Please do not be discouraged by the lectures

of last Tuesday and today, because this is a very crucial tool,

a very fundamental tool, and once we know about that,

we will just use it for solving all the practical problems

that we are considering in the near future.

So please do not be discouraged by that.

Last time we have considered

the signal-value decomposition.

The idea of the signal-value decomposition was that

I basically can decompose any matrix

into a product of three matrices.

At least three matrices have the following properties.

U is a Poissonal matrix.

Sigma is a diagonal matrix

with positive values on the diagonal of decreasing earlier

and another Poissonal matrix V

that is here given in the product as it's transposed.

And using this vectorization that is implemented

in all the packages that we are basically using

for our experiments, like MATLAB or Maple or Mathematica,

they provide this powerful decomposition

and you can use it by using this command

and in MATLAB especially you can just write it this way,

saying that you have the triple U S V

and you compute the three matrices

by just using this command here.

And it's important to note that we can decompose all matrices,

square matrices, rectangle matrices,

singular matrices, regular matrices into U sigma V transposed.

Very fundamental and now we have considered a few properties.

Which properties are important for us?

For instance, we can read the rank of the matrix

by looking at the singular values.

We also have introduced the concept of the numerical rank.

What does it mean that matrix is close to be singular?

Now we have a much better understanding of that.

We have the spectral decomposition of the matrix A.

We have seen that the sum squared singular values

is the verbatimous number and we also build a relationship

between the singular values and the eigenvalues

that we know from basic math lectures.

Then we have considered the two-norms of matrix.

That is related to the large singular value.

Here the concept of the condition number.

I don't want to repeat that in all details.

Now we are considering four important optimization problems.

Why did I pick out these four optimization problems?

Because these types of optimization problems

will reappear over and over again

if we do many language processing.