Yeah, welcome everybody. As you see, Professor Honegger made his promise true and he handed

this lecture over to me. And I have a short plan here. First of all, this is the URL where

you can download the videos and they are hosted by the Fachschaft Informatik, not by the Rechenzentrum.

So it's not the same URL where you can download the previous lecture recordings, but it's

a different one. And as you can see today, it's me who is presenting the lecture. Next

Monday, Marco Bögel, the next three appointments, Professor Honegger has some urgent schedules.

But next week Marco Bögel will fill in. Then the first day next week we will call off,

so there won't be a lecture on Thursday. So you get to sleep a little bit longer on that

day. And the Monday after that, Professor Honegger will return and he will hopefully

do all of the lectures for the remaining lecture term.

Okay, now I need a bit of help. You talked, so far I know you have been talking about

singular value decomposition. How much did he actually tell you? Maybe you can help me

and we can also think about what we heard in the last lecture. Who remembers?

The last slide was slide 10.

Last slide was slide 10. Sounds reasonable. Which would be this one, which is the formal

definition of SVD. And he told, so I figure he told you something about what a matrix

does to a unit ball, how it deforms it. Yes, no? Yes, yes, that's good. And did he talk

about what the decomposition does? No. Maybe we just talk about it again. Who has heard

about this in IMIP already? Ah, that's not too many. Okay, good. Good, then let's talk

about that. Who was the candidate of the day in the last lecture?

There is no candidate of the day. There is no candidate of the day.

So we're all candidates, right? You're all candidates of the day. Okay.

I would have had only one question for the candidate of the day. Do you know what the

candidate of the week is? Okay, good. Let's talk a bit here about SVD. I put this up for

some, so I'll start writing here, okay? So let's summarize what you've seen last time.

So you've seen that if you think of the vectors that all have length one, let's say here is

one and here is one, and let's see if I can draw this. Ah, it's getting better with every

attempt I try. So this is the unit ball, and this is the set of vectors that all have length

one, yeah? And you've seen that if you apply some kind of matrix A, so let's say you apply

a two by two matrix A, and then this matrix has a certain effect on your vectors, and

it will probably do something like this with your vectors. So it will somehow deform them,

and you can think of a matrix as some operation that deforms your unit vectors, okay? And

now you've seen that you can decompose any matrix into three matrices. Why would that

be useful? Yes?

Because inversion gets easy, for example. Inversion gets easy, for example. Exactly.

Did he talk about inversion, how to do that? Did you already hear that last time? Kind

of. I see nodding. Yeah? So I'm still going to repeat it. So please interrupt me. If I'm

telling stuff that you've already heard, interrupt me. So you can hear it the second time if

you look at the video recording. You can hear it a couple of times if you look at the video

recordings of the past anyway. Okay, good. So you can now decompose it into three matrices,

and the nice thing here is that sigma is only a diagonal matrix, and the other two matrices,

they are often normal matrices. So in principle, you can think of these two matrices as rotation

matrices. And now you can decompose basically this thing here into three steps. So what

you do is you get your unit ball, and then you apply your first matrix to it, and this

is the matrix U, and what U will do is it's a rotation matrix. So U will do nothing else

but rotate your set of points. If you rotate your unit ball, and it doesn't do any scaling.

If you rotate your unit ball, there's not much to see because your unit ball, after

a rotation, it will still look like this. So this part is interesting if you have something

else than a unit ball. Then the next matrix, this is a diagonal matrix with some entries

on only on the diagonal, and what this will do is it will perform a kind of scaling. So