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So welcome to week six of mathematical
methods for quantitative finance.

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We're going to continue this week talking
about linear algebra.

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last week we got the basics down, vectors,
matrices and

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some of the arithmetic rules for working
with vectors and matrices.

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And now we're going to look a little bit
deeper into factorizations

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of matrices and the types of problems you
can solve with them.

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So this week's

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menu, I'll start off with a few more a few
more properties or types of matrices.

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So there's a, a transpose, is an operation
you can do on a matrix.

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And a permutation is another operation you
can do on a matrix.

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And then I'll talk a little bit about
vector spaces and sub spaces.

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And then I really want to get in to sort
of the

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meat of this weeks lectures which are
going to be operations.

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So, when you want to do something to a
data set

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Often that something can be written down
as a matrix

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and then I can complete that operation
just by doing, matrix multiplication or

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some potentially more complicated formula
involving matrix manipulations.

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And the example I'm going to use to, to
show

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how that can be done is variance,
covariance matrix which

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in section four I just start calling
covariance matrix

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because the full name is kind of difficult
to pronounce.

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Then in lecture five I'm going to
introduce

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yet another type of matrix called an
orthoganal matrix.

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And this is a matrix that has a very

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nice property that it's inverse is also
its transpose.

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And then I'm going to look at

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some factorizations of matrices that
basically involve

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factoring, the full matrix into orthogonal
matrices and then one more matrix that has

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some sort of special property.

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So in the singular value factorization I'm

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going to end up with two orthogonal
matrices.

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So these are very easy matrices to work
with because they

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have this nice property their inverse is
equal to their transpose.

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And then the third matrix is going to be a
diagonal matrix which that means that

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if, if I look down the main diagonal of
the matrix, those elements are non zero.

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But any other element has to be zero.

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And so again, this is a very easy matrix
to work with because

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it's inverse is just the reciprocal of
each one of those diagonal elements.

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And then I'm going to use the singular

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value factorization to motivate something
called the eigenvalues

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and eigenvectors and then finally in
section eight

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I'm going to look at solving something
called a least.

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Squares problem.

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And so this is if I have a, a scatter
plot.

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values so I have x values on the x axis, y
values on the y axis.

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And then I want to put a best fit line
onto that plot.

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The, the slope and the y intercept at that
best,

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best fit line you calculate by solving a
least squares problem.

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And so I'm going to go over how you can
use

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a matrix factorization called a QR
factorization to solve that problem.