Good morning. In the previous lecture, we studied one of the four methods to work out suitable similarity transformations for solving the Eigen value problem. That was through plane rotation. Today, in this lecture, we considered the second method to work out suitable similarity transformation. This is also related to geometrical ideas and this particular method is based on reflection. So, today we will study householder transformation and tri-diagonal matrices. First, we considered the householder reflection transformation in a geometric sense and then we work out how to find the householder method for tri-diagonalizing a given symmetric matrix. And next, we see what to do with that resulting tri-diagonal matrix, symmetric tri-diagonal matrix. So, first consider this. In a k dimensional space, consider two vectors u and v in this k dimensional space, both having the same magnitude. If u and v have the same magnitude like this, then we proceed to find this particular vector w, which is the unit vector along the direction of difference of u and v. That is, u is this, v is this. This vector is the difference u minus v and we divide it with its magnitude to find the unit vector w in this direction. Now, this small vector w, unit vector in this direction is perpendicular or orthogonal to this plane or hyper plane that actually bisects the angle between these two rays showing the two vectors. Now, with this w in hand, let us construct this matrix h k, which is k by k matrix formed by subtracting the matrix twice w w transpose from the identity matrix. This matrix is called the householder reflection matrix. This has a lot of interesting properties. This is a matrix, which is symmetric and orthogonal at the same time. Symmetry is easy to see here, because identity is anyway symmetric and w w transpose is symmetric. So, this difference is symmetric. To check for orthogonality, we can find out whether h k transpose h k is identity. To see that, it is actually quite simple h k transpose h k and symmetry we have already confirmed, we have already verified. So, in place of h k transpose h k, we can simply write h k. So, h k into h k identity minus twice w w transpose into the same thing. As we open this product, we get i into i that is identity minus i into twice w w transpose and again twice w w transpose into identity. So, total 4 w w transpose minus minus plus 2 into 2 4 and we get w w transpose w w transpose. Since matrix multiplication is associative, so whichever order you multiply these 4, it does not matter. So, if you multiply in this first, then you find that w transpose w is unity, because w is a unit vector. Then what remains 4 w w transpose, which is the same as this. So, that will mean that h k transpose h k is identity that establishes the orthogonality of this householder reflection matrix. Now, what does this symmetric and orthogonal matrix do? Why it is called reflection matrix? You see that consider its action on two vectors, one along w and the other perpendicular to w or orthogonal to w. Orthogonal to w will mean a vector in this plane, which is shown here as plane of reflection. So, take any vector x, which is orthogonal to w that means, which is in this plane. So, when you apply it, apply h k on x, you find that i k minus twice w w transpose x, what you will get? Identity into x is x and this one as you open, you will find first w transpose x and from the very definition of x being orthogonal to w, w transpose x will be 0. So, what will remain? Identity into x, which is x that means, a vector orthogonal to w that is in the plane of reflection gets mapped to the same vector itself, there is no change. On the other hand, how does w itself gets mapped? Get mapped h k w, as you apply this on w, you find that identity into w will give you w, but this fellow will give you twice w, w transpose w is 1. So, you will get w minus twice w that is minus w that means, w itself when operated upon by h k gets mapped to its negative and the vector in the plane, plane of reflection gets mapped to itself. That is the way a reflection takes place, this plane of reflection operates like a mirror. So, if there is any other vector, which has some component on the plane and some component perpendicular to it, then we can consider it like this. Applying h k over y, which has 2 components along w, which is perpendicular to the plane and perpendicular to w, which is along the plane. The component, which is along w gets mapped to its negative and the perpendicular one remains as it is. So, this is typically the action of a mirror reflection. So, this is why this matrix is called a householder reflection matrix. In particular, it will map u to v and v to u, because they are mirror images of each other with this plane as the mirror or plane of reflection. Now, this concept and this particular matrix, how do we utilize in reducing the symmetric matrix to a form more suitable for solution of the Eigen value problem. In this case, we will try to make it tridiagonal. How do we use that? So, that brings us to the point of householder method. Consider an n by n symmetric matrix and so on. Symmetry is shown here a to 1, a to 1 and so on. Now, take u in that context, in the reflection context, take this u to be the vector a to 1 to a n 1. This transpose makes this row vector a column vector through transposition. Now, if the matrix is n by n, then this vector u is an n minus 1 dimensional vector, because the top entry a 1 1 we have left out and that vector u starts from a 2 1. That means, this vector this much is taken as u and then v is taken to be a vector of the same dimension, but it is having a first entry which is the same as norm of u. So, whatever is the norm of this, that turns out to be the first entry of v, all other entries of v are 0. That means, the top entry of v will be the norm of this vector from a to 1 to a n 1 and all the other n minus 2 entries of v will be 0. So, like that we construct the vector v.  The householder matrix h k in this case k is n minus 1. So, it will be an n minus 1 by n minus 1 matrix, h n minus 1 we call it. Then out of that h n minus 1 which is an n minus 1 by n minus 1 matrix, we develop this larger matrix by inserting a 1 here a 0 row above and 0 column on the left of this h. So, this is p 1. Now, p 1 is its own transpose because it is symmetric and it is orthogonal also. So, then what we do is that we apply this orthogonal similarity transformation p 1 transpose a p 1, since p 1 transpose is same as p 1. So, we have just written p 1.   This multiplication you can see immediately we have got p 1 0. So, 0 row 0 column and h n plus 1 n minus 1 here.  Now, as you conduct the block operations in again only problem solution methodologies you will quite often come across this block operations. So, first keep this as it is and we multiply these two a 1 1 scalar into 1 plus u transpose row vector into 0 column vector that is 0. So, you get a 1 1 next a 1 1 into 0 row you get a 0 row plus u transpose into this. What is u transpose into this? That will be the transpose of this. So, u transpose into this will be the transpose of this and what is this? This matrix is its own transpose. So, this is simply h n minus 1 u and through the property of the householder reflection matrix that we have seen just now this is nothing but v. So, therefore, u transpose this will be v transpose. So, you get v transpose here. Next the lower row block u column vector into 1 that gives you u only plus a 1 into 0 that is 0 you get u. Finally, this big block this is a scalar this is a row vector this is a column vector. Now, we have got the trailing n minus 1 by n minus 1 matrix here column vector u into row vector 0 that is a 0 matrix plus a 1 into h we write it. Finally, this multiplication 1 into a 1 scalar a 1 plus 0 row into u column that is 0. So, you get a 1 1 a 1 1. Next 1 into v transpose row vector v transpose plus 0 into whatever. So, you get v transpose here 0 into a 1 1 that is 0 that is a column vector plus h n minus 1 u that is v. So, you get v here and here you will get 0 into v transpose that is 0 plus h into a 1 into h. So, what you have got you have got in the first column you have got a 1 1 and then the vector v. Similarly, in the first row you have got a 1 1 and then v transpose and what is the structure of v that we started with first entry of v is full size of u and all the other entries of v are 0. That means below the second entry from the top everything else will be 0. So, that is what you get here. So, now we rename see in the whole process a 1 1 has remained unchanged a 1 1 has not been operated upon by anything because the first column and first row of p 1 is same as identity. So, a 1 1 has been left unchanged. Now, we rename a 1 1 as d 1 and whatever is the first entry of v we name it as e 2 below which everything else is 0 and out of symmetry that same e 2 will be sitting here on the right of which there will be all 0s and this 2 2 diagonal entry we now call d 2. In the next step this block will remain unchanged the way in the first step this remained unchanged a 1 1 in the second step this much will remain unchanged. What we do in the second step in the first column below the top 2 entries everything else has become 0. Next round in the second column below the top 3 entries we want to make everything 0 this is the process to make it tri diagonal. So, what we consider is that below the 2 top entries whatever is the vector sitting we call that u 2 and then construct a similar v 2 which has the same magnitude as u 2 and all but the first entry all but the first all the other entries are 0 and that size of the matrix vector u 2 and v 2 is n minus 2. Then we construct the next household transformation matrix of size n minus 2 by n minus 2 and enhance it with an identity matrix of size 2 by 2 here equivalent number of 0s here and here to complete the size then apply that on the previous result this and this will keep unchanged that leading 2 by 2 block of a 1 and you will get the next step which will have this much d 1 d 2 d 3 e 2 e 3 in the correct places and the first 2 columns and the first 2 rows have been made process up to the extent that below the sub diagonal and above the super diagonal we have got 0s in