cache first for required files. If found, the website loads faster. It lessens the server's burden by serving cached files instead of generating them repeatedly. Ultimately, caching enhances user experience by providing faster loading times and smoother website navigation. Adding a plugin is probably one of the most simple ways to optimize caching on your WordPress website. Cached plugins simplify implementation, optimize performance, offer advanced techniques and include cache management features for efficient and targeted caching on your website. Let's briefly look at six of the most common caching you will come across. Number one, page caching. Page caching saves a fully prepared version of a webpage, so when somebody visits the website, it can be quickly displayed without needing to be created from scratch. Number two, browser caching. When you visit a website, your browser saves certain files so the next time you visit that site, your browser can use the stored files instead of downloading them again. This helps to load the website faster since the browser doesn't have to fetch all the files from the server every time. Number three, server caching. Server caching involves storing commonly accessed website data. When a user requests that data, the server can quickly retrieve it from the cache instead of generating it again. And number four, CDN caching. Content delivery networks store copies of website files on servers located in different parts of the world. When a user visits the website, the files are loaded from the nearest server, reducing the time it takes for the content to reach them. Number five is object caching. Object cache is used to save on trips to the database. It stores all the frequently used website elements in memory for quicker access, making the site load faster and reducing server work. And lastly, number six, database caching. Database caching focuses on caching the raw data obtained from database queries, while object caching caches processed and ready to use elements of the website. Some plugins to explore are W3 total cache, WP fastest cache, and Lightspeed cache, but there are many others available in the plugins directory. In this example today, we are going to install and activate the W3 total cache plugin and then enable some of the most important settings. So let's make our way to plugins and then click on add new and then search caching. And in this case, we are going to click on install now next to W3 total cache and then activate. And once activated, we can make our way to settings. So when you get here, you will notice at the top, it is welcome to the W3 total cache setup guide. So they provide you a guide to follow. But you can also click on skip this setup guide if you wanted to. Otherwise, you can click on next and go through all of these types of cache one by one. Let's go ahead and test the page cache. And we are off to the fastest time. And as you will notice, they have also recommended the disk enhanced, which is actually the fastest time. Then we can click on next and select test database cache. Now at the bottom below recommended, it actually says leave this feature disabled as the server database engine may be faster than using disk caching. So let's move along and test object cache. And here we will notice that the fastest time is 1.85. So we will select that and click next, test browser cache, ensure that is enabled. And then it also says select lazy load. And we will talk about that in a sec. And when this setup is done, we can visit general settings to learn more about extra settings and features. So let's make our way to general settings. Let's take a closer look at some general settings. First up is page cache. The first thing we want to do is to ensure that this is enabled. Adding page cache can improve performance and reduce server load. And please note some hosts also provide page caching. Next, we are going to look at minify or minification. It reduces load time by decreasing the size and number of CSS and JS files. And it automatically removes unnecessary data from CSS, JS, feed page and post HTML. We can therefore go ahead and enable minify. I have read through the documentation so I will go ahead and say I understand the risks. And leaving minify mode on auto will suffice. Then we can scroll down to opcode cache. Opcode cache is available to pro version users and allows you to store compiled PHP code. Now we move on to database cache and object cache. Let's first talk about database cache. As you know it was recommended during the setup that it might be better to keep this disabled on the specific website. And it also states best used if object caching is not possible. Object cache is possible so we have therefore enabled this feature as it further reduces execution time for common operations. But please note database cache and object cache is not recommended when using shared server hosting. Up next we will ensure that browser cache is enabled. As it reduces server load and decreases response time by using the cache available in a visitor's web browser. As mentioned cdn caching is also important. And to implement this you need a cdn provider. So make sure you sign up for providers such as stack path, cloud flare or others. Let's move on to the Google page speed section. Here you have the option to connect to the page speed insights API which lets you analyse the performance of your website. If you authorise this it does require you to connect your gmail account.  When lazy load images is selected images and other media elements only load when the user reaches that section which will reduce website load time. You can also consider disabling the three options below. When you are in your dashboard you can use your left sidebar to fine tune more specific settings for different types of cache. Lastly it is worth mentioning that sometimes the cache won't realise you have made changes or updated your site and instead of loading the new version of your site it will load the old version. You might then consider clearing your cache to ensure the newest version of your site loads as a form of troubleshooting. You will find options to clear cache at the top of your dashboard. I trust you now have a better understanding of caching and that you will find the right plugin that works for you. It is also worth mentioning it is best to enable caching after you have finished building your site. Visit Learn WordPress for more tutorials and training material.


