Birne Binegar: Okay, so.
Birne Binegar: Thank you for joining today's lecture your zoom.
Birne Binegar: Today, what i'm going to talk about.
Birne Binegar: or an application of the eigenvector liking value problem it's used to simplify and otherwise more complicated matrix making into a much simpler matrix.
Birne Binegar: So.
Birne Binegar: The target matrix is going to be what we call a diagonal matrix so let me.
Birne Binegar: tell you a little bit about that now, so we say that a square matrix is diagonal yes it's non zero entries only occur along the main diagonal from the upper left lower right, let me give you a picture so here this matrix D in the bottom of the slide here is i'm in diagonal form.
Birne Binegar: it's only non zero entries occur in that diagonal from the upper left hand corner, to the lower right hand corner.
Birne Binegar: what's nice about diagonal matrix he says, first of all, they have far fewer components to deal with right general an n by n matrix is going to have in the squared components.
Birne Binegar: Well, for a diagonal matrix you only have you only non zero entries or the one along this main diagonal.
Birne Binegar: So that makes it simpler, but also it's multiplicative properties aren't going to be much easier, as well suppose I have to diagonal matrices.
Birne Binegar: If I multiply them together well what happens here is using the usual rules of matrix multiplication where you take.
Birne Binegar: That products at the first row of first factor with columns of The second factor, most of those are going to evaluate to zero.
Birne Binegar: And, in fact, if you have to diagonal matrices the only non you only entries it survived the multiplication or again going to be diagonal entries.
Birne Binegar: All the other entries with zeros so a diagonal matrix times a diagonal matrix is going to be a diabolical matrix and moreover.
Birne Binegar: It the entries of that product of diet, you know matrices they can compete we really simply that i'm having to take dot products have real vectors with column vectors.
Birne Binegar: You simply have to multiply entries together that first diagonal entry of the product is the first I have an entry of D times the first diagonal entry of the prime.
Birne Binegar: The second diagonal entry of the product is going to be the second diagonal entry of D time the second dive on gap prime etc.
Birne Binegar: that's not only true for matrix multiplication but also when we multiply basis vector or vectors by diagonal matrix these things are much simpler as well.
Birne Binegar: For example, if you take a standard basis vectors let me work this out and detail for two by two example let's say a is equal to one two.
Birne Binegar: Well that's non diagonal let's say it's 230, so this is a diagonal two by two matrix whose diagonal entry in order are two and three let's take the standard basis factor.
Birne Binegar: there's the first one and here's the second standard basis vector written that's calls if I multiply either one of the standard basis vectors by this.
Birne Binegar: diagonal matrix something nice happens eight times, he one, for example, is going to be 2003 times one zero now we do that product explicitly will have two plus zero is to zero Plus zero is zero and what we get is effectively two times.
Birne Binegar: Two times at first standard basis vector and, similarly, when we multiply the second standard basis vector by that dive will matrix.
Birne Binegar: 01 well when we compute this product explicitly it works out to be well first entry is going to be 002 times 00 plus zero times 100 times zero is zero plus three times, one is three and that just equal to three times, he too so look at this miracle.
Birne Binegar: When I have a diet little matrix and I multiply standard basis factors, what happens is those standard basis vectors end up being eigenvectors for the diagonal matrix and in fact they're eigenvalues.
Birne Binegar: I can value, corresponding to this first standard basis vector is exactly the first diagonal entry of a the eigenvalue for the second standard basis factor is exactly the second.
Birne Binegar: diagonal entry of a and so.
Birne Binegar: Not only is matrix multiplication in general, simpler, but multiplying standard basis vectors is especially easy.
Birne Binegar: In general, what you're going to have well I think i've written it down on the slide here, if you have a diagonal matrix types of standard basis vector is just going to be the ice.
Birne Binegar: The ice diver entry of D is going to be the eigenvector though or the eigenvalue corresponding to the standard basis sector.
Birne Binegar: Now that only makes the action COMP computing the action of a diagonal matrix on standard basis factors easy, but even more general directors as well because of what you could do.
Birne Binegar: Take an arbitrary vector with entries V1 through the end expand this vector in terms of standard basis vectors you get an expression like on this top line, but then you can.
