Welcome, in this lecture we will study diagonalization and similarity transformation. First we will discuss the issue of diagonalizability which I initiated in the previous lecture and then we will go through two very important topics in the algebraic identity problem. One is the canonical form and the other is the special advantages that we can take off the symmetry of a matrix first diagonalizability. Consider an n by n matrix which has n linearly independent eigenvectors v 1 to v n with corresponding eigenvalues lambda 1 to lambda n some of these may be repeated for that matter that is lambda 2 and lambda 3 can be equal need not be all different. These are all linearly independent then only we will talk about n different eigenvectors. If we have the matrix which has n linearly independent eigenvectors then consider this if we pack all these n eigenvectors into one n by n matrix with these vectors as columns then we have this n by n matrix here which we are denoting by s and then we examine the product a s. Now, what will be the first column of this product a into the first column of this matrix s that is a v 1 since v 1 is an eigenvector with eigenvalue lambda 1. So, a v 1 is lambda 1 v 1 similarly the second column of the product will be a v 2 which is lambda 2 v 2 and so on till this. Now, we claim that the same product we would get if we multiply the matrix s from the right side with a matrix with a diagonal matrix having the eigenvalues as the diagonal entries. Let us verify this a little carefully what will be the first column of this product it will be this matrix multiplied with the first column of this that will be v 1 into lambda 1 which is this plus v 2 into 0 plus v 3 into 0 and so on. Thus we find that the first column of this product is also lambda 1 v 1 similarly the second column of this product will be v 1 into 0 plus v 2 into lambda 2 which is this plus v 3 into 0 and so on. That means the second column is also same similarly we will find that all the columns till the n s column will be found identical to this. And this matrix this diagonal matrix with the eigenvalues of a sitting in the diagonal positions we will call as lambda and then this gives us s into lambda that means we have got a s equal to s lambda. Now, if we post multiply this equality with s inverse then from this side s gets cancelled here we get s lambda s inverse that is this that means that a can be expressed as s lambda s inverse for lambda is the diagonal matrix with eigenvalues sitting at the diagonal locations. Similarly, if we pre multiply both sides of this relationship with s inverse then we get s inverse a s is equal to lambda that means that the matrix a in the new basis s gets diagonalized. And this process of changing the basis of a linear transformation so that its new matrix representation is diagonal is called diagonalization. In this process the transformation the mapping gets decoupled among its coordinates. What was the necessary condition for this to happen the necessary condition was just this that is the matrix has n linearly independent eigenvectors or we can say the matrix possesses a full set of n linearly independent eigenvectors. So, this was the only requirement for this diagonalization to be possible. So, we can say this about diagonalizability of a matrix a matrix having a complete set of n linearly independent eigenvectors is diagonalizable that is the n by n matrix can be diagonalized if it has a full set of n linearly independent eigenvectors. The converse is also true that is a diagonalizable matrix possesses a complete set of n linearly independent eigenvectors. If you want to prove this then you will say that we already know that there exists a basis s in which the matrix representation will be diagonal. And in that case what you do you pre multiply with s and then you say that we already have this relationship a s is equal to s lambda. That means this and this will turn out to be equal and from that you can figure out that a v 1 is lambda 1 v 1 a v 2 is equal to lambda 2 v 2 which will mean that all these linearly independent vectors v 1, v 2, v 3, v 4 etcetera are indeed eigenvectors that will convince you about the existence of n linearly independent eigenvectors. So, this statement and its converse both are true that is if a matrix possesses a full set of n linearly independent eigenvectors then it is diagonalizable with that same eigenvectors sitting as the columns of the similarity transformation matrix giving the new basis. On the other side if the matrix is diagonalizable then you can claim that it does have n linearly independent eigenvectors a full set. Now, a few important reminders, a few important points which are actually quite simple, but they are sometimes confused. So, note these small simple statements and do not get confused. One is all distinct eigenvalues will directly imply diagonalizability because if all the eigenvalues are distinct then for every eigenvalue there must be an eigenvector that is necessary for the definition of eigenvalue itself that means an eigenvalue must have one eigenvector associated with it if not more. So, if all eigenvalues are distinct that means n distinct eigenvalues are there and each eigenvalue will have associated with itself one eigenvector. So, that means n eigenvectors are guaranteed and that implies diagonalizability, but on the other side diagonalizability does not imply distinct eigenvalues because it is possible that the matrix is diagonalizable even with repeated eigenvalues. In that case a repeated eigenvalue will have that many eigenvectors associated with it as its algebraic multiplicity. So, from diagonalizability we cannot conclude distinct eigenvalues, but from distinct eigenvalues diagonalizability can be directly concluded. However, if the matrix is not diagonalizable then from the first statement itself we will know that there is certainly some multiplicity mismatch and for the multiplicity to have mismatch the algebraic multiplicity must be greater than 1. So, these points we need to remember when we deal with matrices. Now, we note that diagonalizability is not possible for all matrices and that gives rise to two questions. What simplified representation is possible for all matrices and for what kind of matrices diagonalizability is guaranteed to be possible. The first is the issue of canonical form and the second is the question of symmetric matrices. These are the two important topics that we study in this lesson. First canonical forms, there are three forms to which a matrix can be reduced for linear transformation can be expressed through change of basis. They are known as the canonical forms, Jordan canonical form, diagonal canonical form and triangular canonical form. First the Jordan canonical form which is always possible. A Jordan canonical form is the simplest possible form, simplest possible matrix representation of a linear transformation which is possible for all transformations, all matrices and the form is like this which is composed of diagonal blocks. These are blocks having small square matrices. Above these diagonal blocks and below these diagonal blocks all the other entries of the matrix are 0 and these diagonal blocks are also specialized in their shape. Along the diagonal in such a block say the r th block j r we have the eigenvalues, same eigenvalue that is one Jordan block is associated with a single eigenvalue. Though for a particular eigenvalue more such blocks are possible. Now one such Jordan block looks like this in which along the diagonal entries the corresponding eigenvalue will be there and just on the super diagonal there will be ones. Everything above the super diagonal and everything below the diagonal will be 0. So, this is the typical shape of a Jordan block. Such blocks sitting as diagonal blocks in this block diagonal matrix will make this Jordan canonical form. Now unlike the previous case for diagonalization which we were studying earlier where we look for a suitable matrix S such that we could say this in which this lambda is diagonal. Here we will look for this j sitting in place of this lambda. So, the reduction will not be always possible to this diagonal form, but it will be possible always to this Jordan canonical form. So, the associated similar transformation matrix S will have bunches of its columns. If j 1 is 3 by 3 then the first bunch of 3 columns in the matrix S will be called S 1 which will have 3 columns. Similarly, if j 2 is 5 by 5 then corresponding bunch of vectors here will be denoted as S 2 which will have 5 columns and so on.  