Birne Binegar: All let me apologize for Friday I had a flu shot Thursday afternoon.
Birne Binegar: And I had a fever 101 Friday.
Birne Binegar: yeah.
Birne Binegar: The boosters can be fantasy every little thing my thing but anyway i'm recovered i'm wearing a mask simply just to be on the safe side, but.
Birne Binegar: We should be good to go day so they'll deliver right lecture.
Birne Binegar: Last week, what we've been working on is how to compute each.
Birne Binegar: One didn't speak days for an application of our ability.
Birne Binegar: The very first.
Birne Binegar: Application I need I need a problem.
Birne Binegar: To give you some context for this, there are many problems and applications that don't look like they have any.
Birne Binegar: All nicolette was fundamental population population, I like matrix Let me give you one example just do so, we keep your attention, as we work through this.
Birne Binegar: Today pleasure lecture certainly route.
Birne Binegar: One example, the thing that you.
Birne Binegar: That, for example, an engineer if.
Birne Binegar: you're interested in maybe an automotive.
Birne Binegar: tackle this you try to balance it off, but what do you think about what its natural axis of rotation, are you have a symmetrical outfit it'll spin naturally around one access or another, but when you have a irregularly shaped object to.
Birne Binegar: The point at which you can actually.
Birne Binegar: sort of spirit and have a balance just on your fingertip like you like basketball pretty easy um any point work, but unless the bench for artists see where the natural balance.
Birne Binegar: Be it turns out there's a there's a matrix that you compute using three dimensional coordinates xyz you get an index for each of those coordinate and what you do is you construct a moment of inertia matrix.
Birne Binegar: Of the three by three matrix or readers these basically correspond to your three directions so X, Y amp D Thank you, Sir matrix you can still be certain factors associated.
Birne Binegar: And those those back your story badly to directions you you'll have natural location spin and nothing on tumble it will just be simply rather than sort of a compound.
Birne Binegar: So that's one of the things you'll be able to find once you know how do I get back.
Birne Binegar: There are other applications as well, now that just are not necessarily correct days they could be function and, for example, if you wanted to understand chemistry.
Birne Binegar: Why the hydrogen atom as orbital electron eat overhaul electron ever yes formation of the.
Birne Binegar: destination of the periodic table that's also finding it finding value, but now up so much of the middle of the bakery but have a differential offer when they correspond and and the idea that they're.
Birne Binegar: At.
yeah.
Birne Binegar: So.
Birne Binegar: Just to give you a picture of how diverse.
Birne Binegar: The.
Birne Binegar: next topic will be in an application, they will just talk about how we frame the problem badly and then sold.
Birne Binegar: So we begin.
Birne Binegar: With a simple mathematical equation.
Birne Binegar: quality.
Birne Binegar: problem statement is like the dark moments, where the truth and my truth and associated to this makers are going to be certain numbers a label or by Lanka, but i'll call them.
Birne Binegar: And they're also be certain factors.
Birne Binegar: With a dimensional vector in size, but he wrote the matrix and what you want is for the factors.
Birne Binegar: that apply to all the equation matrix multiplication by the bakery.
Birne Binegar: can be fully scalable if you say after I the number lamp so.
Birne Binegar: This population that really like just like every component of that vector V by the thing number one.
Birne Binegar: But the calculation on the right hand side is more complicated, you have to take the Doc product of each row of a with would be and then consolidate result in better So these are two different calculations and what what demand when one person is that true finding value.
Birne Binegar: That, I think that he.
Birne Binegar: satisfies the equation this.
Birne Binegar: Time we're both we wanted something a little bit.
Birne Binegar: worthless they certainly because.
Birne Binegar: The zero vector be on both sides, I get an identity but sort of a trivial identity, once you find it the directory.
Birne Binegar: entity says 008 4.0 vector can have.
Birne Binegar: Any number 900 that's that case we're not interested in doesn't really.
