>> Hola, amigos.
Welcome back.
We just finished our first exploration of
an authentic real world modeling problem
that gives rise to a standard
eigenvalue problem.
That was the coupled pendula problem, and we
use the McCusker apparatus as an entry point
to transform a measurable phenomenon,
something that we could observe and measure,
into an algebraic problem of the form square
matrix A unknown vector x, unknown scalar lambda
for that given matrix A find all eigenvalue
eigenvector pairs such that A times x is equal
to lambda times x. I do want to reiterate that
eigenvalues and eigenvalue theory is really kind
of a second part of linear algebra.
Specifically, we spent a ton of time,
in fact the majority of our introduction
in linear algebra was on the first problem type,
which takes the form known vector
A right hand vector B is known,
and we want to find all values
of x such that A times x is equal
to B. This problem I would call a
statics problem if I was an engineer.
As a mathematician, I say it's a
problem involving algebraic equations,
but the idea is that we are measuring
properties, we are trying to solve
for something that exists without time.
Another way to say this in the
context of real world modeling is
when I'm solving the problem
A times x is equal to B,
that's equivalent to taking snapshots in time.
So I take a picture at one time, I
look at properties of that picture,
I let dynamics play out, then I take a
picture at a different point in time,
and I compare the first picture to
the second picture, and I ask myself,
what are the differences between
the things that I'm observing
at those two specific instances in time.
So imagine throwing a ball in
the air, and the first picture,
I take a picture of the ball right as
it's being thrown, and the second picture,
I take a picture of the ball went to the top,
and then I ask myself, what's the difference
in height in those two pictures, those are
algebraic equations because I'm not trying
to solve for the entire path of
the ball, only for the differences
in position at discrete points in time.
Our approach to linear systems problems, as well
as least-squares problems really focused on how
to solve this problem algebraically.
As we transition into the eigenvalue
theory or the second major problem type
in introductory linear algebra, the main
idea is to tackle, is to have tools to deal
with what we call differential equations.
An engineer might call those dynamics problems.
The point of those is to take problems exist
in the real world that change over time,
and turn them into algebraic processes.
Now, in our analogy, if algebraic problems
are equivalent to studying snapshots in time,
differential equation problems or dynamic
problems are equivalent to looking at movies,
sequences of events that play out over time.
And when we solve those, we don't just want to
talk about the differences in two time periods,
we actually want to create a function
to model the entire behavior
or phenomenon over that movie.
So as time plays out, can we model
what's happening in the sequences
of images that are captured in the videos?
Those two problems kind of fit
nicely into statics and dynamics?
One of my major goals is to help you get a big
picture view of what's happening in this theory.
So as you suffer and focus on studying the
small detail, the minutiae of this theory,
you can make explicit connections between
the hard one, knowledge and wisdom
that you're building for specific technical
content with a much broader view of this theory,
and start to prepare yourself to apply
this theory in action if you'd like.
With that in mind, let's return
to the commutative diagram
of how eigenvalue theories
get applied in practice.
On the left-hand side, we'll focus
on the idea of dynamical systems,
or as a mathematician would
say, differential equations.
The whole idea is that we start with a real
world problem in the coupled pendula problem
that was two pendulum swinging that
are connected together with a spring.
That involves a continuous dynamic phenomena
and something that changes over time.
In our case, we want to figure out
the position of each mass on the end
of each pendulum at any time during observation.
The solution of that would be a
function to describe that or the ability
to pinpoint the exact location of that
mass during any time in our observation.
Going from looking at the problem to having
that solution is not an easy thing to do,
because the human eye can't slow time down.
And usually those dynamics occur quite quickly.
The idea of the mathematical modeling process
is to translate that real world problem,
a continuous problem, into an algebraic
problem via the process of mathematization.
For the coupled pendulum situation, we
actually spent a ton of time mathematizing.
That is a small introduction to the cost, the
actual work and intellectual effort required
to translate meaningful problems
into mathematical problems.
It is nontrivial.
We're going to talk about that in just
a sec, but I do want to call that out.
Once we have our ideal mathematical model for
the continuous problem that we started with,
in this case in the form A times x equals lambda
times x, and we're searching for scalars lambda
and vectors x that satisfy that,
we then do mathematical analysis,
which is basically applying the theory
that we're supposed to learn in math class,
the stuff that math class is really focused
on to generate a meaningful
solution to our ideal problem.
Once we have those meaningful solutions, so
an eigenvalue theory that would be eigenvalue
and eigenvector pairs that
solve that original problem.
The claim is that we should be able to,
if we're lucky, interpret and verify
that our solutions give us meaningful
insights to the original solution
that we wanted for our real world problem.
With this process in mind, I
want to make a few remarks.
Number one, mathematizing
problems is nontrivial.