And then we will have a into this entire matrix S is equal to that same matrix S multiplied with this Jordan canonical form of the matrix A like this. Now here if we try to see what is there in this bunch of vectors S R then we will have a form like this in which the number of column vectors in S R will be the same as the number of columns or rows in this square block J R. First of these entries first of these columns will be an eigen vector of the matrix corresponding to eigen value lambda. And next we will have other vectors W 2, W 3, W 4 etcetera the requisite number to fill up the number of columns which are called generalized eigen vectors. They are not eigen vectors, but they are in some sense resembling to the eigen vectors and they are called generalized eigen vectors. Now what are those vectors we will find out for that what we do is that we consider the product A S R that will be a bunch of columns taken from the correct location from this product.  So, the non 0 component of this will be only S R J R. So, A S R will be S R J R. So, block by block if we equate like this then we will find A S R equal to S R J R then using the expression for S R from here with columns V W 2 W 3 etcetera we can write like this A V A W 2 A W 3 etcetera in the product for A S R on this side S R is this and J R shape is this that will give us V multiplied with lambda as the first column. In the second column there will be two terms V multiplied with 1 and W 2 multiplied with lambda. So, that way if we equate column by column then the first column equality will give us A V equal to lambda V. Second column equality will give us A W 2 is equal to 1 into V plus lambda into W 2 that is this. Third column will give us A W 3 equal to 0 into V 1 into W 2 and lambda into W 3 that is this and so on. So, from here the first one is already familiar to us this is from where we determined the Eigen value then once Eigen value sorry Eigen vector. So, Eigen vector we determined from this equation once Eigen vector V is determined the next generalized Eigen vector that is the first generalized Eigen vector actually is after immediately after the Eigen vector that will be found as A minus lambda I into W 2 equal to V. Once W 3 is found we can find out W 3 like this from this relationship and so on as there are number of columns in S R. So, compared to that number one less generalized Eigen value Eigen vectors will be found because the first plot is taken by the Eigen vector itself. After that if we try to find one extra generalized Eigen vector we will find that the system of equations that come out will be inconsistent. So, that exactly that many generalized Eigen vectors we will find as many are really required to fill the block. Now, with these Eigen vectors and generalized Eigen vectors sitting in columns we will have the full matrix S which in this kind of a transformation basis change will not transform A to diagonal form in general, but it will reduce it to the Jordan canonical form and this canonical form is always possible for all matrices all square matrices. The second canonical form is the diagonal canonical form which we have already seen, but now we will have another quick look at it to see what is its relation to the Jordan canonical form. In the Jordan canonical form which is like this if all Jordan blocks are of one by one size then what happens if this is not such a big matrix, but if it is a one by one matrix that means it will be just a lambda and there will be no place for that super diagonal one and that is the case of the diagonal canonical form. If you associated points we can quickly note first is that the diagonal form is a special case of the Jordan form with each Jordan block of one by one size. This will immediately mean that with the absence of those super diagonal ones you have only diagonal entries in that canonical form and that means that the matrix is diagonalizable and that will also mean that each SR bunch will have a single vector the Eigen vector itself sitting there. So, that will mean that the similarity transformation matrix S is composed of all Eigen vectors that is n linearly independent Eigen vectors as columns and that means all linearly independent Eigen vectors exist in this case when the Jordan blocks are of one by one size or the matrix is diagonalizable. And in that case if you try to find generalized Eigen vector corresponding to an Eigen vector already found then you will find that none of the Eigen vectors will admit any generalized Eigen vectors. The corresponding equation will turn out to be inconsistent. This will also mean that for every Eigen value the geometric and algebraic multiplicities are same. There is a third canonical form which is of a very important practical significance and that is the triangular canonical form. We have already come across triangular matrices in our study of systems of linear equations. Now here when we said triangular canonical form of a matrix then we are actually referring to the triangular form of the linear transformation. That means that we are talking about converting a given matrix to the triangular form through similarity transformation. Now special significance of triangular form arises from the triangularization which is always possible. What is triangularization? Triangularization of a matrix or of a linear transformation is basically the change of basis of the linear transformation in such a manner that its matrix is in the triangular form. That is apply some suitable similarity transformation on the matrix such that the form of the resulting matrix in the new basis turns out to be like this with all zeroes here. Non-zero entries only on the diagonal and above below that everything else is 0. This is the triangular form. Now the practical significance of it is that this is always possible and this is possible always through orthogonal similarity transformation. The Jordan canonical form is always possible but for Jordan canonical form the S that you need is not necessarily orthogonal. Triangularization you can always conduct with orthogonal S and change of basis through orthogonal transformations has a lot of practical advantage. The advantage is both analytical as well as computational. So we find that the triangular form is always possible. In particular if the eigenvalues are all real then it is always possible through orthogonal similarity transformation. Even if eigenvalues are not real even if you have complex eigenvalues for the real matrix even then you can take recourse to complex arithmetic in your calculations and you will be always able to triangularize the matrix with unitary similarity transformation. Whatever holds for orthogonal similarity transformation in the case of real eigenvalues in case of complex eigenvalues the same will be true with unitary similarity transformation that is S in that case will not be necessarily orthogonal but it will be unitary which is the complex analog of orthogonal matrices. Now if you insist on working with real arithmetic only and orthogonal similarity transformations only even though there are complex eigenvalues then you can almost do the triangularization except that for a pair of complex eigenvalues you may be left with real diagonal blocks of 2 by 2 size like this. Because this is actually equivalent to the triangularized version which is this and you will never be able to reduce this matrix till this point unless you allow complex arithmetic in your calculations. But then with this kind of diagonal block setting you will be able to recognize that you have a pair of complex eigenvalues there other than that the rest of it you can do even if there are complex eigenvalues. Now if you can reduce through orthogonal similarity transformation a matrix to this shape you have not completely solved the eigenvalue problem in the sense that you have not been able to determine the eigenvectors completely but eigenvalues are all there along the diagonal. So that way if you are first interested particularly in the determination of eigenvalues then with much less amount of computation you can reduce it to triangular form and get the eigenvalues. Once eigenvalues are determined there are some methods which help you in finding the eigenvectors with less cost. Now other than these 3 canonical forms there are 2 other forms which are actually not canonical forms but which have some important advantages when we talk about computational methods for solving the eigenvalue problem. One is the tridiagonal form and the other is the Hessenberg form. These forms are advantages in the sense that that can be obtained through a pre determined number of arithmetic operations. That is reduction to these 2 forms is not iterative they can be done with a constant number of calculations. The reduction to other forms that is Jordan canonical form or diagonal form or triangular form they might need iterative iterations