The Red Sox defeated the Yankees in last night's wild card game right here at Fenway Park. The score of 6-2 means the Sox will advance to the MLB playoffs. The City of Boston wakes up to a big victory today. Dave Ranin, a longtime Fenway Park tour guide, attended the game last night and woke up early today to head back to the stadium. It was very cool. We're just the excitement anyway of a Red Sox Yankees game is always there. But then add all the pressure of last night, do or die, win or go home. It just made for a very fun atmosphere at Fenway. Here in Fenway, businesses recover from a late night for Sox fans. All of this while cleaning up the streets in preparation for work today. Bianca Sapola, a Red Sox fan and Boston resident, heads to work in the Fenway area sporting her Sox gear. I think today the Fenway area is going to go crazy and I think it's going to be like this throughout postseason as the Red Sox move on. Even just yesterday, working in the area, it was very early where all the Red Sox fans started to come out and get rowdy. So I'm excited to see the energy and excitement here today over the win. The energy is definitely high in Fenway this morning after last night's big win. If they can play the way they did in June and last night, they'll be fine. And so that's what I'm hoping.


The Jack Pendy program presented by America's largest selling cigarette Lucky Strike. SING A SING A SING A SING A Singada Lucky Strike first again with tobacco men. Yes, first again with the men who really know tobacco. Independent buyers auctioneers and warehouses For a recent impartial survey shows that more of these independent tobacco experts smoke Lucky Strike regularly than the next two leading brands combined. So let this overwhelming smoking preference of the experts lead you to real deep down smoking enjoyment. Just light up a lucky, puff by puff, you'll see. LSMFT, LSMFT. Lucky strike means fine tobacco, and this fine lucky strike tobacco means a world of smoking enjoyment for you. Yes, puff by puff, pack by pack. You'll like Lucky Strike. The Lucky Strike program starring Jack Benny with Mary Livingston, Philaris, Rochester, Dennis Dane, yours truly, Don Wilson. Ladies and gentlemen, let's go out to Jack Benny's home in Beverly Hills. It's morning and Rochester is in the kitchen. Hmm, look what time it is. I better start getting Mr. Benny's breakfast ready. Now let's see. I think I'll give him some ham and waffles. He's crazy about ham and waffles. Hmm, we're out of ham. And I had the last six waffles. Let's see, what else? Oh, I know. I'll fix him a plate of bacon and eggs with french fried potatoes. Uh-oh, can't do that. I had the eggs. Come to think of it, I had the bacon and potatoes too. Well, I better take a look in the refrigerator again. Hmm, I wish the boss would buy a refrigerator with electric light in it. This candle keeps melting the ice cubes. Uh-oh, what's this in the corner? Well, the boss is lucky. Here's some orange juice, some pickled beets and milk. I'll pour the milk over the beets. If he eats without his glasses, he'll think they're strawberries and cream. That much is ready. I'll put it on this tray and take it up to him. Better take him the morning paper too. Mr. Benny's still asleep. I better wake him up easy. I'll tickle his feet.  Maybe I'll take off his socks. Let's get chilly, he doesn't take any chances. Socks, corduroy, pajamas, earmuffs, and a derby. What? It's only me, boss. You didn't have to tip your hat. Oh, oh, good morning. Good morning, Rochester.  