So welcome to week six of mathematical
methods for quantitative finance.
We're going to continue this week talking
about linear algebra.
last week we got the basics down, vectors,
matrices and
some of the arithmetic rules for working
with vectors and matrices.
And now we're going to look a little bit
deeper into factorizations
of matrices and the types of problems you
can solve with them.
So this week's
menu, I'll start off with a few more a few
more properties or types of matrices.
So there's a, a transpose, is an operation
you can do on a matrix.
And a permutation is another operation you
can do on a matrix.
And then I'll talk a little bit about
vector spaces and sub spaces.
And then I really want to get in to sort
of the
meat of this weeks lectures which are
going to be operations.
So, when you want to do something to a
data set
Often that something can be written down
as a matrix
and then I can complete that operation
just by doing, matrix multiplication or
some potentially more complicated formula
involving matrix manipulations.
And the example I'm going to use to, to
show
how that can be done is variance,
covariance matrix which
in section four I just start calling
covariance matrix
because the full name is kind of difficult
to pronounce.
Then in lecture five I'm going to
introduce
yet another type of matrix called an
orthoganal matrix.
And this is a matrix that has a very
nice property that it's inverse is also
its transpose.
And then I'm going to look at
some factorizations of matrices that
basically involve
factoring, the full matrix into orthogonal
matrices and then one more matrix that has
some sort of special property.
So in the singular value factorization I'm
going to end up with two orthogonal
matrices.
So these are very easy matrices to work
with because they
have this nice property their inverse is
equal to their transpose.
And then the third matrix is going to be a
diagonal matrix which that means that
if, if I look down the main diagonal of
the matrix, those elements are non zero.
But any other element has to be zero.
And so again, this is a very easy matrix
to work with because
it's inverse is just the reciprocal of
each one of those diagonal elements.
And then I'm going to use the singular
value factorization to motivate something
called the eigenvalues
and eigenvectors and then finally in
section eight
I'm going to look at solving something
called a least.
Squares problem.
And so this is if I have a, a scatter
plot.
values so I have x values on the x axis, y
values on the y axis.
And then I want to put a best fit line
onto that plot.
The, the slope and the y intercept at that
best,
best fit line you calculate by solving a
least squares problem.
And so I'm going to go over how you can
use
a matrix factorization called a QR
factorization to solve that problem.