and there is a question of convergence. But these 2 forms though not canonical forms but they are useful important forms of matrices to which we try to reduce the matrix through similar transformations and these 2 forms are of advantage because the reduction to a tridiagonal form or Hessenberg form can be accomplished with a pre determined number of arithmetic operations a straight forward operation applied only once not iterative. So tridiagonal form as the name suggests has non zero entries only along 3 diagonals the main diagonal, the super diagonal and the sub diagonal and everything else is 0. So this is the tridiagonal form. The reduction to this form for any matrix is that is for those matrices for which we apply this is a matter of a fixed number of calculations depends only on the size of the matrix and there is no iteration involved there is no question of convergence. Similar situation is there with Hessenberg form in which other than the upper triangular matrix one sub diagonal is extra to this stage reduction can be done in a fixed number of arithmetic operations and after that to apply further similar transformations. So as to reduce these entries to 0 that may take a lot of iterations. So Hessenberg form is used typically for handling non symmetric matrices on the other hand for symmetric matrices we will typically try to handle it through tridiagonal form. Next we come to the most important topic of this lesson and that is the topic of Eigen value problems of symmetric matrices. Central to this issue is this very important result and that is that a real symmetric matrix has all real Eigen values and it is diagonalizable through an orthogonal similarity transformation. A similar result is there for complex Hermitian matrices for which this is actually a special case. Since in our course we will be mostly concerned with real matrices that is why in all the discussions I am trying to concentrate on the real versions of the theorems rather than going into the complex version. But the complex Hermitian matrices version is also very similar that would read as a Hermitian matrix has all real Eigen values and is diagonalizable through a unitary similarity transformation and no other change. The steps through which you establish that result is also similar to the one that we are going to go through right now. Now this one sentence actually has got built into itself several smaller statements. First is a real symmetric matrix has all real Eigen values that is first issue is that Eigen values must be all real. How do we convince ourselves of the truth of this statement? So, let us consider this itself as the independent proposition that is Eigen values of a real symmetric matrix must be real. For this what we do? We assume that there is a matrix A n by n matrix A which is symmetric that is for which A is equal to A transpose and its Eigen value is one of its Eigen values is lambda which is h plus i k. Now what do we need to prove? We need to prove that the Eigen value must be real that means this lambda must be real that means its imaginary part k should be 0. That means this is the hypothesis and this is what we want to establish what we want to show. So, what you do? You say that since A has an Eigen value which is lambda that will immediately mean that lambda i minus A is singular. Lambda i minus A into v equal to 0 that means it has an space in which there is a vector v and so on. So, lambda i minus A is singular. If lambda i minus A is singular if this matrix is singular then any other matrix multiplied to it the product will also be singular. So, this product is also singular that is B is singular. Now note what we have put here to multiply with lambda i minus A lambda bar i minus A because we want to establish something in terms of real quantities we want to kill whatever complex imaginary stuff is here and therefore we bring in the complex conjugate. Now use lambda is equal to h plus i k. So, if we insert lambda equal to h plus i k here and expand this. So, lambda is h plus i k lambda bar will be h minus i k.   So, that minus i square is 1. So, you get only this. Now we found that since A has an Eigen value lambda. So, lambda i minus A is singular and therefore B is also singular which is a product of this with something else this is also singular. If B is singular then it has an all space which has at least one vector in it that is if B is singular then there must be a vector to which B multiplies and gives 0. So, let us consider that vector as x some non zero vector x will be there to which B will multiply and give us 0 and then if we pre multiply both sides with x transpose that also will be 0 that will be a scalar 0. So, you get x transpose B x equal to 0 in this relationship in place of B we insert this and here at this insertion point we have used the symmetry of A x transpose this whole thing into x. So, the second part is very easy here k square is scalar and rest is x transpose identity into x that is it. The first one is x transpose h i minus A into h i minus A x the second h i minus A we have left as it is the first h i minus A we have replaced with its transpose this is valid because A is symmetric because identity is any way symmetric if A is also symmetric then h i minus A and its transpose is same. Now, note from here to here we have h i minus A multiplied with x and from here to here we have exactly its transpose that means we have got this fellows transpose multiplied with this fellow that means the norm square that is this part is norm square of h i minus A and this part is norm square of k. So, we have got this equal to 0 now you see norm is a positive quantity norm square is certainly a positive quantity now you have got the summation of two positive quantities or rather non negative quantities two non negative quantities is equal to 0 then since neither of them is like going to be negative. So, for the sum to be 0 each of them must be individually 0 and that means x is a non zero vector so k must be 0 and we come to the conclusion of our proof k is 0 which will mean that lambda is k. So, what does that show that shows that only due to the symmetry of the matrix which has been used at this step we come to the conclusion that k must be 0 that means lambda is real. Now, the first point of the proof is established other than this real Eigen value issue what else is there in the statement in the statement it is that the matrix is diagonalizable that means it has a full set of n Eigen vectors a complete set of Eigen vectors exists for a symmetric matrix we consider this statement separately a symmetric matrix possesses a complete set of Eigen vectors for that what we do is that we consider a repeated real Eigen value lambda of A because if all Eigen values are distinct then we already know that each distinct Eigen value will be associated with at least one Eigen vector which will immediately tell us that n Eigen values will give n distinct Eigen vectors. So, there is nothing to examine there. So, what we do is that we consider a repeated Eigen value which might create some trouble. So, we consider repeated real Eigen value of the matrix and examine its Jordan blocks what we want to establish we want to establish that all Jordan blocks will be of one by one size there will be no place to write that super diagonal one. So, if all Jordan blocks are of one by one size then the result will be a diagonal matrix. So, this is what we want to establish right. So, what we do is that corresponding to that Eigen value lambda suppose Eigen vector is v then we will have a v equal to lambda v and we try to find out we try to determine the first generalized Eigen vector w which must satisfy this relationship. Now, if it has to satisfy this relationship then by pre multiplying both sides of this relationship with v transpose that is by taking dot product or inner product of this with v we will get v transpose this whole thing is equal to v transpose v.  