Birne Binegar: provide any additional what we're interested in this.
Birne Binegar: Is a problem or non zero after and final question.
Birne Binegar: If you could do that you call the corresponding vector eigenvector and we call corresponding number the ID value.
Birne Binegar: And each of the big factor in the number are going to kill your.
Birne Binegar: head so think of a.
Birne Binegar: boat that your number.
Birne Binegar: Two by two matrix here's a two by one called after I multiply you make your kitchen compute the result of a plan B, I did a vector for for.
Birne Binegar: Which is, you can see it's just the entries of the original vector multiplying by cool, so this is four times the visual effects and then you see that the.
Birne Binegar: Does it affect your ID value problem is full I this vector in the number of board but.
Birne Binegar: It only works with work and it actually will do the work for samples, will this vector.
Birne Binegar: Alright, so that's sort of situation to try to replicate the lecture.
Birne Binegar: So.
Birne Binegar: Important part right right now is the fact that we are trying again really shouldn't be possible to replace matrix multiplication I feel.
Birne Binegar: The fact that we can actually solve this problem.
Birne Binegar: For special values have landed special becky.
Birne Binegar: Give a.
Birne Binegar: Little more nomenclature here.
Birne Binegar: it's all.
Birne Binegar: The problem.
Birne Binegar: will call enter the value big factor and.
Birne Binegar: what's generally going to happen is that if you have one eigenvector increase you're going to have input ID.
Birne Binegar: Because you look at both sides of the equation, I could feel about bullfighting wait a minute right one, and it would still be a valid way, and that is also for that it will actually be in a different.
Birne Binegar: perspective, we have happened here, if you have one day.
Birne Binegar: Every skill of both after awesome.
Birne Binegar: You have one zero.
Birne Binegar: you're going to have into.
Birne Binegar: What we do is collect together all the solution.
Birne Binegar: corresponding to give it.
Birne Binegar: All that the Lambda ID.
Birne Binegar: And so, this definition i've got one private lenders and I think that needs at least Wednesday satisfies the equation given number of Lambda.
Birne Binegar: Or what the definition does except class together, along with that one.
Birne Binegar: All the other.
Birne Binegar: That file a civil equation same Lambda.
Birne Binegar: Right.
Birne Binegar: Now we're gonna show you how we solved it.
Birne Binegar: I knew that if I go down problem first thing we'll do is we'll take this the body condition that says.
Birne Binegar: Be is it valuable a.
Birne Binegar: plan.
Birne Binegar: We rewrite this mortal your turn.
Birne Binegar: So I start off this condition is just a basic cases equals scalable issue with these particular ventures and picture number length is this term, to the opposite side and I.
Birne Binegar: boy your athletes, they can be a factor and but what I do with my face the fact of landed on Z and i'm certainly identity.
Birne Binegar: now have a baker time he and another matrix IE right it's all in a.
Birne Binegar: single matrix time.
Birne Binegar: So here's a way we'll think about this in this.
Birne Binegar: poster the for me to deal with matrix equation before, this is a simple linear this coefficient matrix is the vector zero.
Birne Binegar: don't is back what we call linear algebra a homogeneous.
Birne Binegar: coefficient danger factor variable equals zero ideas we you know.
Birne Binegar: get your own writing inside actually.
Birne Binegar: what's going to happen you're going to see coefficient agent you're not a itself rather shifting.
Birne Binegar: With all that land issues with daydreams oftentimes label it as up.
Birne Binegar: At all I need is a minus man I.
Birne Binegar: Even that maybe the sky with what really yes, I have a matrix.
Birne Binegar: 1234 then eight minus lenses.
Birne Binegar: Well, I can write that down and immediately all the ship by Lambda complimentary but let's work it out 123 or minus than the time you made it to 00 Roundup.
Birne Binegar: By these guys at 1.234 my lamp so all this matrix yes it's the original matrix a were just long as a diagonal upper left to the lower right time shifted the entry by matlab.