those first 2 columns and rows. Like that we keep on conducting steps with smaller and smaller householder matrices in the trailing part and the leading part we will have the identity matrices of gradually increasing size. After just such steps till this point it has been converted to tri diagonal and remaining fellows are full and as we go on conducting this kind of steps at the end of n minus 2 steps we will have this complete transformation p 1 to p n minus 2 which will result in a completely symmetric tri diagonal matrix which will look like this. Let us see a quick example is a 5 by 5 matrix now this part is what we called a 1 there. Now, in order to reduce it to a symmetric tri diagonal form we would like to have first 3 0s in these locations. So, we take u as this vector 4 1 2 1 and we want v in which the last 3 entries are 0 and what is the first entry? First entry is the size of this vector u. So, what is that size this square plus this square plus this square plus this square. So, we will have 16 17 18 and 4 22 so root 22. So, this will become our v and with this u and this v it is easy to find out w the difference of u and v and divided by whatever is its magnitude. So, with that we find w and then we work out twice w w transpose subtract it from identity and that matrix is our 4 by 4 householder transformation matrix that 4 by 4 matrix will be sitting here let us call it as 4 0s here 0s here when this matrix is multiplied on this side and this side to this matrix. Then the transformation that will take place will make this root 22 0 0 0 similarly here root 22 0 0 0. Then we will find that this much is secured and then whatever is here this 3 dimensional vector will be then taken as the next u and the next v will be taken as something 0 0 that something will be the size of this and then through that similar process in which the householder transformation matrix in this case will be I 2 H 3 0 0 this 0 matrix is of size 3 by 2 this is of size 2 by 3 and so on. When this is multiplied on this side as well as on this side you will get something here 0s here the third step here will make this as 0 and whatever happens on this side will happen on this side also. So, you get a symmetric tridiagonal matrix like this now the question is that after we have reduced the matrix to this symmetric tridiagonal form what do we do with it that is is the solution of the Eigen value problem of a symmetric tridiagonal matrix anyway simpler compared to the original symmetric matrix answer is yes. There are several ways one can handle this kind of symmetric tridiagonal matrices one way we consider now and the other way we will consider in the next lecture. There is a very interesting piece of theory which tells you how to work out the characteristic polynomials of sub matrices of this that is leading one by one sub matrix leading 2 by 2 sub matrix leading 3 by 3 sub matrix and form a sequence out of these characteristic polynomials. And then try to solve the Eigen value problem based on those interesting properties. So, what will be the characteristic polynomial of this so for that we have to find out the determinant of lambda i minus t. So, this is the characteristic polynomial lambda minus d 1 lambda minus d 2 etcetera sitting in the diagonal places and minus e 2 minus e 3 etcetera sitting in the off diagonal places note that d is indexed from 1 to n and e the sub diagonal and super diagonal entries which is one less in number they are indexed starting from 2. It could have been indexed as e 1 to e n minus 1 that would be equivalent to this, but in this analysis we have indexed it from e 2 to e n. So, there is nothing called e 1 in this analysis fine. So, with this characteristic polynomial you find that the characteristic polynomial of this leading one by one part is simply lambda minus d 1. So, we call it p 1 p 1 of lambda that is the characteristic polynomial of the leading one by one sub matrix from t. So, we call it p 1 that is simply lambda minus e 1. So, then for the leading 2 by 2 sub matrix we have got the characteristic polynomial from here lambda minus d 2 into lambda minus d 1 minus e 2 square. In this place lambda minus d 1 can we simply put p 1 p 1 of lambda we can. So, we write this similarly we can work out p 3 p 4 etcetera, but let us go one large step and try to determine p k plus 1 lambda in terms of p k lambda and p k minus 1 lambda. So, that will establish a recursion among all these characteristic polynomials of the leading sub matrices. So, as we try to do that let us write down here we are going to write down here the same matrix appearing there, but not up to all the way to lambda minus d n, but up to lambda minus d k plus 1. When you try to expand this determinant from this column what we find we find it is lambda minus d k plus 1 into this determinant minus this thing into the determinant that we will find by crossing out this row and this column. So, let us do that and all other entries are 0 right.  Then plus this thing which is minus e k plus 1 into p k plus 1 minus e k plus 1 into something we try to find out that something what will be that something that something will be the determinant found by removing this row and this column. So, let us do that remaining thing will no longer be this. So, we will be removing this column and this row this determinant should be sitting here and see its diagonal entries are lambda minus d 1 lambda minus d 2 lambda minus d 3 etcetera up to lambda minus d k minus 1 and then next because this guy has taken this place actually after removal of this row and in this row other than this entry everything else is 0. So, that means the determinant that we are asking for is this into this determinant right and this is minus e k plus 1 which earlier that minus minus sign that plus we have we have not made it plus. So, that minus will actually now help and because this minus this is remains minus finally and e k plus 1 comes once more. So, it is square now and what else is here that is p k minus 1 that is the characteristic polynomial of the matrix of one further order less and now what we do further other than this. Now, this relationship this recursive relationship will define up to p n in terms of the older one. So, p 3 will get defined in terms of these two p 4 will get defined in terms of p 2 and p 3 and so on through this relationship at the top we also put a dummy element in this sequence in order to complete the sequence and that is one. So, then we will say this will have 0 roots no roots this will have one root which is d 1 this will have two roots which is we can find out what are those two roots and so on. So, finally, p n will have n roots as we construct this sequence then this sequence has some interesting properties. These expressions or rather this expression this recursive expression which is here helps us in evaluating these polynomials extremely fast. Other than that this sequence of these polynomials of increasing degree has further properties they in particular they have the property called a Sturmean sequence property that property they have if all the e j's all the sub diagonal and super diagonal entries are non 0. In that case this sequence p 0 p 1 p 2 all these polynomials the sequence of all these polynomials has an interesting property. Now, our rest of the process will directly depend on that property, but before that we need to ascertain what we should do if there is some e j which is 0 then that is actually for some j j say e j is 0 some of the sub diagonal and super diagonal entries turns out to be 0 that is actually a good news because in that case we can split the matrix we have 1 d 2 etcetera up to d n we have e 2 e 3 etcetera up to e n. Now, if there is a other things are already 0's if there is some e which is 0 here as well as here then this is actually going to obstruct us from using this succeeding formulation for the complete matrix, but these 2 0's will actually help us in splitting the matrix into 2 parts because then we will have the complete matrix in the form of a block diagonal matrix with these 2 0's sitting here. Earlier if we had these as non 0 then it was such a huge long matrix large matrix n by n. Now, these 2 0's here will decouple the 2 sub spaces completely and we will actually have this as a block diagonal matrix this is one block and this is another block. So, whenever we have e j equal to 0 at that location we can always split the matrix into smaller block such that we can consider each block separately. So, having some e j as 0 helps us in splitting the matrix into small matrices until each such block has non 0 e j's all through. So, we can consider only those cases which have non 0 entries here for which the rest of the theory holds. Now, what is that particular property? The 2 million sequence property says that roots of p k plus 1 interlaces roots of p k. What is that? That means if you have roots of p k sitting at locations 1 5 7 9 then p k plus 1 say 1 5 7 9. If these are the roots of p k then the next one p k plus 1 which has 1 more root that is 5 roots say p 4 has these as roots p 5 which has 5 roots will certainly have 1 root below 1 another root between these 2 another root between these 2 another root between these 2 and the 5th root above 9.  