That is equal to this. Now, the left right side v transpose v is norm v square that is obvious here v transpose a transpose is the transpose of a v. So, I have got a v transpose w and here it is taken as it is, but then since v is Eigen vector corresponding to lambda Eigen value that will mean that a v is the same as lambda v and what is this? This is lambda v transpose w minus lambda v transpose w which is 0 and that means v norm square is 0. What does that mean? That means v is a 0 vector, but that cannot be the case because v is the Eigen vector for which linear independence is a must to qualify as Eigen vector. So, in which direction is the only information. So, you cannot have a I have an Eigen vector which is 0. So, this is absurd that means it is not only absurd it basically means that this gives rise to an inconsistency that means there will be no w no generalized Eigen vector which will satisfy this. That means that the Eigen vector will not admit any generalized Eigen vector that will mean that all Jordan blocks are of one by one size which means that it is diagonalizable. So, we have established two points from this complex statement as real symmetric matrix has all real Eigen values and it is diagonalizable, but till now the matrix S, matrix S related to the diagonalization the similar transformation matrix and be anything the further statement says that it is diagonalizable through an orthogonal similarity transformation that is in the diagonalization process we can use a matrix S which is orthogonal. That means the matrix S which houses the Eigen vectors should have all mutually orthogonal columns. Do the Eigen vectors have to be mutually orthogonal? We say that in two parts first we say that Eigen vectors corresponding to distinct Eigen values are necessarily orthogonal. So, here the proposition is that Eigen vectors of a symmetric matrix corresponding to distinct Eigen values unequal Eigen values are always orthogonal they must be orthogonal to show this.  So, for that we take a very simple means to establish this. We evaluate v 1 transpose a v 2 in two different ways. In the first case we simply take a v 2 as lambda 2 v 2 because lambda 2 and v 2 are the Eigen value Eigen vector pair.  In the second case in place of a we use a transpose that is the place where we use symmetry and in that case v 1 transpose a transpose is the transpose of a v 1, but a v 1 from here is lambda 1 into v 1 and that will tell us that this turns out to be lambda 1 into v 1 transpose v 2. Now note that the same expression evaluated in two different ways without utilizing symmetry and utilizing symmetry gives us two different expressions. So, we subtract on this side subtraction will give us 0 on this side what it will give it will give lambda 1 minus lambda 2 v 1 transpose v 2 that is 0. We have already taken the assumption that lambda 1 and lambda 2 are not equal that means this factor cannot be 0. So, only way this can happen is that this factor must be 0 that means v 1 and v 2 are necessarily orthogonal. So, this is the case for distinct Eigen values what will be the situation for equal Eigen values that is an Eigen value appearing twice giving us two Eigen vectors does do they also have to be necessarily orthogonal not necessary, but then see this what we want to see here we want to establish that corresponding to a repeated Eigen value of a symmetric matrix and appropriate number of orthogonal Eigen vectors can be selected what is the idea behind it. If lambda 1 and lambda 2 are same unlike this case then the entire subspace v 1 and v 2 is an Eigen space. So, if there are two vectors which are Eigen vectors corresponding to the same Eigen value then it is not necessary that they are orthogonal, but then the entire plane formed by them is an Eigen space that means any vector in that plane is an Eigen vector. So, if in a plane we have infinite possible Eigen vectors available then two mutually orthogonal Eigen vectors we can always pick up. So, that means we can select any two mutually orthogonal Eigen vectors for using in the basis that is for filling up the appropriate columns of this matrix s that is corresponding to repeated Eigen values orthogonal Eigen vectors are available the Eigen vectors that we pick up do not have to be orthogonal, but if we want we can always get orthogonal Eigen vectors. So, that is why it says is diagonalizable through an orthogonal similarity transformation that is it is possible to work out an orthogonal similarity transformation matrix with which we can diagonalize the matrix symmetric matrix further what we get. So, you see that in all cases of a symmetric matrix we can form an orthogonal matrix v such that in place of v inverse in this case we were writing s inverse now since we are talking about orthogonal matrix v in place of s. So, for orthogonal matrix v inverse is same as v transpose which is a lot easier v transpose a v is lambda which is a real diagonal matrix further try to pre multiply this equation with v and post multiply with v transpose then you get this. The matrix a can be represented in this manner v lambda v transpose where v is an orthogonal matrix and lambda is the diagonal form of a that diagonal form is always possible and that is always possible through an orthogonal similarity transformation v this gives us a lot of facilities. And this is greatly helpful because symmetric matrices appear in many many locations in the analysis in applied science and engineering and it helps that so nice properties. So, interesting and useful properties of symmetric matrices are there and symmetric matrices appear in most of the applications again and again. So, most of our problems are comparatively easy and the enormous amount of facilities that this representation gives us is here. First of all this expression a equal to v lambda v transpose can written in expanded form like this. If you try to multiply these three matrices and open it in the form of an expression you will find that you get lambda 1 v 1 v 1 transpose because you see if you first multiply these two then you will get what you will get lambda 1 into v 1 transpose plus 0 into this plus 0 into that and so on.   This gives rise to a further lot of possibilities. One is that if a particular matrix huge matrix 4000 by 4000 matrix has Eigen values which are organized in descending values descending absolute values the first 10 are large and compared to them the next ones are extremely small. Then what you can do for that for the storage of that 4000 by 4000 matrix you can simply store the first 10 Eigen values lambda 1 to lambda 10 and throw away lambda 11 to lambda 4000 and the corresponding first 10 Eigen vectors you store other Eigen vectors also you can throw away. Later when you need to reconstruct the matrix these 10 Eigen vectors with their corresponding Eigen values will be able to reconstruct the matrix completely by a summation of not 4000 items like this, but just 10 because the contribution of the rest of them will be extremely small. This is one advantage efficient storage with only large Eigen values and corresponding Eigen vectors rest of the things you do not have to store. In this lesson technique we have already seen the application of this expression that is if the and this works only for symmetric matrix. If the matrix is symmetric then there is a representation like this and after we have determined v 1 then from a if we subtract this then what remains has the same Eigen structure as a except that its Eigen value corresponding to Eigen vector v 1 turns out to be 0 rather than lambda 1. The rest of the Eigen values and Eigen vectors are unchanged that is the deflection technique which helps us in finding a few top Eigen vectors a few top Eigen values and corresponding Eigen vectors. Apart from that the orthogonal diagonalizability of the matrix in the case of symmetric matrices helps us in working out practical algorithms which are stable in which the numerical errors do not grow very fast as iterations proceed. Therefore, whenever there is a choice between applying a general similarity transformation and applying an orthogonal similarity transformation computationally we always apply we always prefer to apply orthogonal similarity transformation and in the case of symmetric matrices orthogonal similarity transformations alone suffice to reduce the matrix completely to diagonal form. In the case of non symmetric matrices first of all diagonalizability is not always guaranteed even when the matrix is diagonalizable reduction to diagonal form is not always possible through orthogonal similarity transformation in fact that is not possible. So, you have to take the help of similar transformations which are not orthogonal in the case of non symmetric matrices in the case of symmetric matrices you can conduct the entire operation with only orthogonal similarity transformation. This is why solution of symmetric matrices Eigen value problem is a lot easier compared to general non symmetric ones. The complete picture of different forms some the raw form and some the final desired form that we have we can see here and any block here in this schematic diagram which is on the right or which is below is typically easier to handle for the Eigen value problem compared to corresponding other blocks which are on the left or above. So, in that understanding we find that compared to a general matrix all other matrices general means which may be non symmetric all other forms are easier to handle and the diagonal matrix is the