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‫Welcome back.

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‫And now to make the course complete, let's also find the eigenvectors, let's choose our polls or our

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‫eigenvalues to be like this.

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‫Lamda one equals minus one and Lunda two equals minus two.

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‫And by the way, I don't put Tilba here anymore because we have established that Lunda Tilde equals

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‫Lunda.

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‫So I'm just going to put Lunda here so you can see that you have chosen two polls that are real polls.

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‫They don't have an imaginary part, so you don't have oscillation in your system.

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‫And using the formulas that you had previously derived, you will find that your K one constant will

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‫be minus two in your K two constant will be minus three.

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‫I know that because this is what I used in this course for the drone.

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‫And so these are our two eigenvalues now, our two poles or two eigenvalues, and that also means that

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‫we have to eigenvectors, we have one eigenvector for each eigenvalue.

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‫So let's first find the eigenvector for this eigenvalue, Lunda one equals minus one.

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‫So we're going to take this a matrix here.

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‫And since we know our K one and two values, then this matrix will look like this one is minus two and

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‫K two is minus three like this.

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‫Then we're going to say that our eigenvector looks like this X and Y, they're like unknown variables.

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‫And then we take our eigenvalue, which is minus one, and we put it into this matrix here.

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‫Minus one here and minus one here and then zero here, and then again, your eigenvector looks like

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‫this X and Y.

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‫So if you write out this equation, if you multiply the matrices by their vectors, then you will have

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‫this kind of relationship.

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‫You will have Y equals minus X. That is, if you multiply the first row of this matrix by this vector,

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‫then you will get Y and then you multiply this row by this vector, then you will get minus X.

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‫And then if you do the same thing with the second row, then you will have here minus two X, minus

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‫three Y equals and then you will have minus Y.

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‫So basically you're the know how one of your elements depends on the other one.

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‫You have the relationship, you know that you're Y equals minus X. You can do the same thing with this

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‫one, but you will get the same result.

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‫You will have two Y equals minus two X and then again you will have Y equals minus two divided by two

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‫and then you will have X here.

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‫So Y equals minus X because this thing here will be minus one, and so your first eigenvector, which

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‫is X and Y, can be written like this if you choose your X to be one, I'm going to put here one.

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‫Then since your Y equals minus X, then Y will be minus one.

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‫So in fact, with eigenvectors, the real numeric elements in eigenvectors don't matter that much.

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‫What matters is how one element depends on another element.

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‫So if I had chosen X to be two, then my Y would have been minus two.

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‫So like this and of course, I can also rewrite it like this, I can have two as a constant here and

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‫then again I will have one and minus one.

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‫So eigenvectors are more about the direction.

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‫And the direction is given to you when you find the relationship of one vector element with respect

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‫to another vector element.

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‫So let's check if this solution is correct.

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‫So let's take this original.

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‫Relationship here.

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‫And let's see if it's correct, this is your a matrix and then this is your eigenvector one minus one,

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‫and then that needs to equal and then your eigenvalue was minus one.

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‫So this would be your matrix here.

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‫And again, you multiply it by the same eigenvector one minus one.

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‫And so if you perform this matrix vector multiplication, then you will have it like this.

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‫Minus one equals minus one.

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‫And then minus two.

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‫Plus three equals.

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‫One.

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‫Because here you take minus two, you multiply it by one and then plus minus three times minus one.

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‫So that will give you this.

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‫And we'll hear minus one times, minus one will give you one.

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‫And in the end, you will have minus two.

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‫Plus three equals one, so one equals one.

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‫So indeed, this relationship holds.

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‫And now let's take this eigenvector where we chose our ex to be two and therefore our wives minus two.

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‫So this is our matrix.

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‫Now we have two here, minus two here equals minus one zero zero and minus one here.

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‫Multiplied by two and minus two here, and so again, if you perform the multiplication, then the first

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‫rule multiplications will give you minus two equals minus two.

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‫And then the second row multiplications will give you minus two times two plus minus three times minus

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‫two.