So, this is the interlacing property which is shown mathematically like this and this property leads to a convenient procedure for finding the eigenvalues. Now, I will skip the proof of this particular property, but I will just give you the line of proof and strongly advise you that in the textbook you go through the proof in the textbook or in these slides which are available in the internet. You should go through the proof quite carefully because the proof has a has an inherent beauty in it. So, the line of the proof is as follows first we consider the case of k equal to 1 that is is the statement true for k equal to 1 and that is trivially true because there is only one root and nothing is there to interlace in the case of 2 then you verify this. So, the statement is true for k in the sense that roots of p 2 interlace the roots of p 1. So, the first entry d 1 is interlaced by the eigenvalues of the leading 2 by 2 matrix d 1 e 2 e 2 d 2 this you verify that shows that the statement is true for k equal to 1. Next you assume that the statement is true for k equal to i then you denote the roots of p i as alphas roots of p i plus 1 as betas and roots of p i plus 2 as gamma as you assume the statement to be true for k equal to i you assume this that is the betas interlace the alphas that is the i plus 1 betas will interlace the i alphas and in the number line you can show the alphas with crosses and betas as bars and the picture looks like this then you need to show that in turn gammas will then interlace the betas i plus 2 gammas will interlace i plus 1 betas and that you establish based on this recurrence relation and changes of signs in the roots of the succeeding polynomials. So, rest of the proof I will omit here in the class, but I strongly suggest you that you go through the proof a little carefully we go rather directly to the procedure. We examine this sequence p 0, p 1, p 2, p 3 up to p n for different values of w for a particular value of w if we know that the p k and p k plus 1 and p k minus 1 will have their locations of roots in this manner see one question we are never raising that whether the roots are real or not that question we are never raising because the matrix is symmetric. So, all the roots are real that is anyway known all the eigenvalues are real that is anyway known. So, if p k has this kind of relationship with p k minus 1 and p k minus p k plus 1 their roots then one thing is very clear if p k w and p k plus 1 w have opposite signs then the number of their roots above w can differ by just one. Why because if this sign that is in this suppose w falls here and p k w has a certain sign and the at the same w p k plus 1 has a sign different from that that will mean that above that value above that w whatever is the number of roots of this and the number of roots of this can differ at most by one because at infinity all the p k's will have plus infinity value infinity will have plus infinity value infinity minus something into infinity minus something into infinity minus something and so on. So, at infinity that all of these polynomials will evaluate to plus infinity so all of them are positive. So, the moment one root is encountered the sign changes for each of them. So, it is impossible that one of them encounters too many roots and the other the next one has not encountered any root because of this interdacing property. So, p k and p k plus 1 to succeeding ones to continuous ones in this sequence having opposite signs will mean that the higher one has one root more than the lower one above w. Now, we find that number of roots of p n above w will be number of sign changes in the sequence from this end to that end because as many sign changes if compared to p n p n minus 1 does not change sign that means p n and p n minus 1 will have the same number of roots above w and then from p n minus 1 to p n minus 2 if there is a sign change then we will know that for p n minus 2 one root less and so on. So, in this entire sequence the number of sign changes at w will tell us the number of roots of p n above w. So, p 0 has no root. So, number of sign changes will tell you at the end how many roots this guy has above w. Now, if we do this operation at w equal to a and then w equal to b then above a how many roots p n has and above b how many roots p n has. The difference of the two numbers will tell us how many roots p n has in this interval a b if at a particular value in this entire examination in this entire investigation the p n turns out to be 0 we know that that value is the root is a root. So, after closing like this that is how many roots in the interval a b we can consider a plus b by 2 and then see out of those roots in the interval a b how many are in the lower half a to a plus b by 2 and how many are in the upper half a plus b by 2 to b and so on. So, like this we can repeatedly use bisection to squeeze each of these roots separately and then further we can use bisection itself to go on squeezing the interval till we find the root to our required accuracy or rather than bisection we can find some other equation solving process after locating the roots and separating all the roots. So, with what interval we start do we start from minus infinity to infinity then it will be very difficult to process the whole thing because bisection will work indefinitely. There is a little trick in starting the process if you want to solve for all the eigenvalues and that tells you this all the lambda i's their magnitude are bounded by this quantity which is the maximum over all rows of the entries of the rows e j plus d j plus e j plus 1 take all their magnitude and whatever is the maximum of the sum over all the rows no lambda no eigenvalue of the matrix can have a magnitude which is larger than that. So, if you take the initial interval from minus lambda b and d to lambda b and d then all the eigenvalues are bound to fall within that and then you can go on applying bisection in order to separate the root and separate the eigenvalues and once you have separated them then solving for them you can apply either bisection itself or some other equation solving process or root finding process. So, that gives you this algorithm first identify the interval a b of interest. Now, interval a b of interest can be either the entire interval minus lambda b and d to lambda b and d if you are interested in finding all the root all the eigenvalues or sometimes the your problem may suggest that you are interested in an eigenvalue in a given domain only in a given interval only you are not bothered with rest of the eigenvalues which may fall outside this interval in that case you take that interval as a b otherwise you take the larger interval in which you are sure that all the eigenvalues will lie. Now, for a degenerate case in which some sub diagonal or super diagonal entry of the symmetric tri diagonal matrix is 0 you split the given matrix and operate separately with the different blocks. For each of the remaining non degenerate blocks or matrices you just do two things by repeated use of bisection and study of the sequence p lambda you bracket or separate individual eigenvalues within small sub intervals and then in these bracketed sub intervals by further use of bisection itself or some substitute some other root finding method within each sub interval determine the individual eigenvalues and when the interval becomes extremely small say the interval size becomes equal to 0.0001 then that means you have actually found the eigenvalue so there is no further need to go into that. So, in this lesson what are the important points that we should keep focus on first point is that a householder matrix is symmetric and orthogonal and it effect a reflection transformation. Second is a sequence of householder transformations can be used to convert a given symmetric matrix into symmetric tri diagonal form and then the eigenvalues of the leading squares of matrices form a two mean sequence which have interlacing structure in it in their roots and this property can be used to separate and bracket eigenvalues and further solve in a systematic manner. So, we have a little time in hand so let us consider a quick example at least half way after which you can proceed on that example. Suppose, we have got this matrix 2 3 2 5 4 3 5 7 way same matrix these are the diagonal entries and the off diagonal entries are 1 0 0 1 2 2 1 sub diagonal entries are also same and all other entries are 0. So, this is a symmetric tri diagonal matrix possibly obtained after a series of householder transformations. So, this is the matrix which we are going to solve for eigenvalues. These 2 0s will allow us to split the matrix in this manner. So, there is a 2 by 2 components and this is a 5 by 5 component. This is actually nothing this you can solve from the diagonal entries of this you can solve from the direct definition itself because that will involve only the solution of a quadratic. This otherwise would involve the solution of a Quintic equation which is more difficult. So, for this we apply this methodology based on the Stroumian sequence property. So, for that we construct these polynomials. First one is trivial second one will be lambda minus d 1 that is p 1 next p 2 will be lambda minus d 2 that is lambda minus 5 into p 1 minus e 1 square that is 1. Next you will have p 3 which will be lambda minus d 3 into p 2 into p 2 minus e 3 square and what is e 3 here? e 3 here is 2. So, e 3 square into p 1 then p 4 will be lambda minus d 4 into p 3 minus e 4 square e 4 is 2 into p 2. Finally, p 5 will be lambda minus d 5 into p 4 into p 3 into p 4 into p 2. So, these things we now try to evaluate for different lambdas to locate the roots of these polynomials. So, for now consider the interval which you need to consider you would have noticed the intervals as 1 the rows as 1 5 2 2 4 2 these are the biggest rows. So, some of that turns out to be 8 that means no eigen value of this matrix can have magnitude higher than 8. So, you consider the interval minus 8 to plus 8. So, at lambda equal to minus 8 you try to evaluate this. This is one for all of them this is minus 10 this is minus 8 into minus 5 that is minus 13 into this that is minus 10 already in hand. So, what you get you get minus 13 minus 5 into minus 10 minus 5 into minus 10 that is plus 50 minus 1. So, you get 49 then you come here minus 8 minus 4 that is minus 12 minus 12 into p 2. So, you have already got minus 4 into this. So, you will find that this turns out to be positive then this turns out to be negative this turns out to be positive. And then you will find that 1 2 3 4 5 sign changes are there that will show this I suggest that you verify and check that this turn out to be positive negative positive and so on. And then you will find that there are 5 sign changes from top to bottom that means above minus 8 p 5 will have 5 roots that means all the 5 roots are above minus 8. Then you consider the case of lambda equal to 8 this is 1 and as you put 8 here you will get 8 minus 2 that is 6 positive. Then you put 8 here 3 into p 1 that is 3 into 6 18 minus 1. So, you get 17 then you come here 8 minus 4 that is 4 4 p 2 4 into 17 68 minus 4 into 6 68 minus 24. So, you get 44 still positive like this you will find that in this case all of them turn out to be positive that will mean what is the number of roots of p 5 above 8 above the value 8 number of roots is the same as number of sign changes here no sign change here. So, no root above 8 up to this we have verified that all the roots are actually above minus 8 5 roots above minus 8 and no root above plus 8. So, we have verified that bound that is all the roots actually lie within minus 8 and 8. Now, applying bisection you try to find out the number of roots above 0 above 0 how many so this is 1 as you put 0 here you get minus 2 as you get put 0 here you get minus 5 into this that is plus 10 minus 1 that is plus 9.  