one in which the Eigen value problem is actually already solved. So, that means if we have a matrix in one of these forms on this side then any algorithm any part algorithm which helps us to move from this end to the right side or below south east side then that is one contribution to the solution of the algebraic Eigen value problem that is from the general matrix we might try to reduce it to either the Hessenberg form or the symmetric form. If we have a symmetric matrix then we try to reduce it to symmetric tri diagonal form. If we have it already in Hessenberg form then we try to reduce it to tri diagonal form which is comparatively easier or we will try to reduce it to triangular form which is more often the case from triangular or tri diagonal or symmetric tri diagonal another round of reduction will take us to diagonal form if a diagonal form exists for that matrix. In this case it will certainly exist and any movement along the arrows will mean that we have accomplished one more stage one more stage in the solution process of this Eigen value problem and all these steps all these reductions must be carried through similarly transformations only that is we must multiply the matrix A on the right side with one matrix and with the left side with its inverse then only it is a similar transformation and that will mean that it is basically the expression of the same linear transformation in a new basis the basis S. So, through similarly transformation only we must do all these transformations straight forward deduction like Gaussian elimination only from one side will damage the Eigen structure. So, the similarly transformations should be applied like this and through this any step is preferable if it helps in the direction like this like this or like this or anyway like this. So, the question arises how to find suitable similarly transformations which help us in moving from this direction to this direction in general that is reduction of the problem how to find suitable similarly transformations there are four standard ways of working out suitable similarly transformations they are based on rotation reflection matrix decomposition or factorization and element transformation. These four methods of finding suitable similarly transformations we will study in the coming lectures or coming lessons for the time being to summarize the important issues that we have seen in this discussion is that generally possible reduction which is possible for all matrices is up to Jordan's analytical form. Condition of diagonalizability and the diagonal form we have studied and we have studied the form which is triangular form which is possible with orthogonal similarity transformation. Note here that in the previous chapter in the book there is an exercise which gives you the steps necessary to show this important result to establish this important result that any square matrix can be reduced to a triangular form with only orthogonal similarity transformation. This is an important result and I will strongly advise that this particular exercise which gives you the steps to establish this important result you must go through. The other useful non-canonical forms are tridiagonal and Hessenberg forms that we have come across briefly and the most important lesson of this particular chapter of this particular lecture is that orthogonal diagonalization of symmetric matrices is always possible and all these reductions must be carried through similarity transformations only. So, I would also advise you to go through some of the exercises of this chapter before proceeding further to the next lecture. Thank you.


Virgo, this is Kelly from House of Virgo. Welcome back to my channel. If you're new, please do hit like and subscribe We're gonna take a look at general Little bit of general to see what other things are going on in your life because I know that you guys have Lives other than just love if love shows up here, we're definitely gonna take a look on that look at that I'm gonna try to touch on every aspect. So this is a channel for Virgos You don't have to have a Virgo son. You could be Virgo son or moon rising or Venus Okay, and there's a link below the video if you're new and you don't know where you can get personal readings from me I do offer them. All right, obviously the first two cards out a lustful fiery spirit There's hope as well here. So I feel like this is really your energy Virgo You are you've got some passion coming in here Something you're getting your hope back.  That's the message and I'm hearing don't give up. So if you are It's kind of like I Don't I It's sometimes it's difficult for me to describe the images that they're giving me but it's like when somebody Makes a trail of gun power powder and then lights it. That's you taken off here Explosive power victory. You're gonna be recognized for something some recognition people are going to They're gonna see you What you're doing and if you're in the spotlight or you've got something Whatever you're creating. I feel like this resonates a little bit with that reading that I gave recently Whatever you're creating, whatever you're channeling, whatever you're trying to do here Like you've got this passion this passion in the drive forget the talent It's the passion in the drive, which is what's gonna bring you victory and recognition here Okay, so whatever you have going on in your life. Just Virgo Keep it keep going towards it keep driving The night of wands energy is basically like someone who It's wheels in motion. It's the fire that sets them off and gets them It's motivation and I feel like once you get started on this then the fire heats up and then you go go go Okay, let's see what else is coming up here for you Virgo Okay, and your secrets are safe Okay, so there is definitely something here that I feel like you're not telling anybody about just yet But you're very focused on what you're doing Right now. I feel like it's a hush hush Look, you've got the black cat. So your luck will soon change. This is amazing This is amazing Virgo, whatever you're trying to accomplish here. It feels like Wow, I don't know career It's a creative endeavor if you are like listening to this and you're like, I'm not really sure what my creative endeavor is Excuse me catching a bit of a cold If you're not a hundred percent sure what that is, I feel like you're soon gonna come across this Right now. You might have a lot of things that you're trying to Improve upon in your life or things you're trying to creative or maybe you just don't ever feel like you're creative person But there's some type of an aha moment. There's some something that comes to you where you're like, that's it That's that that's the ticket. That's the way to go This is the thing that I have to do and then that's where that motivation and that fiery passionate spirit comes and just lifts you It's almost as if you you will be carried to your destiny here See what else is going on. That's crazy Okay, you have the spirit unseen forces watch over you. So that's what I'm saying. You're gonna be carried and lifted Through I don't feel like you're you're gonna be exhausted It's almost as if you're gonna be carried to your destination because it's you'll know Virgo You have to trust you'll know When this thing comes or maybe some of you already know when this idea this Concept when this this thing comes to you. It's all just gonna it's like you're gonna plug in and completely Get it. It's it could connect to something very spiritual is as well But there's unseen forces connected to this that are carrying you through Watching over you as you go through this It's almost like spirits in the driver's seat and you're in the passenger seat with all your ideas and you're directing the path But you don't have to be the one in the car to do it. It feels like you're gonna be carried to this The success whatever is going on in your life. I can't even believe it. It's amazing Okay, you've got the key locked doors can now be opened So this is what I'm talking about like this is so magical and it feels very positive I feel very happy for you. Whatever you've got going on in your life. Some of you Look I With the key, you know, it could be key to a new car or key to a home or key to something Okay to unlock something but symbolically it just feels more along the lines of It's go time. You're gonna know when it's time to go if you're in doubt That's because it's not time to do something yet, it's not time to go forward with Something it everything's just gonna it's like like the big bang everything's just gonna come together at the right time and have all of the ingredients and create all of the Necessary components and all the variables will come together and and it's