Birne Binegar: Okay, I guess 930 will start i'm so sorry about this, we did have about coven run through our household last weekend actually and.
Birne Binegar: So.
Drew Munson: Until.
Birne Binegar: till next Monday i'm.
Birne Binegar: In sort of self isolation but we're all fine here and what i'll do today is i'm going to cover a new topics that's going to use.
Birne Binegar: The calculation of eigenvalues and eigenvectors that we've been discussing, thus far, this week.
Birne Binegar: we'll see an application of that today.
But.
Birne Binegar: Okay, so i'm.
Birne Binegar: Here i'm like a microphone other than mine on.
Birne Binegar: jovi could please mute.
Birne Binegar: You can of course interrupt me if you have questions but otherwise I keep your microphones muted.
Birne Binegar: Alright, so my plan for today is for me to discuss how to diagnose major cities which.
Birne Binegar: Well, let me get started and you'll see the purpose of it and and then we'll decide how she'll show you how to do it.
Birne Binegar: Okay, so.
Birne Binegar: What a diagonal matrix is is a matrix which the only non zero injuries occur along the main diagonal that main Dag and all of that stretch of entries that run from the upper left the lower right.
Birne Binegar: So essentially a diagonal matrix is worth all entries are zero except from that upper left hand corner, down to the lower right hand corner.
Birne Binegar: And are interested in these guys is because well, you can see from the form they're particularly simple and, in fact, that simplicity it's a great value.
Birne Binegar: For example, you have to dive a little make your seat well normally to multiply make she's together it's kind of a tedious calculation, you have to compute.
Birne Binegar: n squared different dot products, one for each row and each column of both sides.
Birne Binegar: And so it tends to be a long calculation for matrix of reasonable size but, if your matrix East happened to be diagonal, then the computation of their product is really simple.
Birne Binegar: Like entries along that main diagonal just get multiplied together you don't have to worry about the other entries in the matrix so, for example.
Birne Binegar: Let me see if I can do it with a mouse this one one entry.
Birne Binegar: When you take the product of D amp D prime, the only thing you have concerned about to get the new diagonal entry it's a product of the corresponding entries of the factors.
Birne Binegar: So that makes matrix multiplication pretty CZ instead of having to do n squared separate Doc products, you only have to compute in ordinary products products of these numbers Lambda one limited a new one Lambda to in YouTube etc so that's one nice property, but in fact.
Birne Binegar: It extends also to when you multiply vectors by diagonal matrix east, let me do a simple example on my pad.
Birne Binegar: Here suppose I have a matrix will get diagonal increase zero and one and I multiply the first standard basis vector.
Birne Binegar: That would be one zero by this diagonal matrix So this is the 00 and only diagonal entries what happened here, yes.
Birne Binegar: When I take that product with that first row to get the first entry over here i'm just going to get one.
Birne Binegar: And then, when I multiply second wrong you to get zero.
Birne Binegar: Alright, so that's one and then what about the second.
Birne Binegar: You say matrix or two What if I multiply the second standard basis factor that the one with the only non zero component, the second one well what this is going to work out to be is you'll get zero in the first slot.
Birne Binegar: This 10010 and then zero plus two two.
Birne Binegar: And so what this is is two times the second standard basis vector, and this is one times.
Birne Binegar: First standard basis vector what these diagonal matrix each do to your very simple standard basis vectors is it treats them like eigenvalues this first diagonal entry is exactly the eigenvector of the first standard basis factor respect to this diagonal matrix this.
Birne Binegar: The action of the same matrix on the second standard basis vector well it reproduces that standard basic vector multiplied by two, which is the second diagonal entry.
Birne Binegar: So the upshot of this is that when you have a diagonal matrix the way they act on your standard basis factors it's simply by.
Birne Binegar: scale multiplication there the standard basis vectors turn out to be eigenvectors of a diagonal matrix and their eigenvalues correspond exactly to the corresponding diagonal entry this eigenvalue here to come, because the.