Okay. I brought you the morning paper too. Oh, thanks. Whose picture is this on the front page? Looks familiar, but I can't seem to... Oh, it's Dewey. Must be an old picture, he's smiling. See, look at those electoral votes. Dewey carried New York, Pennsylvania, and Kansas. Thurman carried Alabama, Mississippi, and South Carolina. Mary Livingston carried the maid company. Ha, ha, ha! See, I just woke up and I'm pretty sharp. Ain't I, Rochester? If you say so, sir. Mr. Benny? Hey, just throw it over my shoulder. Oh, say, boss, you didn't even touch your breakfast. I know, Rochester, but I don't feel like eating anything, especially strawberries and cream. I had a... I had a terrible night. What's the matter? It's that echo. It's worse than ever now. I even hear it in my sleep. Boss, you must be imagining things. What do you mean imagining things? I tell you, Rochester's there and I hear it. If I don't get rid of it soon, I'll go screwing. Screwing only? Rochester. Rochester. I know, boss. You heard the echo again. Yeah, how'd you know? When you jumped in the air, your socks changed feet. Rochester, stop kidding about things like that. Oh, Jack. He missed Livingston's here already. Well, I can rehearse in my bathrobe. Coming, Mary. Hello, Mary. Hello, Jack. Am I the first one here? Yeah. You don't mind if I rehearse in my robe, do you? No, but take off that derby. You look like a plumber's friend. I'm sorry. And Jack, before the gang gets here, how about give me the $10 you owe me on the election bet? The what? Oh, yes, the election bet. It slipped my mind. I meant to give it to you before. Oh, sure, sure. It slipped your mind. Come on, give me the $10. Okay. Here it is. Thanks. Well, that takes care of the Hoover election. Now, how about this one? Now, wait a minute, Mary. You can't hold me to that when he wasn't even on the ballot. What was it, Miss Livingston? He bet that Fred Allen would carry the corn belt. Well, he should have. It's been carrying him for the last 10 years. What a program Fred Allen's got. No jokes and double indemnity. Oh, I don't know, Jack. I think that his program is great this year. Fred Allen is very funny. He is, eh? Yeah. He's improved a lot since my mother started writing for him. Oh, your mother? Your mother isn't with Bob Hope anymore, huh? Incidentally, Mary, I'm kind of excited.  Bob's having you on his program? Why? Why? I'll tell you why. Bob Hope came to me, looked me right in the eyes and said, Jack, I want you on my show because you're the greatest comedian in the world. Bob said that. He said, you're the most versatile. The most talented. The cleverest. The wittyest. The most sophisticated, subtle swab.  How much is he paying you? He said, $5,000. But how can you believe a man that can lie like that? We'll come to some kind of an agreement. Yeah, he'll probably give you a bar of swan soap. I hope not. I'll have to break it in two and give my agent half. Anyway, Mary, what's that? Oh, my goodness. It's Dennis. Dennis, what's the matter? April Fool.  This is no time of year to fool people. Truman did it. Ever do that again.  It stood up. It jumped off and ran home to Max Factor. Mary, this is no, this is no time for jokes. This kid frightened me so. My blood turned cold. Holy, holy. There it goes again. Your hair? No, the echoes. Mary, it's serious. I'll tell you, it's making me a nervous wreck. You know, Mr. Benny, last year I used to hear voices all the time. I'd be in my room all alone and I'd still hear those voices. You did, kid? Well, then you must know how I feel. Didn't it drive you nuts? Oh, no, I was nuts before that. That I can believe. Now, look, kid, before the rest of the gang get here for rehearsal, run over your song. Okay. Oh, by the way, Mr. Benny, why were we rehearsing so early? Because I have a busy week. I also have to rehearse the Bob Hope program. I'm going to be on it Tuesday night. Bob Hope wants you on his program? Why? Why? Because he said that I'm the greatest comedian in the world. What's the matter? Is he crazy or something? Not crazy. He may lie a little, but he's okay. Now, go ahead and sing, will you? Why not? Because I'm not getting anywhere. What? I've been working for you for nine years and I can't even afford to buy anything. Clothes, shoes, food or anything. Now, wait a minute, Dennis. I'm not responsible for what you do with your money. What money? What money? I've been giving you checks every week. What do you do with the money you get when you cash them? Oh!  