like you're just gonna know when to move forward Let me take a look. I'm gonna see one more show Virgo one more card. What is this energy playing out here? Okay The Chimera this feels a little similar to the other reading where imagination clouds your judgment. Don't be distracted Don't be distracted once you get clear like right now. You might feel like oh, I don't know I'm not sure how can I take my thoughts and my dreams like the seven of cups bring it to reality You might still be kind of in that place right now where everything's up in the air Visions of grandeur. You don't know how you can make something real I Wouldn't even focus too much on that because I feel like all the elements like I said are just gonna play in together and everything's just gonna come together for you and So don't try to don't get lost too much in fantasies or daydreams too much because I just feel like that will take you away from your goal whatever you're trying Whatever's coming. It's just really feels more like a waste of time and of course you have the maiden So there is a new relationship showing up here for you in this time period in May So there is going to be a new relationship that also blossoms and this could be connected to something that you are Building or doing you could have Be partnering up with some someone in a creative endeavor or something coming together Okay, so that energy there is feels a little bit like the other day Also, okay, right when I was putting these cards away. They just called me to the caduceus here and Also the chalice so you will soon receive news from afar and Also rejuvenation and fulfillment. There's a blessing. This is the ace of cups in the two of cups energy here Definitely some type of partnership could it be romantic? Absolutely for the month of May This has to do with This news is coming either. It's coming from a person, but it's from far away Okay, it's not from where you currently are so it could come through a letter physical Like snail mail could be an email. It could be a phone call. It could be a text It's coming through by way of news Okay, virtually or Somebody delivers a piece of news to you could come from your mailman Okay, but this news is going to rejuvenate you Okay, whatever this energy is because that caduceus definitely represents some type of partnership healing It could have something to do with your health It could have something to do with like and they're just telling me medicine getting approved maybe from insurance as well, it could be for others Drinking from the The holy cup Getting rejuvenated some of you may be starting a juice business or Partnering up with some type of healing modality could be in medicine or holistic Medicine and again, as I said an ace of cups there could be news about a baby a marriage a proposal Okay, there's you're definitely going to be receiving some type of proposal that so with that with that cup here this kind of It's like it it's a key development on your emotional state Where it brings healing it's healing news that comes in so it makes you feel rejuvenated and Fulfilled you have fulfillment here. This is beautiful. This is beautiful. All right, let's get into the tarot and see what messages you have here See if we can extrapolate on some of these messages here for you so you've got some a Major Like aha, you're just right now.  Show us tarot messages Spare what does Virgo need to know these tarot messages? Okay? There you go. We've got the karma card here So this is definitely an awakening as I was saying This is judgment so clarity a hundred percent is coming to you Your life's purpose is going to be revealed to you And I feel I don't know if there is I don't think there's an eclipse in in May But there might be I could be wrong. I don't often follow the moon. I Know there's a full moon May 16th Hold on a second. Let me look and see No, I'm not looking at any I'm not seeing I'm not seeing any eclipses Okay Some of you when there was an eclipse you had an awakening about what you need to do I feel like sometime around the 20th of May It's something is that's around maybe that week And I feel like That's after the full moon. So um, no, yeah that week of the full moon. I Feel like whatever has been eclipsed in your life, whatever you haven't been able to understand or see Because actually this is a full moon Here and the Sun is behind it Okay, so The light it's gonna shine you're gonna understand so that karma card here This is like everything coming full circle for you Okay, you're gonna get what all the good that you've put out whatever has everything's coming back to you You know the old saying Karma returns what you put out you get in return. This is coming to you Around that week of May 20th, and I feel like that'll be like from May 15th through the 21st So that's like the third week of May things are really gonna open up. Let me see what else is coming Here for you. Okay, there is a connection to a queen of sorts Okay, this is a female. So this would be Libra Gemini Aquarius and This is an energy here of honesty and truth and self-awareness. I do feel like There is somebody here here that you'll have a conversation with in the month of May who's going to Help you make some kind of a finite of final decision. It's gonna help you with something, okay? This could be male or female energy brings truth and clarity So it's definitely going to give you some kind of wake-up call And I don't feel like it's negative at all. I feel like it's it's a revelation. I feel like it's some a Feeling of being born again. That's what that rejuvenation is So it's coming through as like a a huge wake-up and This queen of sorts male or female energy has something to do with this. Okay, let me see Okay, five of wands interesting so Okay So there's gonna be something here to do with challenges and obstacles Okay, maybe even some rivalry. All right. I feel like There's a message coming in here related to an Air sign Where the message is going to be despite the challenges Despite the obstacles this can work. This is going to happen This is I'm hearing the phrase crystal clear Something somebody makes something crystal clear about some type of challenges And That you like it's like you will be able to overcome them That's why you had victory and applause and recognition. This feels like this person's in a supportive energy Okay, supportive to you. Let me see here. What is this karma? Okay. Yeah, so you have the Sun so Scorpio Leo Aquarius Gemini and Libra So the Sun is like exposure. This is positive energy on the karma card. Okay, again, this is success Whatever this thing is that you have gone on in your life that you are seeking some kind of success or a joyous outcome That aha moment that whole situation is gonna come to fruition in May Third week, perhaps if it's not already happening for you now I'm just hearing So I did say the 19th out of the 20th so like the 19th in the 20th I feel like with the Sun light is going to shine That's what I was seeing here. Look Because the light the Sun is here behind the moon and it's illuminating the moon and that full moon energy Where the full moon Will be on May 16th that full moon energy. It's going to I guess I want to say It feels like everything that has been unseen or that hasn't been clear that your intuition has been telling you But you haven't been able to See something manifest. That's going to get exposed. Whatever was unseen is now going to come out into the open That's where the clarity comes All right, let's take a look and see what else is here Okay, and there's that night of ones again, right?   Let me see what else is going on here I love this reading so far. There you go. So this is some kind of financial opportunity Definitely or golden opportunity doesn't have to be financial, but a golden opportunity that is coming to you Okay, that's Like physical it's something you can touch something you can feel it could be Money coming to you. It could be a gift of some sort that you're going after I don't feel like you're chasing material things. You might be but for some of you this feels mostly like an Opportunity is is actually going to present itself for you to be able to Charge And to also be and to to be to freely charge to be able to get where you need to go freely You'll have your passions are gonna kick up is what I feel like is going on here.  