Birne Binegar: distribute that matrix product over the individual terms here, and you can move the matrix through those scale or factors of V1 V2 etc, and so what happens.
Birne Binegar: The diagonal matrix times, an arbitrary vector V is just going to well simply computed all you have to do is take the entries of V and multiply each entry by the corresponding diagonal entry of a So the first.
Birne Binegar: The first component of this product it's going to be birthed.
Birne Binegar: First dive on an entry of D times the first entry of V, a second a second component is going to be the second entry of be time the second diagonal entry of the matrix dx era, so all these computations become much simpler and what usually happens when computations becomes simpler.
Birne Binegar: What you can do with them your understanding of the computation also simple ice as well, so, in short, what we're going to try to do is.
Birne Binegar: If we were given a matrix it's not diagonal we're going to look for a way to find an equivalent matrix matrix D, that is diagonal that might be easier to deal with than the original matrix we are given.
Birne Binegar: Alright, so that leads me to this concept that diamond likability.
Birne Binegar: suppose you have a matrix it.
Birne Binegar: Not necessarily diagonal I so it's not necessarily a diagonal matrix but will say that it's diagonal lines if we can somehow convert it into a diagonal matrix.
Birne Binegar: What that's going to require of us isn't some matrix see that's going to carry out the conversion for us and it's supposed to work like this.
Birne Binegar: You there's an there's an invisible matrix see such that when I multiply a from the left by the inverse of a C and I multiply a from the right, I see itself what's supposed to pop out it's a diagonal matrix.
Birne Binegar: When we can arrange that to happen, will regard see as converting a into a diagonal matrix.
Birne Binegar: and
Birne Binegar: Later on i'm not sure if we're going to get to it this semester, because much of this course has been abridged yeah either to do to sick days or to.
Birne Binegar: snow days or holidays etc well i'm probably going to end up being about five lectures short of the normal course but normally what I would do now, after I introduced this concept of.
Birne Binegar: dying likability is I try to show you that this matrix see that does the diagonal ization of a is interpreted as simply making a change of coordinates but we may actually get to that will do today is also simply show you.
Birne Binegar: i'm sorry, let me get the slides in order here will simply show you how you conduct diaghilev's the matrix.
Birne Binegar: matrix a and what we'll find is the I vector I can value problem that we considered in the two earlier lectures which week that's the crucial step.
Birne Binegar: of dying lighting and arbitrary matrix a and, in fact, once you sell the the eigenvector eigenvalue home for a, you will have effectively also solve the problem of fighting a matrix see that will convert a into a diagonal matrix.
Birne Binegar: Alright, so.
Birne Binegar: Let me proceed with that what.
Birne Binegar: What this is going to be based upon is too simple matrix identities that I could have been induced.
Birne Binegar: Maybe in the second week of the course when we were discussing matrix multiplication explicitly.
Birne Binegar: The specialization i'm going to do here is i'm going to choose one matrix see and i'm going to think of it.
Birne Binegar: As a matrix that I create I taking a bunch of calling vectors.
Birne Binegar: and combining them together.
Birne Binegar: Until I get an invite in matrix.
Birne Binegar: So I want these columns to be.
Birne Binegar: Each have an entries in it, when I have in separate columns but I gotta get is an n by n matrix.
Birne Binegar: So rather than specify this matrix in terms of all its component see one one scene one two weeks cetera i'm gonna look into sort of an ego meteor presentation of this matrix treating it's called as if they were vectors.
Birne Binegar: And what will study in this next couple of slides is what happens when I multiply and matrix like this is an arbitrary matrix.
Birne Binegar: And in fact that's the contents of this first level, so in this lemma a is any matrix you want, but see is a matrix that n by N and we've also represented.
Birne Binegar: In terms of its column vectors.
Birne Binegar: The statement is is that I can do matrix multiplication of the matrix see from the left by an arbitrary matrix a and the result will be.
Birne Binegar: A matrix that kind of written in the same way i'm going to specify a product.
Birne Binegar: A time see i'm going to write it it's going to be another n by n matrix but i'm going to write it in terms of its columns as I did for see itself and what it works out to be is a times the first call.