Birne Binegar: This turns out to be.
Birne Binegar: The most relevant.
Birne Binegar: Relevant may prefer calculation.
Birne Binegar: that's gonna be a little bit confusing because i'll filter this lecture be talking about.
Birne Binegar: All the lawyer says, I just have a matrix times the vector is equal to zero, but oftentimes are going to happen is the relative position in that interview.
Birne Binegar: Rather, the shift.
Birne Binegar: Right so.
Birne Binegar: In terms of this shift in a group.
Birne Binegar: The Lambda is a set of all idea about vectors with IP data plans.
Birne Binegar: But that is just the know things of the shift and make.
Birne Binegar: You have a homogeneous that your system.
Birne Binegar: Now.
Birne Binegar: The solutions that.
Birne Binegar: say.
Birne Binegar: X.
Birne Binegar: Zero is.
Birne Binegar: Is what we call.
Birne Binegar: The North face of the corresponding matrix.
Birne Binegar: And we do all sorts of things about this because we've solved now homework, a new system since the beginning of the Semester.
Birne Binegar: But.
Birne Binegar: what's relevant today it's not.
Birne Binegar: optional phase, a rather than open.
Birne Binegar: So what we're gonna be looking for day is first of all.
Birne Binegar: The numbers Lambda which this page says the best other than cast zero.
Birne Binegar: notice that this debt they're taking the zero vector resolution he's up there that could be both sides are going to be valuing your own identity, full of what where is the fear not.
Birne Binegar: satisfied it.
Birne Binegar: Okay, so.
Birne Binegar: first thing i'm going to show you.
Birne Binegar: Is that is.
Birne Binegar: That is that yours.
Birne Binegar: would be what would be an iPad Daddy hey i'm Lambda.
Birne Binegar: The fact that it actually itself well i've actually proved it already on this line right because our general theory of.
Birne Binegar: A matrix was a it's always a solution instead of the homogeneous linear system, you get using the given matrix of the coefficient matrix so let me show you a little bit more directly, that I would say Atlanta identity matrix a.
Birne Binegar: First of all.
Birne Binegar: understand the motivation, so if you have an ID.
Birne Binegar: Meeting, you have a vector.
Birne Binegar: So the.
Birne Binegar: back for.
Birne Binegar: A with.
Birne Binegar: Isaac value.
Birne Binegar: First of all.
Birne Binegar: The.
Birne Binegar: body for the zero vector because that solution is.
Birne Binegar: And it doesn't do us any good, and second, of all he is.
Birne Binegar: A lamp.
Birne Binegar: Yes, gotta live in the north face this shifted make.
Birne Binegar: Right so.
Birne Binegar: Is this this is forgiven eigenvalue the set of all that your your property well that we.
Birne Binegar: believe this property because it subspace always have zero what we want is the subspace is larger than just doing.
Birne Binegar: Today, great.
Birne Binegar: This that is always close.
Birne Binegar: The door right and the idea here so as a five year old follow the.
Birne Binegar: yellow tape measure how dare you and then because we have to be the plan to either make it as painless as can just create scalable full of Laughter
Birne Binegar: Then you reverse the order of each doctor and you see that a fine day, it is equally landed on a flat fee with me this factor hey i'm this also it better, a plan.
Birne Binegar: and similarly under that condition if I have to start off if I get vectors.
Birne Binegar: together well, maybe he distributes over.
Birne Binegar: Because V1 and V2 or it better hey.
Birne Binegar: A time he wants to be replaced by one, at a time and.
Birne Binegar: Time.
Birne Binegar: Hello Linda and you see that now the one who is it that you're painting with light.
Birne Binegar: So.
Birne Binegar: The main object that we're going to compute here is actually not a particular eigenvector they were creating.
Birne Binegar: And what is the page that we're trying to find.
Birne Binegar: You don't want to explicitly write down every time.
Birne Binegar: So.