I'm trying to turn up my I Think I have my volume up as loud as as loud as I could have it up Hopefully that's hopefully that's loud enough, but it feels like you're gonna be given It feels like an opportunity to be able to move forward the way you want to move forward To have the opportunity to go in the direction of your dreams and your plans. That's what I feel like is happening here So there's a gift of some sort being offered to you Okay, let me see So definitely receiving some kind look at that Queen of Pentacles. There you go. That's you. That's you receiving that gift here Okay, your abundance You're going to have the opportunity the resources will be provided to you to go towards the direction of Whatever you're trying to create or whatever you're trying to do here Okay, now I will say that we did see a little bit of love there. So let's jump in and see What that, you know, the caduceus Which is two of cups and the ace of cups. Let's take a look and see what that energy was Playing out here. What can what can spirit tell us about the love energy that's coming in? Love energy for Virgo. What is this love energy coming in here for Virgo? This feels like a new beginning Or something that's going to make you make your heart hitter-patter Hitter-patter love Okay, hold on a sec. All right, let's see. Tell us about this new love energy coming in here For Virgo Oh, look at that. Okay, you have the six of cups. So there is a past reconciliation Happening here with someone where something was very lovely and very nice The six of cups is definitely an energy of nostalgia Gifts again could be related to what I was just saying. Somebody could be bringing you a gift sincerity Innocence reunion Okay Harmony Appreciation loyalty Good memories someone that you had good memories with in the past Lovely memories sweet enjoyable playful innocent memories with this is a return of a past Let me see here Wow Temperance so you have Sagittarius energy. So this is some Things coming into balance here with someone from the past situation moderation This is this is a very nice temperament here that is showing up here this connection showing up very well balanced and equal and Like these are just such beautiful cards here Oh, I just some of you might have connected with someone Around Easter I feel there might have been a connection. Some of you have children coming in to your past Are coming back around children coming home. I'm hearing Okay, we're really going to get it. I'm going to take a look at this here This is reminiscing as well About your children and if you have children that is Or about when you were a child and how I feel like How easy or Balanced things were this is also thinking about a sibling You're they're saying if one of your siblings has passed on There's an angel there watching over Wow, just when I said that the star look how beautiful these cards are here So now we have aquarium energy. There's a lot of healing Coming here some of you so you know the star generally means renewal hope Inter clarity a miracle Having faith being in the spotlight so there's definitely um, I feel like there's a someone from your past returning or um Coming into your life where there's an even Even tempered even maybe somebody's name is steve or steven or steve steven watching over you But I feel like there's a healing. This is like a miracle. This is like beautiful This is like a miracle. It's a miracle. I'm hearing someone from your your past returning to your life Okay, or um, so when I say it it could be love romantic it could be um A family member it could be a friend a childhood friend Or someone that you knew or maybe even someone that you had an interest in when you were younger And now there's kind of like a reunion happening here in may between two people and you know, it's kind of like A fondness for each other gets discovered and it's almost like a miracle for some of you. I feel like Maybe somebody you knew a long time ago and It something never came to fruition. I feel like it's coming to fruition in may Look at that Have the ace of cups again right down here and also The emperor so aries energy here Okay, for some of you you you do have a father figure coming back into your life Um or an aries coming into your life For some of you there's um rekindling and a healing With a with maybe a spouse For others, um, there's a proposal with someone from your from your past. There's marriage on the on the table There's new love blossoming here for some there's a pregnancy. Somebody's having twins or there's another child Maybe there's a grandchild Okay, um, there's some information.  I'm hearing for some Or two two babies An even tempered Child Now like in other words like a pregnancy of some sort is going to go well or reconnecting with a grandparent or grandchildren This is amazing Hold on Some of you are Healing a past relationship With someone that you had children with Um there for some it's just What if there was any turmoil or struggle with them that's over with Okay, and now there's like civility and just reminiscing about when times we're good I'm being able to be friendly with one another Let's see what else they have here for you Okay, so here you have strength Look at all this major Akana. So leo energy here um I I'm getting a sense of like unconditional love here is coming in Somebody loving you or it loves you unconditionally for others if you don't have children with someone in particular at all Um, this is just somebody here coming forward who sits in the position of the emperor where this person is Very well respected mature Likeable um this might be somebody From your past this could be somebody that you knew that's been thinking about you and tries to reconnect with you again Has the confidence reaches out This person has very deep unconditional love for you and a lot of um, you know Respect for you Okay If this is your Listen, remember I said it's a miracle. Okay, because I know some of you like that's No way, but I'm telling you it's a miracle. This is Completely unexpected what's coming through here from somebody with this love connection Okay, the ace of cups twice and then you had the caduceus So it's a healing. Yeah, it's going to be a shock actually It's going to be a complete shock Okay, things because things are changing. It's a new chapter that's starting here with somebody Okay, this person's coming in to talk to you. Absolutely Um, very curious about you. How have you been? What have you done with your life? Um How have things gone? What have you been up to the past 30 20 10 years? Five years whatever it might be um Or however long if you knew this person in the past So there's definitely a lot of communication That's going to come about but this begins. There's like a new love start happening here I mean it it could be connected to like I was seeing what's going on here some type of partnership or proposal Um, it just feels to me like you guys have a lot of good stuff coming up, you know May is spring obviously the flowers are blooming. It's new beginnings Happening here for you guys We also have um Taurus energy I'm picking up look I shuffled again And there's a night of wands yet again. So it's both things, you know Whatever creativity or things that you're trying to start in your life that are you know coming up for you I 100 feel like you're going to It's like you're going to be injected with this this positive energy Going forward like things are going to get moving That lusty fiery passionate spirit comes back to you and It's also it's like hitting everybody, you know, and I feel like also with this person Whoever this person is Whatever was broken down before Uh, if there's a past history with this person, it's an opportunity to rebuild again Just as your life is also You're having the opportunity to rebuild again that aha moment where you everything I feel like everything is just coming together. The enthusiasm Is off the charts here with somebody and in your life It's all it like the energy is touching down in every area of your life here Let me see here. Okay. So there could be a proposal a reunion of some sort Happening here with some of you. There's like a brand new beginning Coming in and I love that. Let's get some more specific messages And see What else does you know what I'm going to pull A couple of different cards here The linormant What else does Virgo need to know this energy coming through here? So Okay, of course you have malady Yeah, so you might be in a position right now where you're just kind of like, uh, I haven't been feeling well Maybe you're not feeling well And I think I mentioned before make sure you're taking good Why can't you see this make sure you're taking really good care of yourself if you're feeling sick You know or if you have a cold or anything like that make sure you're taking really good care of yourself Because you've got a surprise you've this everything that's coming in here for you big surprise This is going to be like what you're out there fishing All you're trying to catch is a couple fish and you pull up some clams with pearls in it This is what I feel like is going on Whoa, wait a minute Some of you are going to be getting, um, okay, they're saying $6,000 $6,000 is coming to you and you're not expecting that Okay, um There might be some of you that are needing to You're ill and you know, maybe you're ill because you don't have enough money or Um to get well. I feel like some type of money is going to come in as a surprise Okay, that's going to help you Get in a better physical mental state Your mental might be affecting your physical or your physicals affecting your mental.  