WEBVTT
00:00:08.000 --> 00:00:14.000
Everything I say today is going
to be for n-by-n systems,
00:00:12.000 --> 00:00:18.000
but for your calculations and
the exams two-by-two will be
00:00:17.000 --> 00:00:23.000
good enough.
Our system looks like that.
00:00:20.000 --> 00:00:26.000
Notice I am talking today about
the homogeneous system,
00:00:25.000 --> 00:00:31.000
not the inhomogenous system.
So, homogenous.
00:00:36.000 --> 00:00:42.000
And we have so far two basic
methods of solving it.
00:00:40.000 --> 00:00:46.000
The first one,
on which we spent the most
00:00:43.000 --> 00:00:49.000
time, is the method of where you
calculate the eigenvalues of the
00:00:48.000 --> 00:00:54.000
matrix, the eigenvectors,
and put them together to make
00:00:53.000 --> 00:00:59.000
the general solution.
So eigenvalues,
00:00:56.000 --> 00:01:02.000
e-vectors and so on.
The second method,
00:01:00.000 --> 00:01:06.000
which I gave you last time,
I called "royal road," simply
00:01:04.000 --> 00:01:10.000
calculates the matrix e to the
At and says that the
00:01:08.000 --> 00:01:14.000
solution is e to the At times x
zero,
00:01:11.000 --> 00:01:17.000
the initial condition.
That is very elegant.
00:01:14.000 --> 00:01:20.000
The only problem is that to
calculate the matrix e to the
00:01:18.000 --> 00:01:24.000
At, although sometimes you can
do it by its definition as an
00:01:22.000 --> 00:01:28.000
infinite series,
most of the time the only way
00:01:26.000 --> 00:01:32.000
to calculate the matrix e to the
At is by using the fundamental
00:01:30.000 --> 00:01:36.000
matrix.
In other words,
00:01:33.000 --> 00:01:39.000
the normal way of doing it is
you have to calculate it as the
00:01:37.000 --> 00:01:43.000
fundamental matrix time
normalized at zero.
00:01:40.000 --> 00:01:46.000
So, as I explained at the end
of last time and you practiced
00:01:44.000 --> 00:01:50.000
in the recitations,
you have to find the
00:01:46.000 --> 00:01:52.000
fundamental matrix,
which, of course,
00:01:49.000 --> 00:01:55.000
you have to do by eigenvalues
and eigenvectors.
00:01:52.000 --> 00:01:58.000
And then you multiply it by its
value at zero,
00:01:55.000 --> 00:02:01.000
inverse.
And that, by magic,
00:01:57.000 --> 00:02:03.000
turns out to be the same as the
exponential matrix.
00:02:02.000 --> 00:02:08.000
But, of course,
there has been no gain in
00:02:05.000 --> 00:02:11.000
simplicity or no gain in ease of
calculation.
00:02:08.000 --> 00:02:14.000
The only difference is that the
language has been changed.
00:02:12.000 --> 00:02:18.000
Now, today is going to be
devoted to yet another method
00:02:16.000 --> 00:02:22.000
which saves no work at all and
only amounts to a change of
00:02:21.000 --> 00:02:27.000
language.
The only reason I give it to
00:02:23.000 --> 00:02:29.000
you is because I have been
begged by various engineering
00:02:28.000 --> 00:02:34.000
departments to do so --
-- because that is the language
00:02:33.000 --> 00:02:39.000
they use.
In other words,
00:02:35.000 --> 00:02:41.000
each person who solves systems,
some like to use fundamental
00:02:39.000 --> 00:02:45.000
matrices, some just calculate,
some immediately convert the
00:02:44.000 --> 00:02:50.000
system by elimination into a
single higher order equation
00:02:48.000 --> 00:02:54.000
because they are more
comfortable with that.
00:02:52.000 --> 00:02:58.000
Some, especially if they are
writing papers,
00:02:55.000 --> 00:03:01.000
they talk exponential matrices.
But there are a certain number
00:03:01.000 --> 00:03:07.000
of engineers and scientists who
talk decoupling,
00:03:05.000 --> 00:03:11.000
express the problem and the
answer in terms of decoupling.
00:03:09.000 --> 00:03:15.000
And that is,
therefore, what I have to
00:03:12.000 --> 00:03:18.000
explain to you today.
So, the third method,
00:03:16.000 --> 00:03:22.000
today's method,
I stress is really no more than
00:03:20.000 --> 00:03:26.000
a change of language.