Birne Binegar: eigenvalue of this diagonal matrix and the second call is equal to two that's why you get that factor.
Birne Binegar: Alright, so that makes it this property, the way act on standard basis vector actually makes it easy to compute the action of this diagonal matrix D on any better right, because we can take a vector.
Birne Binegar: Save components to be one to be N, and then we let me some mouse here.
Birne Binegar: That would component, they want the end can be expanded in terms of the standard basis vectors than the matrix multiplication can.
Birne Binegar: distribute over this summit vectors and what will happens when he hits any of the standard basis vectors is just going to pull down all out its corresponding diagonal entry and then you'll still have the component of the original vector but now.
Birne Binegar: And the end the calculation is just a simple there's no real matrix multiplication well hidden well all we really do is to compute this.
Birne Binegar: product of the diagonal matrix with an arbitrary vector D, we simply have to know how the X diagonally on the standard basic factors, and then we get a quick computation of the action the matrix D on the vector the.
Birne Binegar: Alright, so the moral of these two examples, or that when we're dealing with matrices sometimes we must it's going to be easier.
Birne Binegar: At least computation wise and.
Birne Binegar: Easy computation to often lead to intuitive clarity as well that things are going to be much easier for us if the matrix start off with is diagonal.
Birne Binegar: But unfortunately that's not always the case, so what we're going to talk about today is how do we make it a matrix diagram or how do we convert a matrix into a diagonal matrix so that it's easier to deal with.
Birne Binegar: Alright, so that leads to the following definition will say that a square matrix is diagonal Isobel if we can find another delivery matrix see that, by the way, has to be in vertical because you'll notice i'm going to multiply.
Birne Binegar: A by the University, as well as I see if we can find a matrix see such that when I do inverse the times, a time see i'm able to create a diagonal matrix D from that operation.
Birne Binegar: And what follows we're going to regard deep dive it'll make to see us as a diagonal ization of the matrix a.
Birne Binegar: And the main point of today's lecture is to show you how you find such a matrix and then, once you found that you can convert problems about non diagonal matrix ease into problems about diagonal matrices.
Birne Binegar: Now what's going to require to that is we're going to need to in order to find this matrix see it's going to turn out that what we have to do.
Birne Binegar: Is examine the eigenvalues and eigenvectors of the matrix a once we know those we should be able to create the corresponding matrix see.
Birne Binegar: that's going to be constructed from the Eigen that vectors of a and the matrix D that results from diagonal lighting a that's actually going to be constructed from the eigenvalues okay.
Birne Binegar: Okay, so let's proceed.
Birne Binegar: To get this started i'm going to.
Birne Binegar: First, talk about some simple matrix multiplication identities it's gonna we know how to multiply matrices together already, but what i'm going to concentrate on right now is how matrix multiplication X on the individual columns of the matrix.
Birne Binegar: So suppose we have to square and buy in matrices and let me, let me let's think of see this matrix see let's express it instead of writing down all the entries explicitly let's just indicate that columns of this matrix see are given by particular column vectors.
Birne Binegar: Right so think of.
Birne Binegar: These lowercase boldface these they indicate which call of them other n by n matrix see we're looking at.
Birne Binegar: Now what i'm going to do is i'm going to multiply this matrix see by an arbitrary matrix a and the claim is is that the result of this computation.
Birne Binegar: can be carried out by operating directly on the columns of the matrix see it turns out that the matrix a time see will be the matrix whose columns are simply a times the first column and see the matrix a time the second column C, etc.
Birne Binegar: And so we can see how the columns of theory transformed when we multiply by an arbitrary matrix a going to follow this formula.
Birne Binegar: All right, let me show you two by two example that exhibits this behavior here's to arbitrary two by two matrix needs I got their entries labeled by their.
Birne Binegar: positions in each matrix season carried out the usual matrix multiplication by taking Doc products, the first factor that products have rows of the first factor with.
Birne Binegar: column vectors of the second factor there's the full computation meanwhile if I take the same matrix a and I operate on the first call see one to one of the second factor.
