Welcome everybody to the Path of Recognition exercises.

The topic of today is feature transforms.

During the meeting you can interrupt using the microphone or writing in the chat.

So if I see that... yes?

I would just ask you if it's possible to upload the solution because we want to also take notes on the solution.

If it's possible, I don't know.

Yes, yes, it is possible.

So normally when I give the session, I upload the solution after I give the presentation on Friday.

So after this session, I'm going to upload these slides.

And later when the video is processed, then I upload also the video in FAU TV.

Okay, thank you.

Okay.

So yeah, feel free to interrupt with the microphone.

And if I see that it starts to become a disaster, then we raise the hand.

But in the meanwhile, you can just interrupt or use the chat.

Although the chat is a bit more difficult for me because then I am like focused on the slides and then I have to also check the chat.

But after each exercise, I will check the chat.

Okay.

So let's start with the first exercise.

And the first exercise, you can find the solutions in these two books that are recommended on the slides of the lecture as well.

And you can refer to these pages here.

So yeah, so if you want to later study these slides, you can also refer to these pages.

Okay. In this exercise, we will refresh your knowledge about singular value decomposition.

And first we are going to use C. So in the exercise A and B. So this one is composed, this first exercise is composed of three parts, A, B, and C.

And A and B, we will only see some equivalences that are going to help us to understand the exercise C that is the main objective of the exercise that I will explain later.

So the first exercise. So first, let's go through a reminder of what is singular value decomposition.

So singular value decomposition allows us to write any matrix as the product of these three matrices.

And these three matrices, so any matrix of size n by m is not restricted to that they have to be symmetric. So any matrix.

And then what is U, D, and V. So U is the eigenvectors of A by A transpose. And then, so because of these sizes, A and A transpose is an n by n matrix.

And the columns of the U matrix are going to cover the column space of A.

And D, the second matrix is given by this expression where the most important thing for you to remember is that the diagonal of this matrix is, are the eigenvalues of A by A transpose.

So,

okay, then the third matrix, the last one, are the eigenvectors of A transpose by A.

And then the size is n by m.

So, yeah, because the original matrix was n by m, then the A transpose by A is n by m.

And then we have that the columns in B, in this matrix B,

cover the row space of A, the original matrix.

So this is just like a reminder of what is singular value decomposition.

And so in the first part of the exercise, the question is, what is the relationship between SVD of a square matrix A and A by A transpose.

So first, A, so the singular value decomposition of A is given by this expression that is just the definition of SVD.

And then when we use A transpose,

we have to transpose this part.

So this part.

And we use this to distribute this transpose over these matrices.

And we use this property,

where the order of the matrices when applying a transpose are inverted. So now the last matrix is now the first one.

That's why here the order is inverted.

So this is only, so later you will see why we are going to use these expressions, this equality here.

But yeah, so this is for a later use. OK.

And then what we can say of this relationship between the singular value decomposition of a square matrix A and A transpose.