In fact, it's usually one of the
hardest parts of the entire process.
Number two, most math teachers do not
teach this entire thing in their classes.
What they'll do is they'll focus
on a subset of eigenvalue theory
that is useful for introducing the major ideas.
And then the goal is that you fill in all
these other details to make this useful
in your own life, which often requires
years of intense study and intuition on how
to find information and classes
to help fill in all these pieces.
I personally believe that's a bad model.
I think we can do much better.
In order for us to understand what
the theory that we're learning means,
I think we should actually spend the
time as instructors and as educators
to help you piece together the entire
puzzle while you're being reduced
so that you have motivation to
understand why this theory is important,
and to really dive deeply.
That's what I'm trying to do in this course.
Moreover, mastery over theory
is a long, arduous process.
We will just begin that in
this introductory course.
But in order to get to a place
where we're ready to apply this,
it's going to take a lot more
than just an introduction.
We have to get deep into the weeds.
Another way to say this is that
eigenvalue education is nuanced
and subtle, takes a lot of time.
To help us get idea about how this works,
I want to map that out over to an analogy
about the study of human mobility.
So specifically, in order to get an idea of
what we're doing in building eigenvalue theory,
we're going to tell the story about how
humans learn to move about the world,
and then we're going to make direct connections
between the different stages of that story
and the work that we're going to
do introducing eigenvalue theory.
When we think about human life, we say
usually, a single life begins at birth.
And sometime after that, that baby
learns to crawl on their hands and knees.
It's a slow process, they stumble.
Eventually, that baby gets to a place where
they're confident on their hands and knees
and they're ready to take their first steps.
That's where they begin to walk standing up.
After they've done that for quite some time,
they start to move a little bit faster,
learn how to run, and jump, and move quickly.
This process is really the
foundation of human mobility
and really sets the stage
for more advanced movements.
Specifically, after we learn how to master our
own bodies, humans are amazing in the sense
that we start to bring in larger
technologies to allow us to move.
That might include the ability to bike,
to pedal, to ride a scooter, to skate,
technologies that are really designed
to get us to move around faster
and do require some mastery of the less
technical aspects just involving the body.
In the modern era, we then move
from human powered mobility machines
to automated machines, like cars, buses, vans.
Those allow us to move around a lot faster,
but they also require a huge upfront
cost in order to get into those.
After that, we have what we call the
pinnacle of motion for most human beings,
which is flying on commercial planes.
Now, if I was going to walk down the
street to a store, I don't need to get
in an airplane in order to do that.
But if I want to move around the world,
if I want to solve a big motion problem,
these type of tools are actually
indispensable for being able
to do that quickly and efficiently.
Now, there are some humans that use
specialized vehicles to get around,
like astronauts will use a specialized
vehicle called the spacecraft.
We have people that use helicopters, we have
people that use dirt bikes, we have tanks
and all of the specialized equipment
that get into technical applications.
Now, I'm going to claim that the process of
becoming mobile in human life is analogous
to the process that we're going to
learn in eigenvalue theory education.
So, specifically, I would say that life
begins at the moment that we're introduced
to our first eigenvalue problem that
makes a difference in the world.
There are some mathematicians that won't
even talk about this in their classes.
They'll just skip the idea of
applications and go right into theory.
But as Dan Meyer said so well, if mathematics
is the medicine, what's the headache?
In order for us to understand how to move,
in order to understand how the theory actually
applies, we have to have the intellectual need
for that theory, and that in my
opinion, means we need to know
where eigenvalues show up in our own world.
Once were born into the space of needing
eigenvalue theory, my claim is the first step
in order to learn to crawl, to move around
in our hands knees, we need to know how
to solve small problems by hand
using pen and paper analysis.
The reason is we want to start to set
the foundation for more complex problems
by getting our head around how these
things work using our own brains.
Once we've studied our first few problems by pen
and paper analysis, maybe two by two and three
by three matrices, that's where we
start building an understanding of
and fluency with basic eigenvalue theory.
At this point, we're ready to
really apply eigenvalue theory
for small scale real world problems.
And I would call this the end of the
introduction to eigenvalue theory
and the beginning of our
more advanced stages of life.
This is the end of single human
mobility based on the body and the start
of bringing much more powerful
tools to bear on harder problems.
We're not going to spend a lot of time
talking about that in the introductory class,
but I do want to call those out so
that you understand what you're doing
when you set the foundation in eigenvalue
theory for future possible applications.
When we make the transition to -- from
introduction to more advanced theory,
I believe that the first phase of
that is to find larger problems
that eigenvalue theories apply to.
That would happen in advanced
engineering and technical classes as well
as on the job training and engineering firms.
Once we have those larger problems that can
be solved using the eigenvalue theories,
that's where we start getting into what we
call specialized eigenvalue software libraries.