Birne Binegar: We do that computation it works out to be well a one one scene, one month, plus a one to see one in the first slide etc, but notice that.
Birne Binegar: The column vector i've written down here is exactly the first column of a product that computed previously and, similarly, when I take the matrix a and I.
Birne Binegar: multiply the second column with the original matrix see by it.
Birne Binegar: I do, that computation and I reproduce the second call Well, this is not so much approved unless you understand how I actually cut computed this.
Birne Binegar: original columns the original product, the matrix a times the matrix see to get these columns what you do is you fix a column index of the second hand factor, and then you get the.
Birne Binegar: The role entries you just take the individual Rosa the first factor and got it with that same call, so the columns if, once you fix a column index.
Birne Binegar: What a does to those entries is just operate column by column and that's what this.
Birne Binegar: that's what this identity is saying the see how a axon see it's the fighters to the know how a X on each of the columns will see.
Birne Binegar: Alright, so that's one identity.
Birne Binegar: A second identity, I want to show you how.
Birne Binegar: What happens to a matrix see expressed in terms of the column what happens when I multiply it by diagonal matrix.
Birne Binegar: And now, this time i'm going to multiply from the right by a diagonal matrix and the statement is going to be well what multiplying a matrix see.
Birne Binegar: From the right by a diagonal matrix what that does is it takes the columns of the original matrix.
Birne Binegar: And it's simply scale and multiply so it's called vectors by the corresponding diagonal entry at the matrix D, so this first call and let's see.
Birne Binegar: I played a little lower Casey subscript one when I carry out this matrix multiplication the first column what the result is going to be first diagonal entry at a time to the first column of the matrix see Similarly, the second.
Birne Binegar: The second column of the product C times D is going to be the second diagonal entry of the time, the second column vector of seeing cetera.
Birne Binegar: Right, so I have two matrix identities little, let me rewrite.
Birne Binegar: The first one was well let's say see we express this.
Birne Binegar: Which Preston matrix see in terms of it's called vectors.
Birne Binegar: And then.
Birne Binegar: If we multiply see from the right.
Birne Binegar: What we get is a matrix that's going to be a times that first column vector written vertically so.
Birne Binegar: An n by n matrix time to invite one column vector is going to produce an invite one column vector that we can then fill in, and then the last component is going to be.
Birne Binegar: The original matrix a times the last column C written vertically alright so that's one identity we've seen if we write a matrix has columns this is how a matrix.
Birne Binegar: multiplying from the left hand side, how it acts on it, and similarly in that previous seeding slide or, I guess, in the current slide if we.
Birne Binegar: multiply matrix lab this form by a diagonal matrix from the right now what that does is it takes the original calls and see and adjust scale or multiplies them by the corresponding diagonal entry of D.
Birne Binegar: Lambda see in written vertically alright so given a matrix express in terms of column, we have these two identities alright, so let me show you the utility of that.
Birne Binegar: Well, here I actually prove this identity about it, if you work out a two by two example you'll see it's also equally true.
Birne Binegar: And the slide I show in more detail but it's basically you write a typical entry of this product that's, this is the Doc product of the throat see and the jade column D that's how much with the ik entering the product will be, but if you work it out.
Birne Binegar: explicitly what you see is is that see this.
Birne Binegar: entry of.
Birne Binegar: see exactly the.
Birne Binegar: entry in the case column at the original matrix in any rate now this identity folds the jade column of the product is simply the jake die of entry or he times the jade column vector of see.
Birne Binegar: Alright, so.
Birne Binegar: here's my purpose of showing you those two identities, well, you can actually see from the notation part of my intention here.
Birne Binegar: If i'm going to have a matrix.
Birne Binegar: That diagonal ICES a given matrix a what i'm going to want it is and again, do you like this, where C minus first times a time see is equal to a diagonal entry.
Birne Binegar: So i'm trying to flying for a given matrix a appropriate major CC and D feel that I have an identity like this, then i'll regard see as converting a into a diagonal matrix.