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‫And that should equal.

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‫Minus one times, minus two.

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‫So let's see here you have minus four and then plus six equals two and minus four, plus six equals

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‫two.

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‫So two equals two.

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‫You see now your relationship is also correct, even though you have different numbers for the eigenvector

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‫elements.

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‫So when we talk about eigenvectors, the elements themselves don't matter.

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‫What matters is their direction, how one element depends on the other element.

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‫OK, so that was for LAMDA one.

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‫But now let's find an eigenvector for LAMDA to.

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‫Let's find out how one element of the eigenvector depends on another element of the eigenvector when

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‫our eigenvalue is minus two now, not minus one, but minus two.

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‫So this is the Matrix times the eigenvector and now your eigenvalue is minus two.

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‫So I'm going to put minus two here and minus two here and then zero here and zero here.

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‫And again, I multiplied by the eigenvector.

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‫And from here you can see that your Y depends on X like this.

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‫Y equals minus two times X.

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‫You can see it from the first row multiplications, but you will get the same result from the second

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‫row multiplications.

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‫So minus two times X, plus minus three times Y equals zero times X and then plus minus two times Y,

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‫so minus two times X, minus three times Y equals minus two times Y.

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‫And if you put this thing to the other side of the equation sign, then you will have minus two Y,

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‫plus three Y equals minus two X.

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‫So in other words, you will have Y equals minus two X..

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‫So now your eigenvector, our second eigenvector going to put here to.

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‫Our first eigenvector was one and minus one, and now our second eigenvector is like this, if I choose

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‫X to be one.

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‫Then my wife is minus two.

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‫So this eigenvector here belongs to this eigenvalue here along the two equals minus two.

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‫And let's check if it's correct.

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‫So this is your a matrix times this eigenvector equals.

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‫Now you have minus two zero zero and minus two here, and then your eigenvector here will be one and

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‫minus two.

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‫So you will have minus two equals minus two.

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‫That's from the first row multiplication.

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‫And then the second row multiplication will give you minus two times one, then plus minus three times

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‫minus two, and then equals minus two times minus two.

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‫So you will have minus two, plus six equals four, minus two, plus six.

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‫It is four.

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‫So four equals four.

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‫So that means that our solution is correct, our eigenvectors are correct.

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‫So our second eigenvector will be one and minus two.

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‫And of course, if I had chosen X to be, let's say three, then your Y would have been.

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‫Minus two times three.

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‫So it would have been minus six and then your second eigenvector would have been three and minus six,

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‫and it still would have been correct.

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‫Note that you cannot mix eigenvectors and eigenvalues this eigenvector here that belongs to LAMDA two

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‫equals minus two and this eigenvector here, it belongs to LAMDA one equals minus one.

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‫You cannot use this eigenvector here for this eigenvalue.

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‫Let's check if it makes sense.

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‫This is your a matrix, and now let's take, for example, this eigenvector one and minus two, but

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‫now we're going to take this eigenvalue minus one.

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‫So it's going to be minus one zero zero and minus one in.

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‫Then here we will have our one and minus two eigenvector.

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‫And so if you multiply the first rose by this eigenvector, then you will have minus two equals.

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‫Minus one times one so is going to be minus one and then you will have minus two times one and then

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‫you will have minus three times minus two.

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‫So you're going to have plus six.

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‫And then that equals to.

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‫There are times one plus minus one times minus two, it will be two, so minus two, plus six is four

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‫and that equals two, which obviously is not correct.

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‫And here this relationship is not correct either.

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‫So you see, you cannot mix eigenvalues and eigenvectors for each eigenvalue.

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‫You have a specific eigenvector.

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‫And well, to be honest with you, eigenvectors are not relevant in our course, but, well, at least

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‫you now have seen the process.

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‫Let it be an extra bonus in the next video.

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‫I want to talk a little bit more about different types of differential equations and then we're going

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‫to start wrapping it up and applying it to our specific controller for controlling the drone.

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‫Thank you very much.