>> What's happening y'all, welcome back.
In this video, we're going
to continue our conversation
about introductory eigenvalue theory.
Specifically recall that when we're
studying the problem, Ax equal lambda x,
we call that the standard eigenvalue problem.
We introduced this analogy in the last
video that we can think about the story
of eigenvalue education, like
the story of human mobility.
In this video, we're going to further refine
our different steps of eigenvalue education
into introductory and advanced steps and give
an overview of the specific focus we're going
to make at each stage in our
introduction to eigenvalue theory.
Recall from the last video, we said that
there were really eight steps in our process
of becoming experts in eigenvalue theory.
The first four of which were
going to be the focus
of the introductory explorations
that we do in this course.
With that in mind, we've already become
aware of eigenvalue problem via the study
of the couple pendula problem that we
did in the first eigenvalue lesson.
From here, the rest of this
lesson is going to be focused
on solving small problems using
pen and paper analysis by hand.
We're going to start with an overview looking at
case studies of eigenvalues of 2 by 2 matrices,
we're going to get a really, really good
idea of the different types of eigenvalues
and eigenvector pairs that
show up in the 2 by 2 case.
We're going to then leverage that information
to study the 3 by 3 case and get some ideas
of the type of eigenvalues and
eigenvectors that show up in that situation.
Given the insights that we build
in both 2 by 2 and 3 by 3 matrices,
we're then going to generalize to a study of
what we call the characteristic equation as well
as properties of the characteristic polynomial.
Once we've built meaningful intuition using
the examples in small problems about the type
of eigenvalue and eigenvector pairs
that show up, and we started to think
about characteristic polynomials and how to
solve that, we are then going to leverage
that intuition to build basic eigenvalue theory.
To start with that process is going
to be to focus on diagonalization.
How do we diagonalize a matrix given
that we have a full eigenvalue basis?
The next thing we're going to talk about is
the concept of similarity transformations.
Similarity transformations
actually set the foundation
for computational algorithms
in eigenvalue theory.
That's a very powerful strategy
that is involved in intermediate
and expert level eigenvalue calculations.
We're then going to look at
eigenvalue and eigenvector theory
for specialized algebraic structures.
Specifically, we're going to take a look at what
happens when we're trying to find eigenvalues
of symmetric matrices that will lead to the
discussion of even more specialized group
of matrices called positive definite matrices.
We'll investigate eigenvalues
and eigenvectors of those types.
These 2 types of matrices show up very often in
an application and set the foundation for most
of the advanced theory, assuming
special structure and matrices.
Once we have this under our belt, we're going
to take a look at minimization principles
and how to interpret eigenvalues
geometrically using the concept of minimization.
We saw that in least squares
and we're going to develop
at analogous approach in eigenvalue theory.
The last part of our introduction will be
to cover the idea of incomplete matrices.
These are theoretically very important, though
in practice, incomplete matrices form a very,
very small subset of the larger type
of problems that we want to solve.
They're worth mentioning so that we're
not surprised if we trip over one.
However, the main thrust is going to be
to prepare us to apply eigenvalue theory
in the context that we are most likely to
see them, which are these other major ideas.
At the end of this process, we're going to
actually leverage the intuition that we build
to fully solve a couple pendula problem and
complete the mathematical modeling process
for the standard eigenvalue
problem that arises in that case.
Because I recognize that the introductory
work actually are prerequisites
for more advanced theory,
let's give a brief overview
of the advanced levels of eigenvalue education.
One of the first things that I would
recommend students to do is to go hunt down
and train themselves on how to apply
eigenvalue theory in more advanced context.
This will require insights into the type
of classes that teach that and the type
of interdisciplinary knowledge
that relies on that information.
Office hours are a great way to get
that as well as online searches.
At this point, there's some really,
really good online searches of people
who have asked those questions
that you can find for yourself.
Once we have access to meaningful large
eigenvalue problems, one of the first things
that comes up is how do we compute those
things if we're not going to do it by hand.
That's what I would call
numerical linear algebra.
It's a very, very popular
field and quite powerful.
And in order to do that, we would use
specialized eigenvalue software libraries.
For those of you that want to learn more
about that, I would definitely recommend
that you take introduction to numerical
analysis and perhaps as a three sequence class.
Later in my career, I plan
to do a sequence of videos.
Each lesson is designed to introduce
a specific algorithmic approach
to computing eigenvalue problems.
There are kind of three very popular algorithms,
one of them is called the power iteration.
Another one is called QR-based algorithms that
take advantage of similarity transformations
to transform an original matrix into a
simplified form, very similar to what we did
in Gaussian elimination, except with orthogonal
matrices on the left and right hand side.
And then finally, there's an
approach called tridiagonalization,
where we turn our original matrix
A into a tridiagonal matrix
and then apply special matrices to
find the eigenvalues of that one.
There are also more advanced algorithms that
take advantage of some of these foundations
and apply preconditioners,
blah-blah-blah, blah-blah-blah, blah.
At this point, I do want to give a shout out to
a resource that is quite useful to get insights
into specialized eigenvalue software
libraries, as well as customized algorithm
to solve very large scale eigenvalue problems.
If you go to any search browser and
type in templates for the solutions
to eigenvalue problems, you'll get a result
at the top, which is a book from SIAM,
Society of Industrial and Applied Mathematics.
Up here, this is the website
of a man named Zhaojun Bai,
he's one of the editors of this book.
If you read this, you're
welcome to do that on your own.
You'll get a sense of what this book is about.
It's designed for practicing
engineers and mathematicians who need
to solve eigenvalue problems and
want to find algorithms to do that.
What's crazy about this book is that the entire
contents are available online in HTML format.
So you can get access to this, if you buy
it on Amazon it's about a hundred bucks.
But this gives insights into the more
advanced levels of eigenvalue theory
and it also hopefully will motivate you to take
the introductory material much more seriously
because once you get the introduction under
your belt, a lot of the theory that is covered
in these more advanced techniques, you'll
be ready to understand the foundations.
With that we're ready to start our
work building intuition in the case
of small eigenvalue problems
involving 2 by 2 matrix.
That's exactly what we're
going to do in the next videos.