Birne Binegar: Now what this identity implies.
Birne Binegar: is well just multiply.
Birne Binegar: Both sides of this equation by the matrix see when see here to inverse it's just going to produce the identity matrix and so all that will survive on the left hand side will be a time T, but when see hits the.
Birne Binegar: right hand side of this equation when it hits D from the Left it's just going to produce the times D, so this identity that we want.
Birne Binegar: is equivalent to this identity, where we multiply the same matrix see either from the left by a or from the right by the diagonal matrix D, and now it will employ these two identities.
Birne Binegar: So what's going to happen on the right hand side that because of the now we can implement matrix multiplication from the right by having it the matrix a act on the individual columns of a.
Birne Binegar: The right hand side here it's going to be a times the first column vector of see a type of second column vector have seen the second column, etc.
Birne Binegar: and, similarly, when we multiply the matrix see from the right by a diagonal matrix D where you'll use this identity and what happens is.
Birne Binegar: The resulting matrix is going to be the ice the ice call it's going to be the ice dial entry of D times the ICT call with see.
Birne Binegar: And now we compare both sides of this identity well, you see that a time see one has to be Lambda one time see one, in other words.
Birne Binegar: That first column C one of see has to be an Eigen vector of a with eigenvalue Lambda one, let me write it out explicitly a time seeing one must equal Lambda one.
Birne Binegar: time see one, which means.
Birne Binegar: See one yes and I conductor.
Birne Binegar: Of a with.
Birne Binegar: eigenvalue.
Birne Binegar: Lambda one and that to not only have the first column, but it's going to be true for all the subsequent paul's as well the.
Birne Binegar: On the right hand side the columns here are going to be a times to call them that see.
Birne Binegar: On the right hand side here the columns of this product matrix are just going to be scaling multiples of the column of fee by the corresponding diagonal entry of D, meaning the ice calm the thieves being multiplied by the diagonal entry of the matrix deep.
Birne Binegar: Alright, so that's going to lead us to an algorithm by which we can diagonal is a matrix what we're looking for is a matrix see that will convert a to a diagonal matrix suppose we have.
Birne Binegar: An eigenvectors have a but just use those matrix to form a matrix see but now, instead of having label nice column by see, let me use the particular eigenvectors a as columns.
Birne Binegar: And then i'll use the matrix to construct the diagonal matrix D all us well for the first call i'll be wonders in eigenvector with eigenvalue Lambda one.
Birne Binegar: And so what i'll use, for the first time, but on entry is going to be the first eigenvalue of the matrix a actually what I really need is the eigenvalue have a corresponding to this first eigenvector the one.
Birne Binegar: The second diagonal entry fee is going to be the Eigen value of a that corresponds to the second eigenvector that we chosen for a etc.
Birne Binegar: And then we're going to have AC is equal to DC by the theorem.
Birne Binegar: and
Birne Binegar: So long as this matrix he is in vertical, then I can take this equation multiply it by the inverse of a C and i'll get the diagonal ization condition i'm seeking a way of converting the matrix a into a purely diagonal matrix.
Birne Binegar: Okay, so that's.
Birne Binegar: Essentially, how we're going to die go like matrix she'd.
Birne Binegar: Given an arbitrary matrix a the goal will be to find.
Birne Binegar: First of eigenvalues and eigenvectors once we found those we can use the Ai conductors to create this matrix see that will convert a into a diagonal matrix.
Birne Binegar: And it turns out that the diagonal matrix that created that way, well, we will know that diagonal matrix as soon as we know, the eigenvalues corresponding to our eigenvectors.
Birne Binegar: matrix see we write the auditing vectors in a particular order doesn't matter which order but we choose some ordering of these eigenvectors and then what will happen is that see will convert a into a diagonal matrix.
Birne Binegar: Moreover, we're going to already know that the diagonal entries of D are exactly the eigenvalues of a written in the same order as we've already ordered the Eigen vectors.
Birne Binegar: Right, let me do an explicit example.