Birne Binegar: So this will be an n by n matrix it can multiply it by one matrix and produce another end by one matrix.
Birne Binegar: So note that a time see one is going to be in n by n Thompson and by one what it's going to kick out is going to be an end by one matrix This makes the.
Birne Binegar: multiplication defined and the exterior dimensions give you the result of the of the product so what's going to end up here is a column vector within entries, and once i've done that the each of the calls and a.
Birne Binegar: I end up with an n by n matrix that's going to in fact coincide with the actual product I would get like taking the rows of a and taking their dot products with the columns of see to get the entire product AC.
Birne Binegar: Alright, so let me give you an example that shows you why this identity is true.
Birne Binegar: Effectively, it says we can compute.
Birne Binegar: The action of.
Birne Binegar: Left multiplication by a matrix a on an arbitrary matrix see like just knowing how a X on the individual calls see.
Birne Binegar: Alright, so here's how it works for two by two example.
Birne Binegar: So I Brittany two matrices agency in very generic notation.
Birne Binegar: and
Birne Binegar: When I multiply them together, that means, for example, to get this one one entry, I have to take the Doc product of the first row of a with the first column of see.
Birne Binegar: A second entry is going to be a second entry in the first row of the product is going to be a dot product first row first factor, and the second column of the second factor, the usual computation.
Birne Binegar: Now look what happens when a multiplies a particular call of the matrix see so what i've got here, seeing one one above see two and it's just that first call a will do that matrix multiplication.
Birne Binegar: And what gets kicked out well look at this within the matrix I end up with is exactly the first column of the matrix product AC and, similarly, when I can see how a actual on the second column of see I get exactly the second column of the matrix product, a time see.
Birne Binegar: So this identity.
Birne Binegar: That matrix multiplication get an arbitrary matrix n by n matrix see can be computed simply by knowing how a X on the call to see that's true.
Birne Binegar: Next, I want to look at what happens when I again specify a matrix in terms of its calm, but this time i'm going to multiply see not from the right, but fun for the left by a diagonal matrix.
Birne Binegar: And here's the here's the corresponding identity, if I have a matrix of all diagonal entries.
Birne Binegar: And I multiply it from the right times, an arbitrary matrix who's columns or lowercase C one through lowercase C and what happens is pretty simple.
Birne Binegar: The result of multiplying these two matrices together it's just that the first column C gets changed to that first diagonal entry times the first column.
Birne Binegar: What will happen to the second column at the second diagonal entry fee will multiply the second column of see and we'll just get now the scale or multiple the second column of the by the second diagonal entry of D, that will be the second call.
Birne Binegar: And so the second identity that we want is when.
Birne Binegar: A matrix see is multiplied from the opposite side by diagonal matrix.
Birne Binegar: The only thing that happens is.
Birne Binegar: The matrix see gets changed by these diagonal entries but it's kind of a simple way.
Birne Binegar: When the two types of second call the written vertically and last column with the last diagonal entry he times the last column vector see written vertically.
Birne Binegar: Okay, so two identities that are are basically identities that all from the usual rules of matrix multiplication.
Birne Binegar: you're on this slide i'll approve that lemon.
Birne Binegar: So a matrix is diagonal if it's only non zero entry, secure and when the row index is equal to the column index, otherwise the entries are all zero so right the entries at the diagonal entry in this fashion.
Birne Binegar: Then, when I multiply matrix together what I do is, I take the Doc product of rows of the first factor with columns of The second factor that that's what this some represents is that Doc product of a real vector I with a column and vector the eighth calm the now.
Birne Binegar: way i've my notation is i'm going to use this both based on to indicate the case call so that's where I fix the index K, to be some fixed number, and I just very the row index to get the entries of that column vector.
Birne Binegar: And then the diagonal entries are the entries of this matrix D are given by well this Lambda icons this Chronicle delta delta I J.
Birne Binegar: And what you can observe from this is what i've computed when i'm circling here so i'm.
Birne Binegar: This formula computes all the entries of the product, see times D, but once I fix that jade column i'll be looking at a particular column at the product and what we see is the entries in that call are just.
Birne Binegar: The entries at the matrix see in the jade column times the corresponding diagonal entry of D it's exactly this this identity here that i've.