And this is where the need
for computation comes in.
There is a point at which the size of a problem
overwhelms our ability to do hand calculations.
Could you imagine if I had 50 pendula
coupled with 49 different springs?
If I were to try to do that problem by
hand, I would die before I solve it.
That's where we bring in much, much larger
tolls which are the ideas of computation
so that we can leverage those tools to speed
up the way that we solve those problems.
Now, I would say when we refine those into
commercial airplane style problems, very,
very large problems, that's what
we call finite element methods.
And in fact, what I would say is
within finite element methods,
there are specialized communities of practice
that require customized eigenvalue
algorithms for very special applications.
These are the spaces where professional
engineers and mathematicians make a lot
of money solving problems because they require
real specialization and technical skills
in order to solve problems
on an industrial scale.
The point of introductory linear algebra
is get us to a place where we are familiar
with the terminology and the
ideas as well as the application.
Once we have that computational theory, it takes
us from small scale to large scale problems,
and then industry pays for mathematicians to
be able to solve those large scale problems.
This is why it's so important that engineers
take prerequisite classes so that when we get
to these larger problems, we have a foundation
upon which we can build our understanding.
I like this analogy for many reasons.
And one of them is that as
we move down in complexity,
the more complex concepts are based
on fluency in the previous step.
So the idea of early introductions actually
sets us up for more complex abilities.
And the same thing is true with mobility.
As we get more and more mobile
and aware of our own bodies,
we become able to leverage more powerful
techniques and tools to move around faster.
Another way to say this is
wrapped up in the idea we have
to learn to crawl before we can walk.
For eigenvalue theory, that means that we
study the basics, we study the introduction,
before we get introduced to
the more complex concepts.
And the more fluency we have
in that foundational material,
the lower barrier to entry we will face
when we go on to more advanced problems,
and we might go to apply
those problems in practice.
The last reason that I really like this is
that mathematicians tend not to talk a lot
about the birthing process, but it is
the most fundamental miracle of life.
Birth is really painful and a huge
investment of energy and societal resources go
in to making sure that that birth happens well.
The same is true for eigenvalue problems.
The painful reality of most eigenvalue
theories is, it is actually just as hard
to create the problems themselves
than it is to solve it.
And at this point in life,
it is actually more difficult
to build meaningful eigenvalue problems than
it is to write code to solve those problems.
That is one of the untold
stories of applied mathematics.
Building meaningful problems is probably one
of the most valuable things that we can do
with our education, and it is one of the
least taught things in formal classrooms.
One of the gifts that I hope to give
my students is insights into how
to build specialized expertise
and interdisciplinary knowledge
so that we ourselves can create our
own problems and bring those tools
into the communities that we serve.
This is my focus in helping you set the
foundation to give you the best introduction
and the best material that I can provide
for you so that you can set a foundation
that you can build future expertise on.
In this case, I'm partitioning all
of the steps to build that expertise
into what I would call introductory
material and advanced material.
To get advanced material, you're going
to have to go on to future courses
and really seek that out in your education.
For our situation, we've already got
a sense of where eigenvalue show up
and we've studied an authentic real world
problem to do so, the coupled pendula problem.
For the rest of our introductory lessons
on eigenvalue theory, we're going to fill
out the rest of this material
and show how to use that theory
to solve our coupled pendula problems.
The entire structure of our introduction
assumes that we begin with a matrix,
and then we're going to develop
theory that will allow us
to solve eigenvalue problems
involving that matrix.
With that in mind, our introduction
to eigenvalue theory is going
to help us answer some fundamental
problems that arise
in the mathematical analysis
phase of any applied context.
Some of those questions include, how does the
algebraic structure of an n by n matrix relate
to the eigenvalues and eigenvectors
for that matrix?
Can we get insights into the types of
eigenvalues and eigenvectors that arise
if we know information about the matrix itself?
Next, we might wonder, if we are
given a matrix, how many eigenvalues
and eigenvectors does that matrix have?
Is it guaranteed that we'll always have a
full set of eigenvalues and eigenvectors
for any matrix that we'll encounter in practice?
We're also going to try to explore what
are the connections between the eigenvalue
and eigenvector pairs, and
information encoded in that matrix.
How are those things related to each other?
And then finally, if we are able to
generate eigenvalues and eigenvectors
for a specific matrix, how can we use
that information in applied problems?
What type of insights do eigenvalue and
eigenvector pairs give us about applying
that information when we're trying
to solve dynamical systems problems?
Our introduction is going to focus on trying to
get good answers to these questions to help us
when we're trying to analyze problems
that arise with a given square matrix
for our real world modeling problem.
In the next video, we'll break down the
different topics we're going to study
in this introduction and get a little bit
more specific to give a global interview
of our strategies developing
eigenvalue foundations.
I'll see you in the next video.