Birne Binegar: So.
Birne Binegar: here's a two by two matrix in order to get this started we're going to have to first of all figure out what the eigenvalues and I conductors of this two by two matrix are then we'll use that data to construct the diagonal lighting matrix see and the corresponding diagonal matrix D.
Birne Binegar: Job let's start first we find the eigenvalues a that's done by looking at the characteristic polynomial and setting it equal to zero define.
Birne Binegar: The roots Lambda that will accomplish that right so here's the definition of the characteristic polynomial is determinant of the original matrix with the with the diagonal entry shifted by a parameter Lambda.
Birne Binegar: So we can compute that determinant using the difference of diagonal products because it's a two by two matrix we're gonna work out to be.
Birne Binegar: Just lastly top of triangular so we've just determine it's just a product of the diagonal entries.
Birne Binegar: And what we see so that determinant is already factored much we computed that the eigenvalues the solution to this equation.
Birne Binegar: are going to be Lambda equal to two or Lambda is equal to negative one.
Birne Binegar: Right so Those are two organizers Lambda equal to Lambda equals one.
Birne Binegar: keep track of this date on our pad a is equal to.
Birne Binegar: 260 minus one has eigenvalues.
Birne Binegar: First, one is Lambda equals.
Birne Binegar: Two, and the second one is Lambda equals a negative one.
Birne Binegar: All right, that's the first half of the eigenvector eigenvalue problem, the next step is to find the corresponding eigenvectors.
Birne Binegar: So now.
Birne Binegar: This Eigen space, corresponding to the first eigenvalue Lambda equals to this is the knowledge base.
Birne Binegar: Of the matrix well when I do is take a then subtract that eigenvalue from its diagonal elements, so if the if it's the.
Birne Binegar: Most base of this matrix, which is a no space.
Birne Binegar: Of on explicitly computing will have two minutes to zero that entry doesn't change that entry change negative one minus two is going to be negative three.
Birne Binegar: Alright, so what happens.
Birne Binegar: let's.
Birne Binegar: Do that right.
Birne Binegar: Ah wow I did it right.
Birne Binegar: What I didn't do is complete the calculation here alright, so this is no space of this matrix but now let's put this matrix in reduce row Echelon form well, that means.
Birne Binegar: Maybe normalizing this first row, so its pivot is equal to one and then using that pivot to clear out that entry, so this is actually the null space.
Birne Binegar: Of 0100 all I did here is I gone to reduce row Echelon form.
Birne Binegar: simplify the calculation of that no space, right now, what we need to do is figuring out the solutions this coefficient matrix times, an arbitrary vector X one X two is equal to zero.
Birne Binegar: But what we see from this matrix and reduce will actual on or is it the first column lacking a pivot it's going to mean that X one is a free parameter.
Birne Binegar: While the second column or works typically the second row it's telling us that the second component of a solution is going to have that equals zero so X two X equals zero X one is free, so a solution vector is of the form X one zero or X one times this basis sector one zero so.
Birne Binegar: that's going to equal the span.
Birne Binegar: Of the vector one zero and.
Birne Binegar: The basis factor for this Eigen space is what we would regard as the Eigen vector.
Birne Binegar: Of the original matrix a with.
Birne Binegar: eigenvalue.
Birne Binegar: Lambda equals two two that's what this calculation has revealed to us.
Birne Binegar: All right now.
Birne Binegar: We had to eigenvalue so we now have to do.
Birne Binegar: Is look at that second login space correspond to the second eigenvalue you repeat the same sort of calculation.
Birne Binegar: let's do that.
Birne Binegar: you're right finishing up that first one.
Birne Binegar: right here, it is so now we're looking at the minus one Eigen space, so I subtract negative one from the diagonal of a and calculate the no space of that matrix well here's just doing that subtraction.
Birne Binegar: And it once I do that it's already in reducing Echelon form, so I can immediately figure out a basis for the solution said it's going to consist of factors, where X one plus two times X is equal to zero or equivalent Lee the first.