Birne Binegar: And i've written written on the pad and putting the box okay so.
Birne Binegar: Now, what I can do is use these two identities to describe more explicitly what it takes to diagonal is a matrix.
Birne Binegar: Alright, so here let's start off with an arbitrary but square matrix and we're looking for a matrix see that's gonna allow us to convert a into a diagonal matrix.
Birne Binegar: What i'm going to use is these two identities and it will tell us precisely how to construct the matrix see and the maitre D that does this diagonal ization of a.
Birne Binegar: All I have to do.
Birne Binegar: Is take the desire condition C minor spurs a time see is equal to D and multiply both sides of that condition, like the matrix see now what's going to happen.
Birne Binegar: Is when you have.
Birne Binegar: See the minus first time a time see is equal to D that's our.
Birne Binegar: diagonal ization condition.
Birne Binegar: See is converting a into D well when we once we multiply both sides of that by see we're going to get well see time see the minus first that the identity matrix and the identity matrix time AC is just gonna produce a see.
Birne Binegar: on the opposite side will have see time to D.
Birne Binegar: And so.
Birne Binegar: The conditionally one it's going to imply that a time see has to equal see times D and now let's employ these two identities.
Birne Binegar: That tell us another way of writing.
Birne Binegar: See times a matrix a or a time to matrix C or C time for diagonal matrix D.
Birne Binegar: exploiting those two identities.
Birne Binegar: What we get yes Well, first of all, they see is equal to.
Birne Binegar: The matrix a times each of the columns of see written in their natural order so that explains right hand side, meanwhile, the left hand side, we can compute that it's going to end up being.
Birne Binegar: A columns of the matrix see, but each has been multiplied by the diagonal entries of D, that the first column of C has been multiplied by the first diagonal entry of D, etc.
Birne Binegar: And so, when we compare the column of both sides of this identity, it means that a time see one is equal to a scale or multiple see one and that scale or multiple Therefore, it is exactly.
Birne Binegar: And I get value a because a time, so this vector is producing a scale or multiple that vector.
Birne Binegar: So.
Birne Binegar: What happened.
Birne Binegar: Because of the matrix multiplication identities.
Birne Binegar: First of all, c minus first a time see is equal D that's the condition we want is equivalent to see a is equal to.
Birne Binegar: See times D, which is telling us that.
Birne Binegar: A times the first call.
Birne Binegar: i'm going to write this matrix down.
Birne Binegar: it's going to equal let's do the first call with this it's going to be the first diagonal entry.
Birne Binegar: The diagonal matrix times that first call, and now we compare the two entries, and we see that a time see one.
Birne Binegar: has to equal Lambda one time see one meaning see one is an Ai conductor of a.
Birne Binegar: With.
Birne Binegar: eigenvalue.
Birne Binegar: Lambda one now that's just looking at the first call, but the second call would be similar would be a time see to.
Birne Binegar: The second column on the left side would be the second diagonal entry be time to second call of see and again.
Birne Binegar: A time see two is going to be a scale or multiple see to because we have this identity both both sides have to agree exactly and, in general, a times the ice call them up see it's going to be exactly.
Birne Binegar: The ice diagonal entry and D times the ice column of see.
Birne Binegar: Well, this is going to be true for all I from one to n.
Birne Binegar: Alright, so.
Birne Binegar: In review the dying realization problem is to convert a non diagonal matrix into a diagonal matrix.
Birne Binegar: What these calculations show is that, for this to happen, we need this identity and this identity in in.
Birne Binegar: In turn, requires that each column of C has to be an Eigen vector of a and each guy a diagonal entry of the diagonal matrix D has to be an Eigen value of a.
Birne Binegar: Moreover, they have to be correlated in this section, the eigenvalue corresponding to that, first, I get back so this see one has to be you know eigenvector and in fact it's eigenvalue has to be exactly equal to the first diagonal entry fee.
Birne Binegar: So this now leads us to an algorithm for computing.
Birne Binegar: Both the matrix see that diagonal lines is a and the diagonal matrix D that a is converted into via this matrix multiplication by see.
Birne Binegar: here's how it works what we'll do is we'll find.