So, minus 28 that is like this as you continue you will find that this turns out to be positive and this turns out to be negative that will show the number of sign changes at lambda equal to 0 for these polynomials the sequence of polynomials is 1 2 3 4 5. So, all 5 roots above 0 so all positive roots. So, this gives you a little further information that all the roots are within the interval 0 to 8 in particular this is a positive definite matrix because all Eigen values are positive. Now, what you will do for bisection you will evaluate the polynomials the sequence of polynomials at lambda equal to 4. So, as you evaluate at lambda equal to 4 you will find that you get 1 2 and then here minus 2 minus 1 that is minus 3 and then at lambda equal to 4 this is 0 minus 4 into p 1 that is minus 8. Then here 1 into minus 8 that is minus 8 minus minus plus 12 that means 4. Finally, here minus 4 minus minus plus 8 is minus 4 plus 8 that is plus 4. So, how many of these sign changes here 1 sign change here 2 sign changes here. So, above 4 you will have 2 Eigen values and below 4 you will have 3. So, you have started bracketing 3 in this interval and 1 in this interval 3 in this interval and 2 in this interval. So, 2 sign changes here at lambda equal to 4 that means above 4 p 5 will have 2 root 2 sign changes. So, in this there will be 2 roots in this there will be 3 roots. Next you will go on splitting this next you will evaluate for finding the Eigen values in this sub interval you will evaluate at 2 and then possibly at 1 or 3 and so on.  And further continuation in the same process will squeeze the root for you. So, I suggest that you continue this process till you find the Eigen values with an accuracy of 0.1 that will give you enough practice and you will find that the method works quite comfortably. There was a small error in the calculation in the board work. So, please note this correction. Here what you saw in the board was this we were analyzing the Eigen value problem of this problem and this is what appeared on the board and in this there is a correction this 49 for p 2 at lambda equal to minus 8 was not right. The correct calculation shows that it should be 129 the result of which was that the next 3 signs were also mistaken and the next 3 signs will be this way minus plus and minus and with this as you will notice that for lambda equal to minus 8 there are 5 sign changes and that means that all 5 roots are above minus 8. And then for lambda equal to 8 there is no sign change and that shows that no root is above 8 and in between through bisection then you will evaluate at lambda equal to 0 in which case you will find that all 5 roots are above 0. So, the first 2 columns in this data for lambda equal to minus 8 and lambda equal to 8 you basically get the verification of the bounds of minus 8 and 8 for all the Eigen values and the third column lambda equal to for lambda equal to 0 shows that all the 5 Eigen values are positive which means the matrix is positive definite other than this everything else is alright in the board work. Thank you.


My name is Mark Springer, Dean of the College of Liberal Arts and School of the Arts at St. Cloud State University. Our students study the arts, humanities, and social sciences to explore, gain insight, and communicate about the human condition. In doing so, they develop new understandings of themselves, each other, and the world around them. Making a difference across Minnesota, the majority of our graduates build meaningful careers across the state. And whether entrepreneurs, artists, communicators, researchers, teachers, or leaders in their chosen field, they contribute to the success of our communities. They are Huskies.


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Be Come on, man. What is up with these guys? My man TJ Okay, I mean come on boys.  We could do better. Y'all letting that go down boys. The boys are letting that go down Pick let's go through punch Need it. Come on No fumble.  I'm right at the line. Just let it happen, dude Just let it like God. I just like I just god damn it, man.    Let's pick It's big March on What I like, how does my Bro, are you any motion doubt you put them in like you stop guard in a tight end and you I Don't understand where the guy that was supposed to be on the running back went like I understand the tight There was nobody on the tight end. That's my fault, but that guy should have been covered I don't know.  And that's why we just play zone every play Like it's just bad like my defense.  This is bad man.   I'll take it Take it the defense. Come on like come on man.    We just got a hold of block It's a three-man rush. Come on boys No I'm just like he gave another touchdown like dude.   That was a good show Well, come on. He's playing good. He's tough man kids tough, dude He's playing really good on offense.   Go ahead big fella I'm definitely feeling the mobile quarterback pretty soon here. I can't lie to you.  Come on. Debbie.  Let's go man. We make it all back. Ah He's gonna let me do let's go.   Come on, man. Jesus Christ Filly running a building love to see that. There we go. Thank you boys.   Oh, come on. I need that. That's what we need We need the overthrow dude. Come on, man.  Yes No way you're bumped into each other right there, man Need it. Come on. I Hate that touchdown. Come on boys. Just hold for me for a second feet That come on, man. How about I beg for a fumble back? How about that? How about that boys right here right here? How about that nothing throw the ball away, huh? Come on.   Oh Fumble ball ball ball boys. Let's go. We got it back huge That's huge The kids at home would say that's huge Tricky tricky perfect Perfect I'm gonna call time out here. I'll do it incomplete pass. Oh my god.   I'm dog.   He made the right read damn there every play right read every play dog That kid is good man.


I'll just share my presentation. Hello everyone. My name is Rajesh Tuddu. I lead the series at Tec Mahindra. Today I'll be talking about the media value in the post-COVID world. So in terms of the ongoing pandemic, there are no theatrical releases that are happening. And many movies releases in different parts of the world have moved over to internet. And they have moved over to various other. It doesn't come as a surprise that the industry lost $30 billion in the year 2020 alone primarily owing to the lockdown. And it is a staggering 70% plus loss compared to previous year. So given this kind of scenario, major disruptions that are being observed in the global media and entertainment industry, one is in terms of even before the pandemic, there has been a significant shift in the way people consume media. There have been on-rise of binge watching. And there has been shift to the medium like Netflix, like Hulu, Prime Video, so on and so forth. So the OTT providers are gaining significantly compared to the theaters or compared to the television broadcasters. And with this event, there is also a growing threat of intellectual property rights violation and piracy. Advent of OTT platforms make things a lot more difficult. There's mushrooming of many channels. And sometimes the content also moves from one particular channel or one particular OTT platform to another. The movement cannot be prevented or preempted. But what needs to be taken care of are the underlying IPR rules and also the IPR protection. And we also know that there is a concept of geo-blocking and the contractual provisions with the various broadcasters ask for geo-blocking to be enforced. That means once the content is licensed to a particular broadcaster or it is licensed to a particular OTT platform in a particular geography, it cannot appear on any other OTT platform in the same geography. If there are infringement of those provisions, the impact is significantly high and the damages are very, very, very, very high. So the geo-blocking emerged as a new challenge for a lot of media and entertainment companies. And added to that, there has been a significant decline in the revenues. So while these major disruptions were observed, what still remains important and what still remains paramount is content tracking, production, and payments, which are very, very important for business continuity. As a result of which, we need to look for next generation solutions which will help the media and entertainment companies to overcome the challenges. So can blockchain help? In our understanding, yes, the blockchain can help the media industry primarily to write the new normal, especially in the COVID, as well as the post-COVID world. Because of the inherent benefits that blockchain brings to the media and entertainment companies, apart from enhanced security, it provides immutability and provenance, a real time view of data, and most important is the improved transparency. So let's see how this is going to help with regards to the mushrooming of the OTT platforms and with regards to the shift in the media consumption. I mentioned that the digital rights management system becomes very, very important as a result. And it is not just having a digital rights management system, but it also having a system that is equipped with features like geofencing, with features like IPR management, which kind of prevents cross-pollination of content and also the sneaking of the content into various geographies and into various channels. And we know that blockchain brings in an inherent capability, which is the one-way hash function using a secured hash algorithm or algorithms of that particular nature. And when you do the