Birne Binegar: component X, one has to equal Nick negative two times X two we get solution vectors of this form or equivalent Lee an arbitrary constant X two times this basis vector minus two so for Lambda equals minus one a similar calculation.
Birne Binegar: tells us that the Lambda equals negative one, the corresponding eigenvector is going to be a vector with components minus two one.
Birne Binegar: Alright, so we've now sold the eigenvector eigenvalue problem for this matrix a.
Birne Binegar: and
Birne Binegar: summarize our results.
Birne Binegar: here's a equals.
Birne Binegar: 260 minus one we have an eigenvalue land that equal to the corresponding eigenvector i'll call it.
Birne Binegar: The subscript two and that's actually the first standard basis vector and for our second eigenvalue minus one.
Birne Binegar: We have its corresponding eigenvector is minus two one.
Birne Binegar: All right.
Birne Binegar: Now what we know if we can apply the theorem.
Birne Binegar: If we're gonna have a matrix see the minus first times a time see is equal to D, we need.
Birne Binegar: See to be made out of.
Birne Binegar: The eigenvectors of a.
Birne Binegar: and
Birne Binegar: D is going to be form.
Birne Binegar: From the eigenvalues a.
Birne Binegar: As follows so see.
Birne Binegar: What i'm going to use for the columns of see are going to be exactly the eigenvectors of age, so let's write them in the order that we found them one zero minus two one.
Birne Binegar: And now.
Birne Binegar: I already know what this matrix D is going to be.
Birne Binegar: matrix D, the organization of the original matrix a it's going to be formed using the eigenvalues of a the way I order those eigenvalues I use the same ordering is how use the calls.
Birne Binegar: So the first diagonal entry it's going to be an eigenvalue of a corresponding to this first eigenvector so that's going to be negative two.
Birne Binegar: And then the second diagram entry the entry down here well it's going to be the second the eigenvalue have a corresponding to that second call of see.
Birne Binegar: And there the eigenvalue was minus one, so I this is right, I can tell you is to, and here the eigenvalues negative one, so, in fact, this identity is going to hold.
Birne Binegar: For the exactly these two matrices.
Birne Binegar: What i'll show you i'll show you that by direct computation now.
Birne Binegar: So let's i'm.
Birne Binegar: Here I constructed, the two matrices following our prescription using eigenvectors have a for the columns that see and using the corresponding eigenvalues for the diary entries of D.
Birne Binegar: So i'm going to check this full identity here so i'm going to compute the inverse of this matrix see using a convenient formula.
Birne Binegar: One we were discussing determinants of matrices we had a formula for calculating the inverse of the matrix each using.
Birne Binegar: The determinants of co-factor matrices for determinants of miners have a here's what that formula works out in the case of a general two by two matrix.
Birne Binegar: The inverse of the two by two matrix with entries a B, C D is one of the determinant of the original matrix times the transpose of its own factor matrix which works out to be the minus B minus C a.
Birne Binegar: So, in the case at hand.
Birne Binegar: The determinant of this matrix see is going to be one minus zero or simply one and then.
Birne Binegar: Looking at what this inversion process does to the entries it's reverses.
Birne Binegar: The positions of the diagonal entries well one is equal to one, so it looks like no change here, but actually what i've done is I switched to these two diagonal entries.
Birne Binegar: And then it also reverses the sign of the off diagonal entry so there's two should have been minus two here.
Birne Binegar: And the zero should been minus two sorry I think i've got that written wrong oh I Oh well, oh I did I did reverse the sign here this negative to has gone to falling what happened to the.
Birne Binegar: One to entry on both sides of the this formula alright so.
Birne Binegar: What i've got here well what i've got on the far right is the matrix shot creator from the eigenvectors of a but i've gotten the Center here is a itself and what i've done on the right here is the inverse of this matrix see we multiply those together and what it happened as advertised.
Birne Binegar: The this triple matrix product see in verse times a time see has produced a diagonal matrix and notice going back to my pad here.