Birne Binegar: First of all, with let's assume we've solved the eigenvector eigenvalue problem for the matrix a we found all the eigenvalues like finding the zeros at the characteristic polynomial and then for each eigenvalue we've calculated.
Birne Binegar: A corresponding eigenvector.
Birne Binegar: Alright, so what we'll do is we'll take those the eigenvectors we found and we'll use them to form a matrix see so what i'll choose for.
Birne Binegar: let's create.
Birne Binegar: A matrix see.
Birne Binegar: From.
Birne Binegar: The eigenvectors of a.
Birne Binegar: So see it's gonna i'm gonna you slightly different location.
Birne Binegar: it's going to be some eigenvector have a and then for the second call us another eigenvector of a cetera and now i'll need a total up.
Birne Binegar: In calls for this n by n matrix so i'll need in eigenvectors of the matrix a to create this matrix see in these are going to be eigenvectors and so a time either of them any one of these is just going to be some eigenvalue the matrix it tough.
Birne Binegar: Okay, so I make us matrix see this way.
Birne Binegar: And then, what I can also do.
Birne Binegar: Is I got these eigenvectors arranged in a natural order some what whatever I chose to be the first one.
Birne Binegar: it's going to have a first eigenvalue whatever I chose to be the second I can vector it's going to have its own eigenvalue as well, what i'm gonna do is i'm going to create a diagonal matrix.
Birne Binegar: Writing down these eigenvalues in the same order is there eigenvectors have been written here so put Lambda one here, corresponding to this first call.
Birne Binegar: Zero everywhere else in those row, and then the second diagonal entry D or use the second.
Birne Binegar: i'll use the eigenvalue corresponding to this, I can vector a day like fill out since I have in eigenvectors i'll fill out the diagonal entry that way.
Birne Binegar: And then what's going to happen here is that.
Birne Binegar: See.
Birne Binegar: A time see is now going to equal.
Birne Binegar: The Times sorry.
Birne Binegar: See times D automatically.
Birne Binegar: Because we know how a X on the see it just multiplies each of these vectors.
Birne Binegar: Each of these column vectors to get multiplied by a but since they're eigenvectors i'm just going to get Lambda Lambda icon ci for any of these columns and this product.
Birne Binegar: But also the same thing is going to happen here when I multiply them call matrix the matrix see by the diagonal matrix D it's going to multiply each of the columns of see by the appropriate diagonal entry D and this this this identity is going to be maintained automatically so.
Birne Binegar: If we multiply.
Birne Binegar: Both sides.
Birne Binegar: By see the minus first, then we get the desired identity, see the miners first times AC is going to equal well see the mind first time to see it's going to equal the identity matrix the identity matrix time D is going to just equal D, so they.
Birne Binegar: Identity we want.
Birne Binegar: That Dr the diagonal ization of the matrix a by this our jewelry matrix see.
Birne Binegar: All right, so that's how you diagonal is a matrix you do so by giving a matrix a you find all the eigenvalues or.
Birne Binegar: And you find all the to eigenvectors you arrange the eigenvectors into an invite in matrix and then what will happen is that a time see is automatically going to equal see time see times D and then.
Birne Binegar: Then we multiply both sides by the inverse of a and we're able to diagonal ice the matrix as explicitly however there's one caveat here.
Birne Binegar: and
Birne Binegar: I was.
Birne Binegar: A little bit cavalier when I said okay once we get to this equation, all we have to do with multiply by the inverse of see except what we've known for a long time now, this is not every matrix has members, so this step here.
Birne Binegar: If I not work if the matrix see that we construct from the eigenvectors pay if it doesn't have an inverse then.
Birne Binegar: it's not going to work, but so long as we can invert the matrix see that we construct in this fashion, then we will be able to die in life, the matrix a.
Birne Binegar: Alright, so i'll talk a little bit more about this this caveat to our diagonal ization of a in a minute let's work out.
Birne Binegar: explicitly a nice two by two example where everything goes smoothly.
Birne Binegar: Right so.
Birne Binegar: looks like.
Birne Binegar: Somehow lost my slides here.
Birne Binegar: We bring them back.
Birne Binegar: no idea where they went let's see.
Here we go.
Birne Binegar: alright.