Birne Binegar: Even if it's only one dimensional it's going to be an implement many factors because.
Birne Binegar: Your face so it's every scale of multiple the non zero.
Birne Binegar: Now.
Birne Binegar: You have three employee many more factors just by choosing different scalable.
Birne Binegar: So basically what we're really trying to nail down what it is we're just trying to find a basis.
Birne Binegar: Every other ideas enter in the business a linear combination of you.
Birne Binegar: Put that those considerations ciphers and just.
Birne Binegar: trying to figure out how we can actually find.
Birne Binegar: Numbers Lambda and after be satisfied.
Birne Binegar: invite you back your ID value condition.
Birne Binegar: So.
Birne Binegar: let's add in by endangered.
Birne Binegar: All the time that I value problem finding all the Members land.
Birne Binegar: It and that will be paid as your associate.
Birne Binegar: And that's going to her even, as I stated, it is obviously the two days, except, first of all.
Birne Binegar: You want to find the numbers Lambda but which you can find solutions have such a system it's going to turn out you have an arbitrary number Lambda here.
Birne Binegar: You won't be able to find a non zero vector that solve this problem only going to be pretty special values of the land that that you'll actually have non zero solutions they'll become apparent in just in just a few slides.
Birne Binegar: And then, once you figure out the.
Birne Binegar: Numbers Lambda What would you have non trivial solution is figuring out with those illusions darn that that's something we already do.
Birne Binegar: This condition here the idea factor condition is equivalent to this only the homogeneous linear.
Birne Binegar: what's I haven't been so many places civil laughter by him and was it number four.
Birne Binegar: Is Jackie numerical matrix until this is become a homogeneous linear system for which we did identify the solution.
Birne Binegar: By agent is accurate reviews your actual form of unifying three parameters and they read articles.
Birne Binegar: and your solution all that all that crap will do that.
Birne Binegar: Probably on.
Birne Binegar: Wednesday today we're going to focus on just figuring out what numbers will work.
Birne Binegar: So that's where we begin.
Birne Binegar: we're going to try to find numbers Lambda but this equation as solution where the vector be is not equal zero.
Birne Binegar: that's the only case and so.
Birne Binegar: This equation we've already see every three or four more familiar with certain matrix under factor equal to zero.
Birne Binegar: All of.
Birne Binegar: The X equals zero paid and always going to be a solution, but it's the one we're not interested, or pull that trigger it all ignored and forth over St data or the knowledge.
Birne Binegar: Of quitting.
Birne Binegar: And so the real question you have to find the eigenvalues hey when do we have solutions of this equation, other than X equals zero.
Birne Binegar: Right, that seems.
Birne Binegar: kind of abstract.
Birne Binegar: In the first glance impossible oh that.
Birne Binegar: We have there that will give us the solution.
Birne Binegar: If you recall that oh about the time of the first mid to.
Birne Binegar: even talk to you about it, the inverse of the matrix and I stated, the only fundamental.
Birne Binegar: amazing and I have an inverse well what constitutes a has a denver's one consequence of that is that every look here system, this form has a unique solution.
Birne Binegar: The way you get that is you use the inverse double like both sides by the inversion did you get X is equal to a long time fee, you know, once you know the inversion you basically write down the solution that.
Birne Binegar: Particular equation of this for is the equation or eight X is zero day has been birth, this also has a unique solution, but you can observe that that unique solution has to coincide with X equals zero.
Birne Binegar: And another like condition is that the usual national on board with the matrix is equal to the identity matrix and yet another equivalent statement is to determine at is not equal to zero.
Birne Binegar: So this really is positive and, lastly, where are we in.
Birne Binegar: terms of the way it happens is people are not a matrix has been.
Birne Binegar: Self what's relevant to the current problem is not.
Birne Binegar: The fear directly but rather.
Birne Binegar: The nation so because he Stephen sorry are Quinlan either all true for all false so i'm gonna give you again, a condition, where I feel that I have solution other than actually you're not totally negate the whole fear of.