Birne Binegar: it's exactly the diagonal matrix that was predicted from our diagonals eight for a promo I can vector and I get that from a solution that the eigenvector eigenvalue problem a the matrix see that converts a to a diagonal matrix is indeed.
Birne Binegar: constructed using the eigenvectors of a and the diagonal matrix that a is converted into well that correspond to exactly that a diagonal matrix entries are the eigenvalues of the original matrix a.
Tyler Martinez: Professor I have a question.
Tyler Martinez: yeah so, can you explain again how you know which eigenvalue goes into which entry of D.
Birne Binegar: Alright, so.
Birne Binegar: So i'm going to fill both matrices out call my call so like choose one of the eigenvectors have a.
Birne Binegar: I write it as the first call let's hop over to D will get that first diagonal entry by using the eigenvalue they correspond that eigenvector.
Birne Binegar: So what's going to happen is the columns of C and D are going to be naturally correlated what you're going to use eigenvectors for C.
Birne Binegar: And then the corresponding column D what you'll use is the corresponding eigenvalue so.
Birne Binegar: This goes in the sea, this goes into the corresponding column D, this is going to go into the matrix see this is going to go into the corresponding column.
Birne Binegar: D, so you just that you just work out you to some ordering of the eigenvectors and that's because you have this connection between the columns of seeing the columns a D you'll also get the right ordering of these eigenvalues that way as well.
Tyler Martinez: Okay, thank you.
yeah.
Birne Binegar: All right, so um there's one caveat to this process.
Birne Binegar: and
Birne Binegar: That is.
Birne Binegar: i'll show you the step that occurred.
Birne Binegar: If we have a bunch of eigenvectors have a.
Birne Binegar: And i've run format matrix D using the eigenvalue well they i've diagonal entry at a let's suppose I chosen that so that always coins plans to the.
Birne Binegar: I can value of the ice calm at the matrix see if that true this identity will hold that it goes back to those matrix multiplication identities, the tricky part is.
Birne Binegar: Is that, in order to get the second equation C minus first time a time see is equal to D, I have to multiply this identity by the inverse of see but as we've seen.
Birne Binegar: Make your seat do not always have in verses and so that's why I put this condition in both in both fun because I can't get from this valid equation to this equation, I want, unless I know that the matrix see is in vertical.
Birne Binegar: So the one caveat to this construction to the to our being able to diagnose a matrix.
Birne Binegar: And diagnostics and matrix a is when we can in our construction at the matrix see we might not be able to construct and convertible matrix in which cases to kind of break down.
Birne Binegar: But what i'm going to show you next is like right exactly where it breaks down how we can spot that a little bit more easily.
Birne Binegar: So let's move forward tool or read here we've worked through this example and now, what I want to remind you of is.
Birne Binegar: How can we make sure that a matrix see that we construct using the I can vectors of a given matrix a, how can you make sure that that's actually an embeddable matrix.
Birne Binegar: And so I remind you of a theorem the whole fundamental theorem up in but convertible matrix these that gives us criteria for when a matrix has an inverse.
Birne Binegar: Well, we had a bunch of criteria, the one I want to focus on here is this last at this fourth criteria that the rank of not the matrix see has to equal the size of the matrix so.
Birne Binegar: You like that condition.
Birne Binegar: So condition.
Birne Binegar: For that see is convertible.
Birne Binegar: Well that's that the rank of the matrix see has to equal, and so this is an already n by n matrix So what we need is this matrix to have what we call full rank and, of course, the rank is also the dimension.
Birne Binegar: Of the column space.
Birne Binegar: Of see and feel that's going to equal, the number of basis factors.
Birne Binegar: For.
Birne Binegar: For the calm state of see and So what do we need we're going to need in literally independent call vectors right because the.
Birne Binegar: dimension of column space it's going to be determined by.
Birne Binegar: The number basis factors we need well those basis vectors we're going to need exactly enter them.
Birne Binegar: And what what effectively, that means there's been a need the column.