Birne Binegar: We are good and.
Birne Binegar: Good right at the diarization algorithm alright, so what i'm going to do in this next slide just work out this example.
Birne Binegar: Okay, so in this example of a two by two matrix that's not dying, you know i'm going to find a matrix see that's actually going to die, or is it.
Birne Binegar: The first step it's going to be to solve the eigenvector like in value problem for the matrix a that we're trying to die.
Birne Binegar: So first thing we do is, we find the eigenvalues a that means looking at the characteristic polynomial.
Birne Binegar: The determinant of a minus Lambda times the identity matrix that's going to produce a polynomial and the printer Lambda which set that equal to zero, to find the roots.
Birne Binegar: And it's pretty obvious that, for this example the eigenvalues are going to be the numbers two and the numbers negative one, those are the values of Lambda for which the characteristic polynomial vanishes.
Birne Binegar: Right.
Birne Binegar: For step the eigenvector eigenvalue problems, the plane, the eigenvalues.
Birne Binegar: Then, for each eigenvalue we're going to find the course corresponding eigenvector.
Birne Binegar: dope for that first eigenvalue land that equal to.
Birne Binegar: Let me use the pad to keep our.
Birne Binegar: result.
Birne Binegar: available to us to 00 minus one was a lot without her eigenvalues Lambda is either equal to two or negative one, and now we're supposed to focusing on Lambda equals two and we wanted to eigenvector.
Birne Binegar: Or the Eigen space.
Birne Binegar: He subscribed to the set of all vectors two dimensional vectors for which.
Birne Binegar: That are eigenvectors for a with eigenvalue to that now that amounts them now solving a linear system.
Birne Binegar: With coefficient matrix.
Birne Binegar: So we want.
Birne Binegar: That it's going to be no space.
Birne Binegar: Of the matrix we create by subtracting from a the eigenvalue.
Birne Binegar: along its diagonal entries alright so i'm i'm computing here like compute that this matrix let's subtract the number two from the dive on entries of a and what we get is a minus two times I it's going to equal.
Birne Binegar: To minus two zero that doesn't change that doesn't change here's a diagonal entry minus one minus that diagram that eigenvalue is going to be negative three.
Birne Binegar: So this is.
Birne Binegar: This is the matrix we want.
Birne Binegar: But we quickly reduce to reduce relational for what we get in this matrix with component.
Birne Binegar: You can see that I can re normalize.
Birne Binegar: This row, so that its pivot is one big lead to that and I can also use that first pivot to clear out the entry here let's going to be the we do show allegedly entry form at this matrix and now we can.
Birne Binegar: solve the corresponding linear system, a minus two times I X is equal to zero, and you can more or less read off the solution from the form of this coefficient matrix and reduce rational or first row says.
Birne Binegar: X is equal to zero, and the second row since zero is equal to zero.
Birne Binegar: This column, the first column doesn't have a pivot so that makes X one a free parameter.
Birne Binegar: And to our solution vectors going to have three parameter, and the first slot zero in the second slot or X one times one zero all right so.
Birne Binegar: here's a basis specter for the solution set the linear system, corresponding to this matrix and our.
Birne Binegar: This is our eigenvector.
Birne Binegar: The corresponding.
Birne Binegar: To the eigenvalue.
Birne Binegar: Lambda equal to that which started off with.
Birne Binegar: Alright, so that i've done this calculation, also on the slide and what we end up with is.
Birne Binegar: An eigenvalue two and a corresponding eigenvector one zero.
Birne Binegar: But back on the pad here we have another eigenvalue to deal with as well, so i'll deal with that on the next slide so similar calculation.
Birne Binegar: defined the Eigen space we calculate the north space a particular matrix the matrix we get by subtracting from the diagonal elements of our original matrix.
Birne Binegar: The eigenvalue that we're interested in.
Birne Binegar: That works out to be this matrix luckily, in this case, it turned out to be already in reduce real actual form we solved the corresponding linear system and its solution said it's going to be a free parameter X two times the vector would components minus two one.
Birne Binegar: And so the sector minus two and the first component one, and the second component is going to be the eigenvector of the original matrix a corresponding to the eigenvalue negative one.