Leave him alone, Mary. He knows what he's doing. I'll get the scripts ready and then... Well, that must be the boys' nouns about time. Come in. Hiya, Jackson, Dennis. You delectable Cupid, are you? Well, if it is a little old Phil Harris, the breath that made Milwaukee famous. Now look, Don, you too, Phil. Look, when I call a rehearsal for a certain time, I expect you to be here. Wait a minute, Jackson. What's your rush? Take it easy. Relax. You'll have a show Sunday. What? And suppose you do have a lousy one. Nobody's going to turn off the radio. I come on right after you. Phil. You got better insurance than Fred Allen. Phil. Phil. Pencil head. It just so happens that I was on the air 15 years before you even had your own show. I know. And what happens? The minute I'm on my own, they embed television. They want to see me, too. Look, Dad, you better iron out those wrinkles if you want to stick around. What? Them blue eyes ain't going to do it alone. Well, Phil. Phil, I'll admit one thing. You're unnatural for television. The screen is square and your head will fit right in there. How is that, Don? Don? Mary, where's Don? His girdle broke and he can't get out of the den. How we ever get a show put together, I'll never know. Now, look, we came here to rehearse. Oh, for heaven's sake. Don, will you answer the phone, please? Sure, Jack. Hello? Yes, this is Jack Benny's home. What? Oh, that's the final vote. All the precincts are in, huh? Every vote has been counted and it's official now? Good, good. I'll tell him. What is it, Don? Luckies are first again with tobacco men. Oh, was it a close race? Oh, no, no. More independent tobacco experts smoke lucky strikes than the next two leading brands combined. Excuse me, but who are these tobacco experts? Mary, we've got to get... Well, here's what Colonel Crockmorton P. Day of Blediola Springs has to say. Oven smoking luckies for an ion of 3,000 years. 3,000 years.  Dum, dum, dum, dum, dum, dum, dum. What's the matter with you guys anyway? You come over to rehearse, we can't even get started. Look, if I don't have a good show Sunday, I'll hear from my sponsor, Mr. Riggy-O. Riggy-O, lady. There it is again, the echo. Now you're all here in this room. I know you heard. I dare not. Neither did I. Who heard anything? Kids. Kids, you're sure you're not ribbing me, are you? No, Jack, we just don't hear it. Well, I hear it. I've got to do something. I even hear it in my sleep. Well, Jack, why don't you do like I ask you to? I've been after you for three weeks to go to a psychiatrist. Yeah, I guess you're right, Mary. I'll have to go to some doctor. Well, I know a wonderful psychiatrist.  OK, Mary, the quicker the better. But what about the rehearsal? Oh, we can rehearse tonight. All right. We'll rehearse tonight, kids. Come on, Mary, let's go. What's the number of the doctor's office, Mary? 11-07. It's right down the hall. This must be it here, room 11-07. Yeah, look what it says on the door. Dr. Nelson's psychiatrist. Come in, lie down for a while. Let's go in. Just a second. Jack, stop trying to look shabby. The doctor has one price for everybody. I'll have to pin my sleeve back on. Oh, well, let's go in. Nurse, I'd like to see Dr. Nelson. Very well, in just a few minutes. But first, I'll have to have some information about you. Well, my name. Oh, just a second, sir. I have to write this down. I want to get a pencil. There. Now, your name? Jack Denny. Occupation? Radio comedian. Are you currently employed? Yes. Yes, I am. And now for your financial status. What is your bank balance? Well. You better get another pencil, nurse. Mary, please. We'll go on to the next question. How tall are you? 5'11". Your weight? 