Birne Binegar: If if we have a matrix that's convertible the columns that see have to be linearly independent so.
Birne Binegar: In other words.
Birne Binegar: We have a maybe a six criteria see is in vertical.
Birne Binegar: If, and only if all it's called.
Birne Binegar: Literally independent.
Birne Binegar: So.
Birne Binegar: In the case at hand we're going to be creating a matrix see using Ivan the vectors of a of a of a given matrix So what we need.
Birne Binegar: yeah and so, in order to get.
Birne Binegar: In vertical matrix.
Birne Binegar: See constructed from.
Birne Binegar: Our diarization algorithm says we should be constructing this matrix see from the eigenvectors of age, so this in vertical matrix she's it's constructed from.
The eigenvectors.
Birne Binegar: A as columns.
Birne Binegar: So the question is, is this matrix see is going to be in vertical only if it's columns are all linearly independent, but the way we're intending them constructed matrix is using eigenvectors have a to get the columns so what you need.
Birne Binegar: What we're going to need is in.
Birne Binegar: Literally.
Birne Binegar: Independent eigenvectors.
Birne Binegar: And so that's going to be.
Birne Binegar: The the basic criteria for when matrix a is going to be diagnosable we can always satisfied the identity AC is equal to see times D, we can always.
Birne Binegar: satisfy this identity, simply by creating creating the matrix see from the eigenvectors a and then great you matrix D I just tearing up the eigenvectors corresponding to.
Birne Binegar: eigenvectors have a with their eigenvalues, but the problem is to actually get an inverted matrix see so that's this identity hold because that identity that we want CA the minus firstly is equal to D that only works when see is that in vertical.
Birne Binegar: And the argument we made in the last couple slides then is is that.
Birne Binegar: We can satisfy this identity, when we create see from eigenvectors and D from eigenvalues and we have enough linearly independent eigenvectors so that, in fact, this inverse matrix these exists.
Birne Binegar: And so what happens in.
Birne Binegar: it's the following a matrix a is going to be diagonal angel if, and only if it has enough linearly independent eigenvectors it needs its many linearly independent eigenvectors, as we have columns in the original matrix.
Birne Binegar: Alright, so now so that's that's the basic.
Birne Binegar: principle I, which will die analyze matrices.
Birne Binegar: Alright, so.
Birne Binegar: I think what i'm going to do is.
Birne Binegar: i'm going to stop here today, even though it's four minutes early it's also.
Birne Binegar: My understanding so weekend before Easter is the Friday before Easter so i'll let you guys go maybe four minutes early, but what we will do on Monday.
Birne Binegar: Well we've seen.
Birne Binegar: A is.
Birne Binegar: Reliable.
Birne Binegar: If, and only if.
Birne Binegar: A has so this is an invite in matrix and has at least well it had can only have it most and as well if all if, and only if he has in.
Birne Binegar: Literally.
Birne Binegar: Independent eigenvectors.
Birne Binegar: And then, what i'll do on Monday.
Birne Binegar: This will simplify these criteria so answering this question does a half in linearly independent eigenvectors well to figure that out you're gonna have to find all the eigenvectors of a and count them.
Birne Binegar: And what i'm going to do on Monday is give you some simpler criteria, so you can detect whether or not a matrix is dying realizable or not, without having to do an explicit computation of all eigenvectors and eigenvalues to simplify.
Birne Binegar: This criteria.
Birne Binegar: For special cases.
Birne Binegar: So, for example, it's going to turn out that if your matrix is is symmetric meeting is equal to a transpose, then the matrix is always diving likable.
Birne Binegar: And that particular example, while seemingly special you have to work, the matrix that looks the same when you convert rosen the columns that turns out to be pretty common in physics and an engineering and so it's good to know that we have a diet.
Birne Binegar: A symmetric matrix that this diagonal ization is always going to work for us alright, but i'll stop here today and we'll pick up in class i'll be returning to host you on Monday we'll pick up with this discussion on Monday okay so have a good weekend and I will see you on Monday.