Birne Binegar: Alright, so what i've done on the last few slides here is i've given you a matrix a.
Birne Binegar: 260 minus one, I think, and I found an eigenvalue to with eigenvector.
Birne Binegar: The work just right at the to the subscript here is indicating the eigenvalue and it's the vector on.
Birne Binegar: One zero.
Birne Binegar: And we had a second eigenvalue Lambda equals minus one and it had an idea vector.
Birne Binegar: given by the negative one.
Birne Binegar: which was.
Birne Binegar: minus two one all right, so we now have the ingredients from which we can construct a matrix see that will diagonal is a fact, will get a matrix see such that see the miners first time a time see is actually equal to a diagonal matrix.
Birne Binegar: way we get see.
Birne Binegar: Is we take the eigenvectors we found.
Birne Binegar: And we use them.
Birne Binegar: to form the calls and this matrix see.
Birne Binegar: that's going to be one zero and the second comes home it's going to be the second eigenvector minus two one.
Birne Binegar: it's gonna In fact we also going to know that the diagonal matrix that see convert to a into it's going to be exactly the diagonal matrix where we use these two eigenvalues as its entries.
Birne Binegar: What we do for the matrix D is, we have a first calm it's eigenvalue yeah the eigenvalue correspond this eigenvector have a is to so i'll put two in that first diagonal entry.
Birne Binegar: And then, for the second diagonal entry in the slot down here well it's got to be an Eigen vector of a corresponding to the eigenvalue minus one, so I guess I changed notation here i'm going to use this I use this eigenvector for the second column see, I have to use the second eigenvalue.
Birne Binegar: For the second diagonal entry a and then the other entries of Dr census diagonal just can be zero.
Birne Binegar: Okay, so I claim that having found.
Birne Binegar: The eigenvalues of a and their corresponding eigenvectors I can solve this problem, simply by using the eigenvectors as the columns of see and using the eigenvalues as the as the diagonal entries a D.
Birne Binegar: Let me demonstrate.
Birne Binegar: That this actually worked in this example.
Birne Binegar: So there's the i'm repeating this construction of this matrix see as well, and what Well, this is going to pop out automatic now okay so suppose I create see exactly the fashion i've described like listed the eigenvectors of a and i've arranged them at the columns of the matrix see.
Birne Binegar: Now, since we're working with two by two matrix ease, I have a nice formula, by which I can compute the inverse of see I don't have to do any row reduction.
Birne Binegar: I can use the Co factor expansion formula that can be applied to two by two matrix sees this is effectively one over the determinative a times to transpose up the Co factor matrix of a but the Co factor matrix of a it's pretty simple.
Birne Binegar: The co factors are all always going to be determinants of one by one matrix so you so just going to be the entries of of matrices and, in fact, for an arbitrary two by two matrix you can get it to inverse like.
Birne Binegar: Let me work on the slide here what you end up doing to get the inverse of an arbitrary two by two matrix is you reverse the.
Birne Binegar: The order of the diagonal entries, and then you reverse the sign of the art diagonal entries.
Birne Binegar: And so, using this formula, I can.
Birne Binegar: First of all, I can compute the determinant of C one times one minus two times zero using the difference of diagonal products, so this ad miners BC it's going to work out to be exactly the number one.
Birne Binegar: And what happened.
Birne Binegar: To get the entries of the matrix private of this inverse as I reverse the diagonal entries well they're both equal to one, so they don't appear to change.
Birne Binegar: But I also have to reverse the sign of the off diagonal entries so negative two is going to go to two and zero is going to go to negative zero or zero.
Birne Binegar: So there's the inverse of see computed without using any row reduction.
Birne Binegar: Now all verify that see the miners first time to a time see ends up being exactly this diagonal and the diagonal matrix we create using the eigenvalues of a.
Birne Binegar: dark computing first I multiply guess first two factors together they worked out to be productive first two factors is the matrix 240 minus four I multiply again and what happens is I get exactly the diagonal matrix that was predicted.
Birne Binegar: As long as I create the matrix see using eigenvectors of a and I create the diagonal matrix D by using.
Birne Binegar: The eigenvalues have a written in the same order as i've written these columns here so that first diagonal entry is the eigenvalue corresponding to the eigenvector I used for the first column of see.