forensic watermarking and when you enable the forensic watermarking with the hash functions, it will result in a very good track and trace capability, and which will also help to identify the original content from the fake content. It will also help to differentiate the original content from the fake content. In addition to that, given the innate nature of blockchain to facilitate microtransactions and micropayments, that particular innate feature of blockchain can be leveraged where micropayments can be enabled for digital artists. So in a sense, we believe that media companies leveraging the blockchain adequately will be in a better situation to grow 10x in the next three to five years. So is there in any other technology that can provide benefits that a blockchain can provide to the media and entertainment companies? The answer is no, because in terms of given the wide nature of the benefits, there is no other technology that comes closer to blockchain. Media and entertainment companies have a multitude of systems. If I talk about the systems, they have a content management system, they have an affiliate management system, they have a conditional access system, and various other systems. But the problem with these systems is they are very, very siloed, and the data from one system does not move to the other system in a seamless fashion. As a result of which, there is no exchange of information, and that lack of exchange of information in a seamless fashion brings in many, many challenges for media and entertainment companies. So that presents an opportunity for blockchain implementation, where blockchain can be implemented as a trusted middleware that links all these operating systems in a media and entertainment company, aggregates the information from these systems, and presents it as a single source of truth as a result of which, because of the lack of information, there is people do not end up violating on their contractual obligations. So the benefits are multitude, whether it is automating the contracts, facilitating transparency in royalty payments, lowering the transaction costs, addressing the fake news, reducing the copyright infringement. And so on and so forth. There are many, many benefits that blockchain can provide or can endow to the players in the media and entertainment industry. So let's see in terms of how does it cut across the value chain within the media industry, primarily from a content value chain perspective. So the opportunities are there in royalties, where blockchain-based platform can enable near real-time micro payments. The opportunities are in contract management, whether it pertains to digital media resale, or whether it pertains to automating contract creation. Opportunities are in digital advertising with respect to audience data sharing, ad campaign reconciliation reporting. And more important, most important, from a rights management perspective, where it facilitates record keeping on a universal basis and most important in a trusted and immutable fashion. So what is it going to lead to? It is going to lead to provenance with respect to the ownership of digital assets. It will lead to enhanced licensing and monetization. It will lead to increased efficiency in audience data sharing, reducing the time spent in reconciling the campaign data. So there are many, many, many benefits that are there from the many, many benefits that are there from integrating or from localizing or from internalizing blockchain in the day-to-day operations for a media and entertainment company. So let's see in terms of what the players are doing in the industry. I have been able to collate some examples here. There are both large players as well as small players who have started this particular journey with blockchain and then they only intend to enhance their adoption whether it is in the ongoing pandemic or the post-COVID world. So if we have good examples from Comcast, Disney in 20, as early as 2018, Comcast has announced a blockchain insights platform with NBC covering channels like Disney, Channel 4 and the others, primarily from a perspective of sharing the sharing, I mean, primarily from a perspective of establishing the authenticity of the viewership, establishing the authenticity of audience without sharing data. And what has it resulted in? It has resulted in focused advertising, a better planning and an ability to reach the target audience, the intended target audience. We've seen this with respect to Spotify which has come up with a media chain, primarily to work on developing next generation technology which with the name of connecting artists and the other right holders who own a particular music track or who are the rightful holders of the music track and it'll help to identify, it'll help to establish the ownership which becomes very, very easy on the service that is hosted on the Spotify's media chain. And there are examples from people like PRS, SACM and ASCAP, these are the royalty collection agencies which have built a blockchain network primarily to manage music copyright or the information pertaining to the music copyrights. So there are leading industry platforms like blockchain-based contracts and rights management system. It is considered it as a platform which is a single, I mean one source or which provides a single window kind of an approach and which caters to all the aspects that are important from a day-to-day operation in media and entertainment industry, whether it pertains to contracts, the rights management, track and trace and royalty payments. Based on my experience, there's not much of digitization that has happened in media and entertainment industry. You will have many, many production houses which still maintain their contracts either in the Word documents or the PDF documents and they are contained in the inboxes or in the folders of some key people in that particular entity. And the rest of the people in the organization do not have access to these contracts and important provisions pertaining to geo-blocking or important provisions pertaining to royalties or payments are contained in this particular contracts. So there is no mechanism to harness the information that is contained in this contracts and facilitate easy access to that particular information to different user groups within the organization. And I've also seen with respect to royalty payments, there's not much of automation. Many, many royalty payments are still managed on Excel and there is no exchange of information between the various parties who provide inputs into those royalty payments. And same is the case with the rights management or with track and trace. So the BCRMS platform is a one platform which addresses all these aspects. It starts with digitization of the contracts and it not only facilitates the digitization of the contracts but extraction of the key contractual provisions and conversion of those key contractual provisions into smart contracts which can be used across the organization and they provide very good inputs to the royalty calculations on an automated fashion and also tracking the digital rights management in an easy fashion. That's what the platform does. And with the advent of NFTs, the non-fungible tokens and for some of those who are not initiated into the exciting world of non-fungible tokens, they're basically tokens which establish the provenance of ownership of a digital file. It could be a music file or it could be a picture. It could be a video file. And what NFT does is it helps people to have the ownership on digital collectible or crypto collectible in an undisputed fashion. And what we are doing with respect to or rather what we are seeing as a trend in the industry is to have these NFTs integrated into the platforms like blockchain powered contracts and rights management system, the platform that I presented to you so that it not only tracks the rightful owner but it also establishes the provenance of that particular ownership along with an associated non-fungible token. And this is being integrated into the blockchain platform. So what do we gain to accomplish or what does the industry gain to accomplish by integrating the functionality of a non-fungible token into a blockchain platform? It is from a perspective of two aspects. One is it is helping people to tokenize the IP, not only helping people to tokenize IP but also to facilitate what we call as the fractional ownership of digital assets. So it is going to bring in the monetization capabilities with an underlying non-fungible token. For example, if there is a turning movement in a particular episode or in a drama series and you could take that particular turning movement and then you can mint underlying non-fungible tokens and you can possibly put it up for sale in a NFT marketplace. It could be audiovisual clips of popular movies or it could be Grammy clips, Oscar clips as the underlying non-fungible tokens. And obviously it also helps people manage the incentives primarily from a perspective of the subscribers or the buyers of the movie clips or the music labels and will also help to drive royalties especially in the context of fractional ownership. So in a sense to sum it up, there will be multitude of opportunities in the post-COVID world. The key is to spot those particular opportunities based on the problems or the burning problems that are there or the problems that have been around and which couldn't be solved through non-blockchain implementations. There's an opportunity to solve those by incorporating or by internalizing blockchain and various other blockchain powered platforms in media and entertainment industry. With that, I will bring my presentation to an end. I'm happy to take any questions that the audience may have. If there are any questions, please go ahead and put those questions for me in the chat screen. I'll be happy to pick them up. I'm seeing some comments. Thanks, Satish. Thanks for your kind comment. Okay, I'll take this, take it as no questions.  If you need any further information, please feel free to reach out to me at mycoordinates rajesh.dudu at techmahendra.com. Thanks for watching my presentation. Please stay safe and wish you all a great day ahead. Thank you. Bye for now.