Birne Binegar: The equivalent what they would be a.
Birne Binegar: solution when your existence of this or don't have.
Birne Binegar: That X equals zero is not the only solution, but you know neutral is one form of a thought.
Birne Binegar: And the terms of a equals.
Birne Binegar: So, in particular, focus on is that we're looking for solutions that that video system better other than the trivial solution, and while we see it that's going to require that the determinant of the coefficient matrix as the people through.
Birne Binegar: Last night stay here a homogeneous linear.
Birne Binegar: A vector X equals zero is going to have solution other than X equals zero if, and only if the determinant of the coefficient interesting.
Birne Binegar: Alright, so let's go back to.
Birne Binegar: Know Oh, I remember it then.
Birne Binegar: This condition here just really is stating it better eigenvalue vision as.
Birne Binegar: A statement about a whole G is that your system for this is a trick on the on the on deck your is equal to something a page that you're on the right hand side.
Birne Binegar: And what we conclude from the lamb or the corollary last time if we're only going to have non trivial to this equation when the determinant of the matrix a minus length of time I zero.
Birne Binegar: Okay, so um.
Birne Binegar: We calculated targets, we know what they look like, at the end of the calculation to basically pull know people in the entry of amazing so this condition here it's going to return all of your entry record by making.
Birne Binegar: Although we open the parameter lamb.
Birne Binegar: When we set a polynomial equals zero, we all know.
Birne Binegar: That the condition defined non trivial solutions of the equation like this it's got to be a polynomial equations in the parameter Lambda.
Birne Binegar: That being.
Birne Binegar: redefined.
Birne Binegar: For a matrix hey we establish this polynomial permanent shifty.
Birne Binegar: And he said this polynomial is actually going to tell us what's the highest value a bar number planet which we have non trivial you know.
Birne Binegar: I would use the new releases.
Birne Binegar: We call this the characteristic polynomial a and we call it a condition that that's all going to.
Birne Binegar: be definitely turn advantages that you can actually I bet.
Birne Binegar: We call that the courage receive.
Birne Binegar: Corresponding.
Birne Binegar: Okay, so and sorry.
Birne Binegar: The way we're going to find the value of n by N, the possible numbers Lambda which we can find that to be challenged by the vision is going to be the root of this equation.
Birne Binegar: i'm sorry my slides.
Birne Binegar: don't compete with.
Birne Binegar: Your polynomial that we.
Birne Binegar: are shifting them later on mate I had no idea.
Birne Binegar: that the current we did a polynomial in and do all invites a tree and the roots of that phone solution that he is a hero, those are going to be back down.
Birne Binegar: Maybe think come clear we do an example.
Birne Binegar: here's a two by two matrix.
Birne Binegar: You have like oh no.
Birne Binegar: Oh, you start off right, you know, both for a minus Lambda times.
Birne Binegar: You see what this additional matrix all it does today is to just shift the main diagonal entry by the mountain native land.
Birne Binegar: But it's still a matrix and even though this is a parameter rather than listen number, we can still forgiveness determinant using the difference of the diet coke products.
Birne Binegar: And we get a polynomial here is explicitly and what we're actually interested in want to find it.
Birne Binegar: Is what are the values of random.
Birne Binegar: Data that you're.
Birne Binegar: Using see that the only way we have a Lambda is equal to zero you either have to have land for Lambda.
Birne Binegar: So these two numbers for a negative two are the high value.
Birne Binegar: All right now that's pretty easy for a two by two matrix because.
Birne Binegar: All in order to agree to are pretty simple all the very least, if you had a polynomial equation.
Birne Binegar: Of the form a.
Birne Binegar: A Lambda squared.
Birne Binegar: c zero such up quadratic equation.
Birne Binegar: These are always solvable using the quadratic formula general you have two groups.
Birne Binegar: Or you have.
Birne Binegar: A plus or minus square root of each player or A to Z divided by two.