165". Your eyes? They're blue, aren't they? Blueer than the lips of a schoolboy at 40 below. The doctor will be ready for you in a few minutes. Have a seat. Right over there by that gentleman in the corner. His name is Mr. Skinner. Oh, by the way, nurse, that Mr. Skinner isn't a violent patient, is he? Oh, far from it. In fact, he's very gentle. Gentle? Yes. He thinks he's a rabbit. Hmm. Come on, Mary. Let's sit down. Jack, what are you so mad about? A smart alec nurse trying to kid me a rabbit. Man, look just as normal as I do. Oh, Jack, there isn't room for both of us to sit on the sofa. Yes, there is, Mary. I'll ask Mr. Skinner to move over. Pardon me, sir. Eh. What's up, Jack? Gosh, you hear that, Mary? What's up, doc?  Yeah, but Dr. Nelson has helped me a lot. He has? What is his treatment? Oh, he uses applied psychology. He tries to make me think I'm something else. Right now, he's trying to convince me I'm a pig. A pig? Yeah, that's all, folks. Oh, say, Mary. Yes, Jack? No, it's a shame this guy is crazy. He's almost as talented as Mel Blank. Say, Bud. Yes, Mr. Skinner. I told you all about myself. Now, what are you here for? Oh, it's really nothing very serious. It's just something that bothers me. You see, I'm a comedian. Oh, on a stage? No, radio. Radio, ladies. There goes again. Mary. Mary, you heard it back time, didn't you? No, Jack, I didn't hear a thing and calm down. But there was an echo. Nurse, you heard it, didn't you, nurse? Well, no, I'm sorry. Mr. Skinner? Mr. Skinner, where are you going? I'm getting out of here. This guy's nuts. I've got a good mind to go after him and punch him right in the nose. Jack, control yourself. I won't control myself.  All right, Jack. Now, sit down. Yes, doctor? Oh, very well, sir. I'll send in the next patient. It's Mr. Benny. What? No, it's Benny. Benny just left. Go right in, Mr. Benny. OK, you wait. You wait here for me, Mary. All right, Jack.  Oh, Mr. Benny. Yes? The doctor may want to examine your head.  Thank you. Doctor, doctor, I'm Jack Benny. How do you do? And now, what can I do for you? Well, doctor, I hear echoes that no one else seems to hear. Echoes? How long have you suffered from this delusion? Oh, for several weeks. Uh-huh. And tell me, how old are you, Mr. Benny? 39. 39? How long have you suffered from that delusion? It's no delusion. I'm only 39. And what kind of a psychiatrist are you, anyway? Instead of a couch in here, you've got a double bed. Certainly. I like to lie down, too. And now, Mr. Benny, do you have any idea of when this echo first started bothering you? Yeah, you see, this summer, I was in Europe. I went to Switzerland. And I heard some Swiss Yodlers. And ever since then, an echo has followed me. Very interesting. Now, does this echo repeat everything you say? Well, not everything. It usually happens on words that end in O. You mean words like folio? Yes. In fact, it happened once on that very word, folio. Folioli. There it is. There it is. Look, you heard it, doctor. There it is. You must have heard it. There's no echo, Mr. Benny. Now, calm down, and we'll get to the cause of your neurosis. Gee, I hope you can. Let's get started with the treatment. Just a moment. I want to look in the next room and see how another one of my patients is getting along. Two and two is four. One and two is three. Three and three is six. Who is that? Mr. Gallop. Of the Gallop poll? Yes. Roper's still working on one and one. Now, Mr. Benny, lie down here and relax. Thank you. Comfy? Yes. Yes, are you? Yes. Now, we'll start with the color association test.  Ready? Yes. Black. White. Blue. Red. Orange. Yellow. Green. Money. Very interesting. Mr. Benny, you seem to have a certain fondness for money. Well, perhaps a little. And tell me, when did you first notice your desire to hold on to money? When I was six months old, I swallowed a dime. Oh, and when did they get it out? Oh, it's still in there. Good heavens, man. Why didn't you have a doctor remove it? Well, how can you trust anyone when you're under ether? Well, Mr. Benny, you've got to have confidence in... Pardon me. Yes, nurse? Oh, Dr. Nielsen, Mr. Skinner came back.  