Okay, very good morning. Welcome to a brand new week before I begin whether you're new to the channel or you watch more frequently Don't forget to like and subscribe so the channel be much appreciated But look, let's get straight into it because a big week ahead We've got a whole ton of earnings coming out from all of the mega cap tech names We've also got the FMC meeting as well happening this week. So let's start with earnings. They're in fact 175 S&P 500 companies reporting this week by far the busiest one that we'll see throughout the earnings season and almost half 12 of the Dow 30 components are also reporting as you can see here Some of the highlights include the likes of alphabet and Microsoft aftermarket on Tuesday We also get Boeing pre-market Wednesday with meta aftermarket and then Thursday a big day because we get the likes of Apple and Amazon and then if we just zoom out a little bit We've got Exxon and Chevron to finish off the week on a Friday So a super busy day ahead on that front and something I'll be going over in much more detail If you want to follow us on LinkedIn or on other social channels We'll be covering all of those in a lot more detail when they come out Otherwise the other things I'm looking at this morning are WTI crews trading a little bit lower We've actually moved below a $94 handle going into the beginning of the week here in mainland Europe and it comes As you can see here after capping a third weekly decline on Friday what we saw last week In fact, that's the longest streak of losses so far this year Again having been predominantly elevated by lots of other things, but namely that of the energy crisis We're now reversing course of that even though the fighting there particularly in the eastern side of Ukraine continues to rage on Concerns predominantly outweighing that though are on the expectation of an economic slowdown Overshadowing the tightness of the physical crude market on the supply side as well something to be aware of is the Libya updated at the weekend and national oil Corporation otherwise known as knock aims to bring back production of around one or bring back to the production to 1.2 million barrels a day in the next two weeks as well The other thing that potentially as well as feeding into this a little bit has been a little bit more of a positive development on the Chinese COVID situation which we continue to watch with great interest COVID-19 cases fell to their lowest in a week even as officials in financial hub is Shanghai and Macau they continue to ramp up testing at this point in time But moving forward into one of the main things we're looking out for this week and undoubtedly the FMC is really has top billing As you can see here the market implied rates World would indicate around a seventy eight point seven percent probability of a seventy five basis point rate hike from the FMC on Wednesday night as they try to really arrest Persistent inflation that would take then the interest rate the federal funds rate to a new target range of two and a quarter to two and a half percent talk of a bank of Canada style one hundred basis points, which was briefly Priced in after we had that red-hot US inflation data print about two weeks ago has dissipated We've had a softer batch of recent economic activity data and indications Or indicators have come out a little bit on the weaker side And that's tempered those expectations a little bit in combination as well as some of the most hawkish members of the of the Fed Namely Bullard and Waller have come out and really talk down a hundred and more committed to seventy five at this point in time So hence the readjustment that we've seen In terms of other things we're looking out for US GDP also comes out this week And of course, it'll be closely watched as I mentioned which is kind of weighing on concerns for oil markets It's this inevitability around the potential for a US but more global level recession Looming on the horizon and the impact that I have on demand and in terms of US GDP on Thursday It's forecast to have risen an annualized point five percent during the April June period according to a survey from Bloomberg speaking to lots of different economists across the street while infantry's probably weighed heavily on the second quarter results The demand picture also darkened As consumer spending business investment and housing may have all taken bit of a step back And of course this is as that cost of living crisis really rages on that's impacting as we've seen from the lights of soft sentiment based surveys like University of Michigan confidence has seen Quite a aggressive fall in the state of the US consumer psyche Other things to look out for in the US this week We've got consumer confidence data happening on Tuesday during a good Wednesday your weekly jobs claims as usual on a Thursday Alongside core PCE pricing next personal income spending and Chicago PMI all happening on Friday So pretty busy week really if you look at that calendar of all the economic data points You got the FMC with their expected 75 basis point hike as well coupled in then with a hundred and seventy five S&P companies reporting with all the mega cap tech names. Yeah, it's going to be quite a busy one for sure Looking over in mainland Europe. What are the main things we've got? We've got German iPhone That's business sentiment out later on this morning It's expected by economists to touch a fresh two-year low Meanwhile your era economic confidence on Thursday is anticipated to drop to its lowest since Feb of 2021 so really in the doldrums of where we were and the most onerous lockdowns that we had across Europe Eurozone inflation is expected on Friday to accelerate to eight point seven percent We're looking at your area inflation rate year-on-year here on this chart You can see and hence there are the move that we had from the European Central Bank of last week who hike rates Of course for the first time 11 years by a margin of 50 basis points the largest then incline We've had since the year 2000 as they've chiefly got in mind trying to tackle this inflation issue and have actually seen one ECB member Kazakhs this morning who said that a large interest rate hike may not be over and two week a euro is a problem The hike in September needs to be quite significant and should be open to larger hikes And actually euro got a little bit of a bump this morning on the back of that comment Albeit from a lesser known ECB council member the other thing we've got from Europe to look out for is going to be that of here euro area GDP That's also coming out at the back end of the week It's out to forecast and show a minimum a bare minimum level of growth in the second quarter This comes amid as well euro area economic confidence data We're going to get on Thursday anticipated to as well drop quite sharply The inflation reading as I said in combination with GDP That's all going to be coming out on Friday will be key ones to watch for the eurozone And then finally over in China something that's probably not so regularly on your radar But worth just being aware of is the China political bureau These are really the decision-makers over top-level policy in China They're likely to meet at some time later in the week still yet to be confirmed the exact details With the property market tanking zero covid policy testing Is very much in full flow at the moment giving a lot of the fresh upticks in cases albeit slight tempering We've had in the last week and in the global growth outlet dimming a lot of people watching out for these talks Are any signals of more monetary fiscal stimulus on their way or whether officials acknowledge or not that their 2022 growth target is unachievable and this comes after that Very minor growth print that we had out of China, which continues to go through quite a steep Deselevation of growth at this point in time That is it. So hopefully that was useful again Don't forget to like and subscribe to the channel if you're new here really appreciate that Otherwise have yourself a good week ahead. Take care


Hi friends, welcome back to my channel and welcome if you are new here I apologize in advance for the lack of makeup But it is just a relaxing day and I wanted to give my face just a little bit of breathing room today You are in store for a w w what I eat in a day to lose weight on the green plan So I am following the green plan on w w Just remember all of the points are exactly the same with the exception of the zero point foods So most of what i'm eating today is going to be very similar in points on both the blue and purple program So if you want to see what I eat in a day to lose weight on w w's green program A little bit of diesel and lola some halls and just some fun. Just stay tuned Hobby lobby hall for you We have everything from office decor to christmas A little bit more fall decor and halloween decor as well for next year So let's get into this hall because this is a huge hobby lobby hall So let's start with halloween all of their halloween was 66 off I don't have any for my office because I haven't had an office up until now So I picked some things up for next year that I thought would just be really cute on my bookshelves So hello books cats and bats trick-or-treat october 31st Normally 1599 and it was 66 off So keep that in mind for prices for halloween 66 off I found this cute little black with black and white bats that says trick-or-treat Normally this was six dollars and again 66 off I found this cute little pumpkin and he's technically a Picture holder, but I'm just going to use it as decor and it was four dollars 66 off And I love this little ghost. Uh, he's so cute Normally 399 as well And then I found this cute little beware which was normally 399 And I thought that that would be cute on my bookshelf And then I also thought this was just the cutest darned cat dish ever Maybe fill it with some treats or something and normally 999 and then of course 66 off So I think this is enough halloween decor for next year for my office. Just so cute I couldn't pass it up for the price. I also picked up a few more fall Thanksgiving type of decor since I just ordered a second bookcase as well So I have this grateful thankful blessed little wood sign with an easel And I just think it's so cute and I don't even see the point. Oh, there we go Normally 599 and all of the Thanksgiving decor was 50 off. So couldn't pass that up I found this cute little galvanized metal Pumpkin with these cute little leaves. I just loved this It goes really well with the decor in my office And it was normally 14 dollars and then half off And then I couldn't pass this up just for something kind of fun This cute little give things pumpkin with a crow on it normally eight dollars and half off So that's just a few little fall things. I have quite a few in my office already I just wanted to kind of fill the gaps. So next I'm going to show you just traditional home decor So my new desk has a couple of shelves. So I found these really cute baskets They have that chicken wire Gray with this fun lining Unfortunately, these were not on sale, but they were only 12 dollars a piece, which is not bad So I picked up two of those and I used my 40 percent off coupon for this candle holder Normally 22.99. So I think I paid, I don't know 12 dollars for it But you can see it's that gray brown kind of rustic look And I have a really pretty white pillar that I'm going to put in this and this is going to go on my bookshelf And then I had to pick this up as soon as I saw it This is the letter c of course for my last name galvanized with that kind of brush rust edge And this was normally 10 dollars and was half off So of course I had to pick that up and then I needed a couple more pieces of greenery So this one was actually Half off and it's normally 15 dollars and it's got this nice white base with this just kind of Greenery I just needed some greenery for my other bookshelf And I thought for half off you couldn't beat it for about seven dollars And then this I am obsessed. This was also half off and this was normally 25 dollars So it was 12 dollars. So it's this cute little chicken wire basket with these burlap Plants and this is going on my shelf. I cannot wait. I already have the perfect spot for this on a smaller shelf in my office So I couldn't pass that up. So that is all the just regular home decor that I grabbed now Let's move right into Christmas. So I found this white wood tree. It has this cute little base with holly and pine cones I just thought it was so cute. It matches the colors in my office Normally 799 it was half off. So I couldn't pass that up I actually saw this at the checkout counter gets me every time And then I found this cute little mary christmas sign.  