Birne Binegar: So call.
Birne Binegar: For formula that you've known since high school, maybe before.
Birne Binegar: But unfortunately, when we okay three by three matrix or or by boarding at all alone will no longer going to be too it's going to be.
Birne Binegar: it's the grid is going to be the same as the dimension of the matrix so four by four by four matrix the characteristic polynomial corresponding all know, the only point to the board.
Birne Binegar: And so.
Birne Binegar: How do you solve polynomial equations what is the degree of the polynomial was greater than two.
Birne Binegar: Well, for that you need a backdrop.
Birne Binegar: So from high school algebra.
Birne Binegar: At the not hired because.
Birne Binegar: Obviously, the Iranian cancer deep off into the conflict.
Birne Binegar: But the story is this anytime you have a polynomial.
Birne Binegar: polynomials always going to be back horizon in terms of linear polycom.
Birne Binegar: layer is simply free one column that's mine is number.
Birne Binegar: 317 factors also.
Birne Binegar: off after 40 degree one in the parameter so that's called it please factor.
Birne Binegar: Unfortunately, to get this factorization.
Birne Binegar: Even if you have polynomial that system.
Birne Binegar: Real numbers on to power, the S.
Birne Binegar: This factorization doesn't necessarily curve of the real numbers you.
Birne Binegar: can take the polynomial.
Birne Binegar: He X equal X squared plus one Well he of X.
Birne Binegar: equals zero well, the only way that happened X squared is equal negative one X is equal, but the bottom line is.
Birne Binegar: negative.
Birne Binegar: assault.
Birne Binegar: By the solutions of P of X is equal to zero, you have to.
Birne Binegar: Do the complex numbers, you have to admit complex numbers as part of your.
Birne Binegar: solution to me, but nevertheless.
Birne Binegar: Real coefficients exact your eyes it's just that you have to back dries it.
Birne Binegar: Complex.
Birne Binegar: At the office.
Birne Binegar: So.
Birne Binegar: Nevertheless.
Birne Binegar: It this is really just be a statement that order itself polynomial equations the real numbers are not sufficient, and you have to sometimes you have to leap up into the complex numbers illusion that's also going to be the case here, but we.
Birne Binegar: We try to find it sometimes you can have, I think, values of the matrix to lose all your equation.
Birne Binegar: Like through.
Birne Binegar: Like everything else in this course we try to keep the computation simple and so just like I avoid using high.
Birne Binegar: to another thing when I insert.
Birne Binegar: entries over maitri just because of a simplification more complicated.
Birne Binegar: So to will try to avoid conflict number.
Birne Binegar: Of decoration it.
Birne Binegar: Everything I say oh it's true.
Birne Binegar: Whether we regard the underlying feel rewarded real numbers like.
Birne Binegar: Just that occasionally you get to run into polynomials in which are the only route to the corresponding on the equations or complex numbers.
Birne Binegar: For the most part can avoid complex numbers in this course, all just try to avoid those kind of examples.
Birne Binegar: If they do occur.
Birne Binegar: It all it really means is that the arithmetic that you perform, so you do have growing up, you still have to multiply rose and gather together, but the underlying a risk taker is going to be the House like numbers, rather than your real.
Birne Binegar: But that's The only difference.
Birne Binegar: But.
Birne Binegar: What i'm trying to say here is that I want to solve all.
Birne Binegar: The ways to do this in general, if the point sounds factorization.
Birne Binegar: And then, once you have that factorization you can feed off of the bathroom thinking itself what the solutions on this case you could be our one are two of our in.
Birne Binegar: Okay, now I can do a more complicated example a three by three matrix and will find all of it.
Birne Binegar: That he really looked at his character.
Birne Binegar: He.
Birne Binegar: Did.
Birne Binegar: And so.
Birne Binegar: There is a matrix i've gotten tied to the permanent all the only because God is like on this main diagonal track.