Birne Binegar: long as I correlated the column some of the first factor with the eigenvalues I use to create the matrix see this identity is guaranteed.
Birne Binegar: Okay, so.
Birne Binegar: that's essentially how you solve the eigenvector eigenvalue problem up to that caveat that I mentioned at the start.
Birne Binegar: To achieve this realization you need a matrix see that you can but which you can convert this inverse and that's not always possible.
Birne Binegar: And what we'll talk about on Monday is we'll find a number of simpler conditioned by which we determine whether or not this matrix see that we can that we construct from the eigenvectors a how we can know, an earlier stage whether or not it's convertible.
Birne Binegar: So.
Birne Binegar: What i'll start with on Monday is the.
Birne Binegar: The basic theorem the so called fundamental theorem of convertible matrices that gives us some criteria for when the matrix see that I construct in this fashion, when it's actually going to have an inverse.
Birne Binegar: What we've seen.
Birne Binegar: For a long time now is that the existence of a matrix inverse is actually equivalent to a bunch of other properties of the matrix.
Birne Binegar: For example, see will have an inverse you see can be reduced to the identity matrix or C will have a unique inverse if, and only if we can always solve on non homogeneous linear systems for every.
Birne Binegar: vector be we put on the right hand side or we're only going to have a convertible matrix see when the only solution of the homogeneous linear system with coefficient matrix see if the only solution that that is X equals zero and.
Birne Binegar: We have two other conditions, we need the rank of seed equal and or the determinant of seed and it zero any one of these five conditions that that's true for this matrix see we construct in this fashion it's going to tell us whether or not see itself isn't vertical.
Birne Binegar: But.
Birne Binegar: The actual condition we work with the most on Monday, though it's going to be this condition for.
Birne Binegar: So.
Birne Binegar: See is convertible.
Birne Binegar: that's only going to happen when.
Birne Binegar: According to the condition for.
Birne Binegar: The rank of see.
Birne Binegar: is equal to, so this is a square matrix with in wrote in and call will need the rank of seed equal and now the rank of see that's identical to the dimension.
Birne Binegar: Of the column space of see.
Birne Binegar: And in fact the dimension of calls face to the theme is going to equal to the number of basis vectors.
Birne Binegar: For the called Space Agency.
Birne Binegar: And the number of basis vectors you're going to have well that's going to be equivalent into the number.
Birne Binegar: Of.
Birne Binegar: Literally linearly.
Birne Binegar: Independent.
Birne Binegar: columns of see because if you have any linearly dependent columns you're going to have a basis for the column space of see.
Birne Binegar: And each one of these literally in the patent pending columns is going to be interpreted as a basis vector and so when you count them up you'll get two dimensional called space which it has to equal in So what is this condition really telling us.
Birne Binegar: For.
Birne Binegar: See to be.
Birne Binegar: In vertical.
Birne Binegar: will need.
Birne Binegar: An linearly independent columns but the way we're making see is we're using eigenvectors of the matrix a to construct the call So what will really need is in.
linearly.
Birne Binegar: Independent.
Birne Binegar: columns well, let me write it this way and literally independent eigenvectors.
Birne Binegar: Of a to us.
Birne Binegar: As called for see.
Birne Binegar: So we use eigenvectors because we want this matrix see to diagonal is a.
Birne Binegar: But this matrix See also has to be convertible so we're going to need that the column is that we use to create see we need them to be linearly independent so each of these eigenvectors that we use to create this has to be linearly independent.
Birne Binegar: Okay, so.
Birne Binegar: i'll continue with this discussion on Monday.
Birne Binegar: The goal of monday's lecture is to not necessarily analyze.
Birne Binegar: All the all the eigenvectors of see and make sure that they're linearly independent.
Birne Binegar: Rather, the goal monday's lecture see is to have conditioned on the matrix a that will tell us whether or not the mate they we have enough linearly independent eigenvectors to create a matrix see that we can invert and thereby diagnose a.
Birne Binegar: But that's the story that will be explained more completely on Monday, so what i'll do now, though, is simply stop lecture wish you a happy Easter weekend and i'll see you again on Monday.
Birne Binegar: you're welcome.