So this was six dollars So not a bad price for that And then I had to pick this up. My middle name is joy and I love this because it is so cute And rustic and it was normally ten dollars. So I paid five dollars for it Couldn't pass it up. You can see the woods kind of rustic. I loved that And then I found him and I thought that he was Seriously, the doshgarn Cutest little snow and I put him over here so you can see him. Isn't he cutie's got little metal arms I just love Love love him and he was normally ten dollars So I only paid five dollars for him and I just think seriously. He's so cute and then of course I had to pick up this joy to the world sign again. It's my middle name Normally six dollars. So I paid three dollars for it And I just think it's cute and all of these are just gonna go So well on the shelves in my office.  It'll probably go on my desk It's a little easel chalkboard and it says days till christmas And you just put a little piece of chalk here on the ledge and you can change that each day I have one for my house as well Um eight dollars normally so I paid four dollars for it And I just think it's adorable And then I had to pick this little christmas bell With this cute little pine cone and holly mix on top you guys Isn't this cute? It's going on my shelf of course in my office And this one was a normally Seven ninety nine. So I paid a three fifty and four dollars. So such a great deal so Cute and then I couldn't pass this up. I was thinking I wanted a christmas tree in my office So I saw this one and he is just so cute with this cute little button burlap bottom and the little star on top It was normally five dollars. So of course two dollars and fifty cents Couldn't pass it up. Also. I found these ho ho ho blocks. I thought that these were the cutest tied together with some jute Normally twelve dollars. So I paid six dollars for that Two more things. I have this cute little snowman now I think he's going to go in my office because he too has that really rustic vibe to him I just think he is adorable And he was normally twelve dollars. So six dollars And then I actually picked this up for my house because this is more The decor of my house. I am very traditional in my christmas colors I think he's so cute. You guys look at him.  He was normally twenty dollars. So I got him for ten bucks and I just think he is so Dang cute. So that is everything that I picked up at hobby lobby. You guys quite the haul It was almost two hundred dollars But you know what I've got all the seasons covered for my office some fun new decor pieces And with all the discounts and coupons you can't pass it up at hobby lobby So that my friends is my ginormous hobby lobby haul So this is my breakfast for this morning bringing in some fall vibes So I have a bowl of fruit here We had some leftover grapes and honeydew from when we traveled back from leavenworth yesterday And I just paired that with some raspberries So i'm going to have a bowl of fruit and then here I have two of the western bagel pumpkin baglet So these are the western bagel alternative baglet. These are so Good you guys. These are amazing. So there are these small Little bagels each one of these little bagels is only one smart point So that is one point bagel the pumpkin ones are so good If you have two of the bagelettes it bumps it up to three points, which is what I did here But these are outstanding all of the western bagels as long as you purchase the alternative see that right there All of those bagels full size are going to range from two to five smart points Mini baglets, which I think is the perfect amount I mean even one of these is a great portion of your breakfast The mini baglets are only one smart point of peace. They have pumpkin everything cinnamon. You name it. They have it I love western bagel. I eat these all the time. I love them toasted I added a little bit of the trader joe's pumpkin cream cheese to it and it is the perfect pairing So if you're interested in checking out western bagel, I do have a 10 off discount code I will put it here on the screen for you if you do their build a box option where you get to select the Types of bagels that you want in your box. You can also get free shipping and use my code for 10 off So it's kind of a double discount But I highly recommend these if you are a bread lover a bagel lover This is going to save the day because the smart points are so low and they are so good So I highly recommend western bagel check out their website any questions. Just leave them in the comments below for me I love these these are limited editions. So You're interested in the pumpkin get them while you can because they will be gone So i'm going to have two of the pumpkin baglets for three smart points I have two smart points worth the trader joe's pumpkin cream cheese on there So good and then I have one full egg and one egg white So my breakfast is two points for the egg because I am following the green plant So I count for eggs and then five points for my two bagels including the cream cheese Zero for my fruit. So my breakfast is a total of seven smart points.  I know she mama. Hi, buddy I know it's raining. It's pouring rain here today. So we don't want to get up when it's raining, huh bud No, lola. Do you want to get up? Hi, baby? Well, good morning.  Probably not No, can you probably not because it's raining, huh? Oh, I know baby. I stress. Yes a big girl Good girl, babe. You guys just relax today because it's raining out. Yeah, you just relax today So I wanted to show you guys these face masks. So there's a company called save sue I'll put it here on the screen for you It is a platform where you can actually receive free products to try and test out and review Well, they reached out to me and asked if I'd be interested in testing out These 24 karat gold collagen eye masks and I said well, yes You know, I love my masks and my face treatments So I wanted to give these a try and I wanted to give you guys a review on these Because you can get a lot of these products for free off of that website I am going to go ahead and link the website below for you as well Go check it out. It doesn't hurt to try free product See if you like it and you can always add it to your list of things that you want to purchase in the future So I'm going to put these on I'll show these to you guys And then I'll review them once I have left them on for some time See how they work. So here's what the masks look like. They are gold They're in that liquid in that little pouch So you want to make sure you wash and dry under your eyes really well And then we'll apply these eye masks and we'll leave them on for 20 minutes And we'll see what kind of results we get. All right, so I put on the eye masks when I first put them on They feel really good. They're very cooling. They are kind of a rubbery Slimy feel. I know that doesn't sound very good, but they feel really good going on your eyes. That cooling just really Feels amazing on your eyes. So you're supposed to leave these on for 20 minutes So I am going to do that Collagen in there supposed to tighten around the eyes Help with any type of begginess that you have from lack of sleep or Whatever you may have the bags under your eyes from so I am excited to see how these work So I'll be back in 20 minutes to share with you guys my opinion of these masks And again, this is from savesu.com. They give a lot of free product out for you to try and review So if you're interested definitely check out the link below, but let's see how these work All right, so it's been the 20 minutes. So this is what these look like I don't know if it's gonna focus on them very well, but they are Fairly thick. So we're gonna go ahead and remove them So there's my eyes. They feel still nice and cold. These actually held their cold Sensation for the entire 20 minutes, which was really nice And as I'm feeling these now all of the liquid or the gel is gone. So these are great. I don't notice any Bad qualities of these eye masks. I don't know if it's making any difference per se, but it feels really good It's nice and cooling. So definitely check out the savesu website see if you can pick up some products to review It's nice just to have some new things to try out, especially when you kind of want a little bit of a Self-care day. So check it out down in the description box below. So here's my lunch for today It's about 11 30. I have one of the california turkey Raps that I made in last week's meal prep if you guys have not checked out that video It was posted last monday one week ago. These wraps are so good So what is in here is an Olay extreme wellness spinach and herb wrap for one smart point Two slices of center cut bacon for one smart point One point worth of avocado one point worth of light mayo and two points worth of Turkey breast.  These are the hip peas in vegan white cheddar So this is what they look like when you pull them out of the bag and that is two smart points So this is an eight smart point lunch and this looks so good Is it lunch time for you guys too? Guess what? Mommy has some bacon left over from her wrap. So that's what you guys get to have today Special day. Here you go, buddy.  