Birne Binegar: Now I can do with determining population.
Birne Binegar: Your co factor expansion along.
Birne Binegar: or explicitly, but I did is I have.
Birne Binegar: When I design that.
Birne Binegar: zero.
Birne Binegar: minus one plus one determining.
Birne Binegar: what's the corresponding by looking at this slide I live in a row column it tastes like a matrix one one.
Birne Binegar: But I don't have to compute this determinant because it's being multiplied by two.
Birne Binegar: And then, I have a signature entry there is.
Birne Binegar: Negative Lambda.
Birne Binegar: Five is minus one to.
Birne Binegar: determine report on lighter load in a row and follow for fun to eventually.
Birne Binegar: 1911.
Birne Binegar: That I have contributions from the third element that.
Birne Binegar: It will be plus one.
Birne Binegar: minus 123 determinant of the quarter funded by.
Birne Binegar: Exactly.
Birne Binegar: And so that's the usual complication determinant sushi plugins are calculating against the you get a polycom as predicted to turn it off three on three matrix the characteristic polynomial in the automobile.
Birne Binegar: So.
Birne Binegar: We saw the equation here they have lamp this all know very well factor.
Birne Binegar: Now.
Birne Binegar: General cubic polynomial does not easy to write a factorization usually have to play around with that or.
Birne Binegar: You can often factor is just my theory, but this one simple way to proceed is at least identify.
Birne Binegar: and fill it out to see here is that this third degree polynomial every term has a factor of Lambda So if you factor, and that is particularly the landing gear all has to be rude yourself to do for Lambda here right inside.
Birne Binegar: here.
Birne Binegar: Oh, but actually happen is a polynomial that up third order but it's a linear factor times that degree two columns.
Birne Binegar: Now this guy.
Birne Binegar: Hopefully you guys can adapt your eyes to this basically land or minus two.
Birne Binegar: plus one.
Birne Binegar: So now i've achieved the full linear factorization of the gift of the characteristics calling for matrix ED and the eigenvalues we now can read off for spotting for this factor zero figure, this is mango minus.
Birne Binegar: That are funding this factor second it done to the first one last factor, the last room saying that equal minus one.
Birne Binegar: Are you done for the matrix.
Birne Binegar: So i'm.
Birne Binegar: summarize.
Birne Binegar: it's all.
Birne Binegar: Better.
Birne Binegar: hey.
Birne Binegar: You find the.
Birne Binegar: path to success first thing you do is you create.
Birne Binegar: or calculate the corresponding oh no meal.
Birne Binegar: factor.
Birne Binegar: yeah.
Birne Binegar: grab.
Birne Binegar: The grab the object so just go ha.
Birne Binegar: land is gonna look something like some overall vision and the minus.
Birne Binegar: Lambda mind just aren't to.
Birne Binegar: Mine and you're going to have a total of and factors, because for invite in matrix this poll know meal is going to be a degree in so you're going to eat a total and factors these numbers these offset from the variable Lambda those are exactly the.
Birne Binegar: Right so you'll get a list of numbers that actually which not you can do the corresponding numbers of the eigenvalues of big.
Birne Binegar: complete the project, you will find.
Birne Binegar: It better and that's going to be done by June, because you're ordering shirts, what if I choose a particular number, it becomes an explicit numerical matrix and you simply solve the usual in your.
Birne Binegar: head this matrix reduce it to zero usual one for write down the corresponding equation fan the solution in terms of the free variable and those sectors that are being multiplied by the three parameters, those are exactly your I can.
Birne Binegar: Alright, so i'm going to stop here today.
Birne Binegar: On Wednesday.
Birne Binegar: On a project with lots of.
Birne Binegar: thought it might be that guy from this part is not the hard part because really.
Birne Binegar: But.
Birne Binegar: In combination with the qualifying.
Birne Binegar: will be better.
Birne Binegar: We get different places, depending on the matrix.
Birne Binegar: Okay.