I think... Call him a carrot and I'll see him later. And now let's see, where were we? Doctor, doc, please, about that echo. Oh, yes. Mr. Benny, I've come to the conclusion that what you're suffering from is nothing more than a somatic hallucination induced by your psychotic neuroses and aggravated by self-induced inhibitions and subconscious phobias of the psychosomatic urges. And there's only one cure for it. What's that? A four-way cold tablet. A cold tablet? Doctor, I'm all confused. Well, in simple language, Mr. Benny, when you first heard those yodellers in Switzerland, they created a great impression on you. And now, when you think you hear these echoes, it's nothing more than a thought desire to return to the scene of a happy experience. Oh. Well, what do you think I ought to do about it? Well, stop working so hard. Take a vacation. Get away from Hollywood. But where could I go? Why don't you try Sun Valley? In Idaho? Idaho? Well, I've always wanted to go there. Just a minute, Mr. Benny. Huh? Didn't you hear that echo? Oh, no. There was no echo that time. That's funny. I could have sworn I heard one. Oh, no, no. You see, it couldn't happen on a word like Idaho. It usually happens on words like I told you before. You know, like folio. Folio? Folioli? There it goes again. There it goes again. You heard it that time, didn't you, Mr. Benny? Didn't you? Didn't you? No, no, no. I didn't hear anything. But you must have. You must have. Calm down, Mr. Nelson. There's no echo now. As you explained, it was just a figment of my imagination. Then why am I hearing it?  Well, thanks for helping me, doctor. I feel a lot better. Well, wait a minute. Wait a minute. Don't leave me now. Don't go. Don't go. Don't go to your name, Mr. Benny. What's wrong with that doctor? Oh, well, I better get out of here. Oh, there's Mary sitting in the corner. Mary. Mary, I'm all cured. And it was the simplest thing in the world. The doctor talked me for a few minutes, and then I... Mary. What's the matter? Mary. Be back in just a moment. But first... That's it. That's it. That's it. That's it. That's it. That's it. That's it. That's it. That's it. That's it. American. An impartial survey covering all the southern tobacco markets reveals this important fact. More independent tobacco experts smoke Lucky Strike regularly than the next two leading brands combined. Yes, Lucky Strike. First again with Tobacco Man. Remember, these are the men who really know Tobacco, the independent auctioneers, buyers, and warehousemen. That's why it's important for you to know that the overwhelming choice of these experts is Lucky Strike. You've heard the survey results. Now, here's what Mr. Sydney Curran, Tobacco Warehouseman, who sells on an average of eight and one-half million pounds of Tobacco a season recently said. At auction after auction, I've seen the makers of Lucky Strike buy a fine Tobacco that's got real smoking quality. Tobacco that smokes up mild, cool, and fragrant. I'll smoke Lucky's 26 years. A Lucky Strike smoker for 26 years. And Mr. Curran, like you, looks to a cigarette for enjoyment. Real deep down smoking enjoyment. So light up a Lucky yourself and puff by puff. You'll see. LSMFT. LSMFT. Lucky Strike means fine Tobacco. And naturally, fine Tobacco means a really enjoyable smoke for you. That's why you'll like Lucky Strike. Ladies and gentlemen, stay tuned in for the Phil Harris Alice Face Show, which follows immediately. And on Saturday night, be sure to hear a day in the life of Dennis Day. Well, Mary, I got to go over to rehearse with Bob Hope now. Have you got a good part? I don't know. I've got the script here. Let's see. On page three, I say, hello, Bob. And then he says, Oh, here on page six, I say, well, if it isn't high ever back. And then Bob says, Oh, but on 12 page 12. Oh, no, they cut that. Oh, well, I'll add live something. Good night. This is NBC, the National Broadcasting Company.