Yeah, don't get up on So I wanted to show you what I got from sensei Because in my new office I have the sensei oil diffuser and I love this I didn't want to burn candles in my office. So I just diffused my essential oils Instead so I do have a couple of other ones that I've had One of which is from sensei one from clobby lobby and one from a Essential oil shop in seattle, washington But I decided since I spend so much time in here now that I wanted some holiday type of oils to diffuse So if you're interested in sensei one of our amazing tried members my subscriber melissa doffer Does sell sensei. So if you're looking for some wax melts burners christmas gifts essential oils There is a link down in the description box if you click that link it is an open party You can go ahead and place your order It'll be shipped directly to your doorstep and of course melissa can answer any questions that you have So I just wanted to show you quickly the oils that I picked up for the holidays So first I went ahead and got the tea tree now. I just love the smell of tea tree in general It is crisp. It is fresh. It invigorates me if I didn't have colored hair I would use tea tree shampoo. It is not good for colored hair It literally strips out your color, but I love the smell of tea tree So I picked up that one and then also I'm really excited about this cedar wood pine I love pine anything and we do have an artificial tree But pine is one of my favorite things about christmas So we often will burn pine candles and I thought that I could Put some of this cedar wood pine in my diffuser and diffuse that Christmas feel in my office and then I also grabbed the holiday collection Of oils and this is three different holiday scented oils So it comes in this adorable little tin box And these are the size of the oil. So the ones that you get are cheers Roasted chestnuts and dancing snowflakes. So they come in this cute little jar With this foam insert and here are your notes in the different sense So cheers raise a glass a festive sparkling pear spiked with lemon sugar and vanilla yummy, and then your roasted chestnuts is Crushed chestnuts vanilla and brown sugar offer a sweet spin on a holiday classic So that would be just warm and comforting and then dancing snowflakes is crisp morning frost Sways over fresh juniper and soft soothing lavender. So that one is also going to have that little bit of that christmas Pine vibe, which I'm really excited about. So I have all of these amazing oils from scentsy to diffuse And I will tell you that their oils are way less expensive than a lot of those other companies do tera young living all of those And their oils are fabulous. You just need a few drops in your room just smells amazing So check out the link down in the description box if you're interested in any type of scentsy It helps out one of our fellow subscribers and tribe members melissa So i'm going to have a cup of tea while I do a little bit of work. I just cleaned my house I'm going to edit some videos. This is my tea kettle. I love this so much. I purchased this at friend mire This is my cute little fall mug from I think ross Harvest blessings pumpkin autumn give thanks and I'm just going to have some tea out of my sip spy Box, this is a subscription tea box. I love it comes to your doorstep every month I do have a link down below for five dollars off of your first box. It is not an affiliate link They just sent me these teas to try out and I love them and I wanted to share five dollars off with you guys So i'm going to be having a 16 ounce cup of tea So i'm going to be doing two packets Of this hot cinnamon spice from harnie and suns you guys it is cold out Merini jareary It is fall and i'm excited for some hot cinnamon spice So i'm actually going to use two packages just because I have a larger mug of tea and of course zero points And for my second cup of tea. So right now there's my hot cinnamon I'm going to have two little packets of this motion matcha. This is amazing. There's no added sugar So it's zero points. It has a very fruity matcha flavor And both of these teas have caffeine or some sort of energy component So it's perfect for that one o'clock 130 afternoon slump. So i'm going to enjoy a couple cups of nice hot tea So here's what i'm going to have for a snack this afternoon. I'm Busy here working on my computer still. So i'm going to have some of the blue diamond bold wasabi and soy sauce nuts I'm going to have four smart points worth of these Which is 23 nuts and then i've just stirred up that cup of matcha tea So this is my four smart point snack So I went to tj max again to grab a couple more things for my office And some dog treats. So I just wanted to show you what I found. I just got a few things this time So I found these really cute pajamas. These have that fun little winter Snowflake feel to them.  So I picked those up. Oh, they're so soft. I'm so excited And then I found this cute little sign. It says thankful bless and pumpkin spice obsessed And it's just my colors for my office. It actually would be really cute in my kitchen as well And I got it on clearance for five dollars. So I thought that that was just absolutely adorable couldn't pass that up I did get a couple of dog treats. I found these turkey Cranberry treats these look perfect for the dogs. They're kind of a Like biscuit shape and they are grain free. They have no chicken and they were quite a deal for $2.99 And then I found these little piggies. I thought these were really cute too. Look at so cute. They're pig shaped It is sweet apple flavor. So this is what those look like And again grain free no chicken and they were $2.99 And then I found this flower or this greenery I bought a greenery for my second bookshelf and then ended up using it somewhere else And then I found this one for my bookshelf. I think it's beautiful. It's so pretty And it was only $6.99, which is pretty darn good. There is room for one more piece of greenery So I'm going to look the next time I'm at hobby lobby and love it love this Then I found this candle. It is cedar and balsam.  So I want this just for decor on my shelf It's my colors. It's the black the white and this nice kind of rustic wood feel as well And this candle was $6.99. So not bad at all And then lastly I wanted one more just kind of tin container not for any specific purpose just want it for decor I like this one.  So I picked that up as well again just for decor No other purpose. So that's my little haul from tj max for tonight's dinner I'm going to be making skinny elfredo. It is my husband's birthday and it is our anniversary today And he loves chicken elfredo. So I've decided that that's what I'm going to make him for dinner But I'm going to make it w w friendly. So let me show you what is in tonight's dinner You're going to need some reduced fat milk. I'm going to be using the kroger carb master milk minced garlic chicken broth Pasta you can either do fettuccine noodles spaghetti noodles Whatever you prefer we are going to go with rotini because I am adding chicken So I want to make it more of a Authentic kind of pasta dish. This is the fiber gourmet light rotini It is 40 less calories than normal pasta. It has 19 grams of fiber 8 grams of protein 23 net carbs in his non-gmo And the best part my friends is no matter what plan you are on You can have two ounces of this pasta for only three smart points traditional pasta five to six Smart points crazy that it is about half of the smart points of traditional pasta The best place to order this is off of the net triton website down in my description box There is a link. It'll take you directly to net triton. You can shop around there are hundreds of Ww friendly items, but make sure you are adding the fiber gourmet pasta to your cart because We're obsessed my husband and I love it. So we're going to be doing the light rotini tonight I have some black pepper and that is supposed to be red pepper flakes.  The first thing that we need to do is go ahead and chop up some fresh parsley We're going to garnish our pasta with that and then I'm actually going to take my chicken here I'm going to trim any visible fat and I'm going to dice it up fairly small And we're going to go ahead and fry that up in a fry pan while we're putting together our pasta Not even under So I have my chicken just cooking here in a fry pan I'm just going to get it nice and cooked through And then I also have a mixture of water in that little bit of chicken broth that was left In the carton after I measured it out for the recipe and I threw that in a pan to cook down our pasta So our pasta is just about done. We're going to go ahead and cook it al dente Drain all of the water but about a half of a cup We are going to reserve that for later and then my chicken is also coming along nicely As soon as it is cooked all the way through I'm just going to transfer it into a bowl and set it aside so we can cook the rest of our pasta here in this same pan After you've removed the chicken from your pan, you're going to go ahead and add in your garlic Let this cook down for about a minute or so just until it is nice and fragrant Once your garlic is fragrant, we're going to go ahead and add in one cup of our chicken broth And we're also going to add in three quarters of a cup of the carb master milk And we're going to let this heat through we'll reduce the heat and add in our yogurt and our parmesan cheese When the chicken broth and milk mixture comes to a boil We are going to go ahead and add in our half of a cup of parmesan cheese And we're going to add in your two tablespoons of non-fat Greek yogurt If you did decide to use that it was optional in the recipe We're going to reduce the heat and let this simmer until it begins to thicken As it's simmering go ahead and add in Your red pepper flakes and remember these are pretty hot so it's up to you How much you want to use I literally just added a tiny tiny bit And then I also went ahead and added some black pepper So we're going to let this simmer until it is nice and thick As your sauce thickens we're going to go ahead and add back in The chicken and we're also going to go ahead and add back in our pasta Now if your sauce is too thick that is what that reserved pasta water is for Mine actually looks pretty good So we're just going to make sure that our pasta gets nice and covered In the sauce and the chicken gets mixed and we're just going to let this warm through So here is the alfredo pasta all ready to go This pan makes only four servings.  So here is my dinner for tonight So I have one fourth of the chicken alfredo One fourth on the green plan is seven smart points And then of course my broccoli and my cauliflower are zero So this dinner of chicken alfredo is seven smart points So here is my dessert for tonight I have eight ounces of the no sugar added cherry pie filling And to that I topped it with one of the chocolate graham crackers Yum, this is so good. I am out of that for you ready whip. How Do I let that happen? So I'm out of whipped cream or I would have put a little bit of that on there But this is so good. This you guys is like a chocolate covered cherry pie. It is delicious So it is four smart points for the no Sugar added pie filling and two smart points for the full Sheet of graham crackers. So this is my sixth smart point dessert Thank you for joining me on another what I eat in a day to lose weight on ww's green plan I hope you enjoyed this video lots of fun halls diesel and lola Lots of fun in general all of my food I gave you the calories and the smart points and remember if you are following one of the other plans You can easily put these recipes into your recipe builder and just remember the points are all the same Minus the zero point foods. So thank you so much for watching if you're new I'd love it if you take a moment and subscribe hit that little bell So you're notified whenever I upload a new video. I'd appreciate a thumbs up It helps my channel out a lot and of course comment down below Let me know what was your favorite part of this what I eat in a day and that oh by the way you guys that dinner So good must must make and pick up that fiber gourmet pasta at nettrition. You're gonna love it Thank you guys again for watching and I'll see you next time. Bye