I think I said that in the beginning Okay, look you've got love keep its arrows I feel like you know, it's funny because I said, okay Well, let's try to take a break from love readings and let's just do some general readings And if love shows up then we'll touch on that love has just been showing up all over your reading So of course can't get away from it. Um So a moray there's a moray here Okay, you've got some love coming in this is like the third time now So for some of you if you're not feeling well, absolutely do some self love self care kind of energy. Okay, don't skip Your vitamins don't skip your healthy eating. Um, if you're feeling low Just get some rest. Don't skip out on your sleep Okay, you definitely are going to be getting like a lump sum coming in It could be a maybe it's tax return or Some unfound money. I feel like this is going to be Money that's coming to you that you you are not even expecting. Maybe you're expecting some but not this amount This is like extra and also um cupid's arrows 100 percent going to be biting Okay, you've got room Okay, let's see Room Okay, they're just saying all right Hold on a second. Oh, okay. I'm seeing you Oh, they're saying the love room or room for love I feel like you're you're gonna be in a room. It's a hotel room. I see you in a hotel room in may With these big windows a hotel room a love room with a lover hotel room with a lover It's definitely but it feels like a um It for some of you it's a b&b a bed and breakfast kind of thing A bed and breakfast I'm just seeing um Some of you are visiting um Grandmom's old house or something Okay falseness Hypocrisy. All right What is going on here with hypocrisy? Oh So, okay the message here.  Um, so there's it's a leo or it's a person who um And this is a very specific message They tell you that their cat does something but it's not true about their cat It could be like the cat's sick or something's going on with a cat Maybe it's a vet. They try to dupe you out of money. I'm not sure but um, there's falseness here that's attached to Someone that has a cat um Or a story somebody okay if if you've lost your cat. I'm just hearing and Somebody tells you that they haven't seen your cat, but they actually do have your cat That's a specific message for somebody I don't know who this person is but They they do have your cat Okay These are weird messages Okay, so you have oh, yeah enemy.  They it's hyperbole.  This feels like someone who um, this is like involving a friendship or a coworker Okay Vanity i'm just hearing somebody who's kind of mean So just watch out for that energy someone who's kind of vain Okay, here comes the money card, right?  Oh look, okay Hmm Okay, you have lord that's landlord so So if initially when I had the room card I felt like initially that somebody's either going to be able to rent a room Of some sort or maybe buy a hotel. There's landlord energy here So for the cat, I do feel like if you're trying to move somewhere or rent a place and they say no pets That actually there's a legal stipulation with that like you you can you're legally allowed to have your pet I'm hearing but this landlord is not is saying that you can't but you can So there may be some type of um, you know Maybe you pay money or something or they they try to Swindle you or like in And try to charge you for like a pet deposit or something and then they're not legally allowed to do that I'm just hearing So there might be a dispute with a landlord and also, um, you might become a landlord you might be um, or becoming Able to sell a home Okay That might be where all the money's coming from. So if you have a home listed on the market or you think it about selling You'll you'll get it and if you're thinking about buying you're going to be approved to buy This is what i'm hearing. Let me see what else is going on here. Look. There you go house And you can't see it. Sorry house hotel So you're definitely going to be it. Yep. That's it. Yep. Yep That house is coming through you're going to be able to get that You're going to be purchasing or selling your home Um a home Okay, or a hotel or a commercial property Definitely coming in here for you with a lot. Remember I was saying I see you in a hotel room It could have been a love room for some of you. Um Piece of commercial property I'm hearing Okay, let me see here So yep, there's the reunion card. Like I said, there's definitely a reunion connected to love coming in here Where you are going to be in like some kind of room like a lover's room with somebody here Maybe a room of your house Okay, or a room of somebody's house Playing out here. Let me get the rest of these cards because I see There's something okay doctor's office um, I see you Going to being in a doctor's room or a doctor's office here In the month of may for some kind of illness. I don't feel like it's anything serious um Something to do also with a prison doctor or maybe somebody's a doctor in a prison here Okay, um Okay, some of you are um reuniting with someone who's um a physician Could be like a lover or a friend or your or you could be a physician You're going to some type of a reunion Or maybe graduation from like whenever ago or you're going to some kind of um You get invited to a party a house party of some sort or an event in a hotel For doctors or physicians. Okay, your landlord's going to prison. I'm gonna tell you that right now um But you got to be really careful with money um If you become a landlord as well or if you're going to buy property Or own property or sell just be very careful With somebody in particular in the month of may Right because somebody's coming through is being false. They're not being Truthful so there may be like a money issue or a legal or a suit Lawsuit or a sell settlement or something like that Coming in here also be careful.  They're saying just a pet a pet's illness Of some sort it might come as a little bit of surprise like maybe they They eat something or something gets lodged. I'm not seeing death here. So don't worry about that Okay, um Also, if you're if you're feeling ill you may be feeling ill as I was saying Um, you'll be surprised somebody's going to come from your past And take care of you Okay Could be a friend. It could be a lover.  There's somebody's going to be by your side Helping you. Um, you're not going to expect this. I mean giving a lot of love love medicine To help you when you're ill. It's very beautiful energy Also be careful at some kind of reunion or an event that you go to um you have There might be somebody around you who's like, um Not someone you would normally commingle with they're kind of there's a jealousy Some type of a jealous energy attached to somebody here That if it's like some kind of school reunion or something along those lines, there's somebody there Who um, isn't is a false friend Okay, so you want to keep your eye out for that? I feel like you're going to be sitting Like at a table or if it's like a banquet or some kind of an event Maybe it's a wedding or family reunion or something you know Try not to sit with that person. I'm just and if you have to just you know Just try to keep your communication with them very minimal not to give too much information Okay, also um question the doctor. They're saying get a second opinion They're saying get a second opinion Okay, um I'm trying to think now. Do I want to do any other messages I think that's enough for may so let's see how this goes If this if this reading if you guys like this reading and you feel like we can continue doing this because it's Pretty general, but it touches on all areas if you want to continue with this Um, just leave me a comment if you like it and I'll just do this We'll do this for a little bit and see how it goes All right, you guys so it's more predictive. Hopefully you enjoyed that. Let me know And if I want to incorporate more cards, I will in the next readings. All right, you guys Love you. I will see you all tomorrow tomorrow's reading. I'm going to do and that will be for the Um, I'm just gonna do them Monday through Friday and so I'm gonna leave you the weekend reading tomorrow All right, you guys love you. Don't forget to hit subscribe. I'll talk to you soon. Bye