‫Welcome back.
‫And now to make the course complete, let's also find the eigenvectors, let's choose our polls or our
‫eigenvalues to be like this.
‫Lamda one equals minus one and Lunda two equals minus two.
‫And by the way, I don't put Tilba here anymore because we have established that Lunda Tilde equals
‫Lunda.
‫So I'm just going to put Lunda here so you can see that you have chosen two polls that are real polls.
‫They don't have an imaginary part, so you don't have oscillation in your system.
‫And using the formulas that you had previously derived, you will find that your K one constant will
‫be minus two in your K two constant will be minus three.
‫I know that because this is what I used in this course for the drone.
‫And so these are our two eigenvalues now, our two poles or two eigenvalues, and that also means that
‫we have to eigenvectors, we have one eigenvector for each eigenvalue.
‫So let's first find the eigenvector for this eigenvalue, Lunda one equals minus one.
‫So we're going to take this a matrix here.
‫And since we know our K one and two values, then this matrix will look like this one is minus two and
‫K two is minus three like this.
‫Then we're going to say that our eigenvector looks like this X and Y, they're like unknown variables.
‫And then we take our eigenvalue, which is minus one, and we put it into this matrix here.
‫Minus one here and minus one here and then zero here, and then again, your eigenvector looks like
‫this X and Y.
‫So if you write out this equation, if you multiply the matrices by their vectors, then you will have
‫this kind of relationship.
‫You will have Y equals minus X. That is, if you multiply the first row of this matrix by this vector,
‫then you will get Y and then you multiply this row by this vector, then you will get minus X.
‫And then if you do the same thing with the second row, then you will have here minus two X, minus
‫three Y equals and then you will have minus Y.
‫So basically you're the know how one of your elements depends on the other one.
‫You have the relationship, you know that you're Y equals minus X. You can do the same thing with this
‫one, but you will get the same result.
‫You will have two Y equals minus two X and then again you will have Y equals minus two divided by two
‫and then you will have X here.
‫So Y equals minus X because this thing here will be minus one, and so your first eigenvector, which
‫is X and Y, can be written like this if you choose your X to be one, I'm going to put here one.
‫Then since your Y equals minus X, then Y will be minus one.
‫So in fact, with eigenvectors, the real numeric elements in eigenvectors don't matter that much.
‫What matters is how one element depends on another element.
‫So if I had chosen X to be two, then my Y would have been minus two.
‫So like this and of course, I can also rewrite it like this, I can have two as a constant here and
‫then again I will have one and minus one.
‫So eigenvectors are more about the direction.
‫And the direction is given to you when you find the relationship of one vector element with respect
‫to another vector element.
‫So let's check if this solution is correct.
‫So let's take this original.
‫Relationship here.
‫And let's see if it's correct, this is your a matrix and then this is your eigenvector one minus one,
‫and then that needs to equal and then your eigenvalue was minus one.
‫So this would be your matrix here.
‫And again, you multiply it by the same eigenvector one minus one.
‫And so if you perform this matrix vector multiplication, then you will have it like this.
‫Minus one equals minus one.
‫And then minus two.
‫Plus three equals.
‫One.
‫Because here you take minus two, you multiply it by one and then plus minus three times minus one.
‫So that will give you this.
‫And we'll hear minus one times, minus one will give you one.
‫And in the end, you will have minus two.
‫Plus three equals one, so one equals one.
‫So indeed, this relationship holds.
‫And now let's take this eigenvector where we chose our ex to be two and therefore our wives minus two.
‫So this is our matrix.
‫Now we have two here, minus two here equals minus one zero zero and minus one here.
‫Multiplied by two and minus two here, and so again, if you perform the multiplication, then the first
‫rule multiplications will give you minus two equals minus two.
‫And then the second row multiplications will give you minus two times two plus minus three times minus
‫two.
‫And that should equal.
‫Minus one times, minus two.
‫So let's see here you have minus four and then plus six equals two and minus four, plus six equals
‫two.
‫So two equals two.
‫You see now your relationship is also correct, even though you have different numbers for the eigenvector
‫elements.
‫So when we talk about eigenvectors, the elements themselves don't matter.
‫What matters is their direction, how one element depends on the other element.
‫OK, so that was for LAMDA one.
‫But now let's find an eigenvector for LAMDA to.
‫Let's find out how one element of the eigenvector depends on another element of the eigenvector when
‫our eigenvalue is minus two now, not minus one, but minus two.
‫So this is the Matrix times the eigenvector and now your eigenvalue is minus two.
‫So I'm going to put minus two here and minus two here and then zero here and zero here.
‫And again, I multiplied by the eigenvector.
‫And from here you can see that your Y depends on X like this.
‫Y equals minus two times X.
‫You can see it from the first row multiplications, but you will get the same result from the second
‫row multiplications.
‫So minus two times X, plus minus three times Y equals zero times X and then plus minus two times Y,
‫so minus two times X, minus three times Y equals minus two times Y.
‫And if you put this thing to the other side of the equation sign, then you will have minus two Y,
‫plus three Y equals minus two X.
‫So in other words, you will have Y equals minus two X..
‫So now your eigenvector, our second eigenvector going to put here to.
‫Our first eigenvector was one and minus one, and now our second eigenvector is like this, if I choose
‫X to be one.
‫Then my wife is minus two.
‫So this eigenvector here belongs to this eigenvalue here along the two equals minus two.
‫And let's check if it's correct.
‫So this is your a matrix times this eigenvector equals.
‫Now you have minus two zero zero and minus two here, and then your eigenvector here will be one and
‫minus two.
‫So you will have minus two equals minus two.
‫That's from the first row multiplication.
‫And then the second row multiplication will give you minus two times one, then plus minus three times
‫minus two, and then equals minus two times minus two.
‫So you will have minus two, plus six equals four, minus two, plus six.
‫It is four.
‫So four equals four.
‫So that means that our solution is correct, our eigenvectors are correct.
‫So our second eigenvector will be one and minus two.
‫And of course, if I had chosen X to be, let's say three, then your Y would have been.
‫Minus two times three.
‫So it would have been minus six and then your second eigenvector would have been three and minus six,
‫and it still would have been correct.
‫Note that you cannot mix eigenvectors and eigenvalues this eigenvector here that belongs to LAMDA two
‫equals minus two and this eigenvector here, it belongs to LAMDA one equals minus one.
‫You cannot use this eigenvector here for this eigenvalue.
‫Let's check if it makes sense.
‫This is your a matrix, and now let's take, for example, this eigenvector one and minus two, but
‫now we're going to take this eigenvalue minus one.
‫So it's going to be minus one zero zero and minus one in.
‫Then here we will have our one and minus two eigenvector.
‫And so if you multiply the first rose by this eigenvector, then you will have minus two equals.
‫Minus one times one so is going to be minus one and then you will have minus two times one and then
‫you will have minus three times minus two.
‫So you're going to have plus six.
‫And then that equals to.
‫There are times one plus minus one times minus two, it will be two, so minus two, plus six is four
‫and that equals two, which obviously is not correct.
‫And here this relationship is not correct either.
‫So you see, you cannot mix eigenvalues and eigenvectors for each eigenvalue.
‫You have a specific eigenvector.
‫And well, to be honest with you, eigenvectors are not relevant in our course, but, well, at least
‫you now have seen the process.
‫Let it be an extra bonus in the next video.
‫I want to talk a little bit more about different types of differential equations and then we're going
‫to start wrapping it up and applying it to our specific controller for controlling the drone.
‫Thank you very much.