And I feel a little guilty
00:03:23.000 --> 00:03:29.000
about the whole business.
Instead of going more deeply
00:03:29.000 --> 00:03:35.000
into studying these equations,
what I am doing is like giving
00:03:33.000 --> 00:03:39.000
a language course and teaching
you how to say hello and
00:03:37.000 --> 00:03:43.000
good-bye in French,
German, Spanish,
00:03:40.000 --> 00:03:46.000
and Italian.
It is not going very deeply
00:03:43.000 --> 00:03:49.000
into any of those languages,
but you are going into the
00:03:47.000 --> 00:03:53.000
outside world,
where people will speak these
00:03:50.000 --> 00:03:56.000
things.
Here is an introduction to the
00:03:52.000 --> 00:03:58.000
language of decoupling in which
for some people is the exclusive
00:03:57.000 --> 00:04:03.000
language in which they talk
about systems.
00:04:02.000 --> 00:04:08.000
Now, I think the best way to,
well, in a general way,
00:04:06.000 --> 00:04:12.000
what you try to do is as
follows.
00:04:09.000 --> 00:04:15.000
You try to introduce new
variables.
00:04:13.000 --> 00:04:19.000
You make a change of variables.
I am going to do it two-by-two
00:04:18.000 --> 00:04:24.000
just to save a lot of writing
out.
00:04:22.000 --> 00:04:28.000
And it's going to be a linear
change of variables because we
00:04:27.000 --> 00:04:33.000
are interested in linear
systems.
00:04:32.000 --> 00:04:38.000
The problem is to find u and v
such that something wonderful
00:04:37.000 --> 00:04:43.000
happens, such that when you make
the change of variables to
00:04:43.000 --> 00:04:49.000
express this system in terms of
u and v it becomes decoupled.
00:04:49.000 --> 00:04:55.000
And that means the system turns
into a system which looks like u
00:04:56.000 --> 00:05:02.000
prime equals k1 times u
and v prime equals k2
00:05:01.000 --> 00:05:07.000
times v.
Such a system is called
00:05:06.000 --> 00:05:12.000
decoupled.
Why?
00:05:08.000 --> 00:05:14.000
Well, a normal system is called
coupled.
00:05:11.000 --> 00:05:17.000
Let's write out what it would
be.
00:05:14.000 --> 00:05:20.000
Well, let's not write that.
You know what it looks like.
00:05:18.000 --> 00:05:24.000
This is decoupled because it is
not really a system at all.
00:05:23.000 --> 00:05:29.000
It is just two first-order
equations sitting side by side
00:05:28.000 --> 00:05:34.000
and having nothing whatever to
do with each other.
00:05:34.000 --> 00:05:40.000
This is two problems from the
first day of the term.
00:05:38.000 --> 00:05:44.000
It is not one problem from the
next to last day of the term,
00:05:44.000 --> 00:05:50.000
in other words.
To solve this all you say is u
00:05:48.000 --> 00:05:54.000
is equal to some constant times
e to the k1 t and v
00:05:54.000 --> 00:06:00.000
equals another constant times e
to the k2 t.
00:06:00.000 --> 00:06:06.000
Coupled means that the x and y
occur in both equations on the
00:06:04.000 --> 00:06:10.000
right-hand side.
And, therefore,
00:06:06.000 --> 00:06:12.000
you cannot solve separately for
x and y, you must solve together
00:06:10.000 --> 00:06:16.000
for both of them.
Here I can solve separately for
00:06:14.000 --> 00:06:20.000
u and v and, therefore,
the system has been decoupled.
00:06:17.000 --> 00:06:23.000
Now, obviously,
if you can do that it's an
00:06:20.000 --> 00:06:26.000
enormous advantage,
not just to the ease of
00:06:23.000 --> 00:06:29.000
solution, because you can write
down the solution immediately,
00:06:28.000 --> 00:06:34.000
but because something physical
must be going on there.
00:06:33.000 --> 00:06:39.000
There must be some insight.
There ought to be some physical
00:06:37.000 --> 00:06:43.000
reason for these new variables.
Now, that is where I plan to
00:06:42.000 --> 00:06:48.000
start with.
My plan for the lecture is
00:06:45.000 --> 00:06:51.000
first to work out,
in some detail,
00:06:48.000 --> 00:06:54.000
a specific example where
decoupling is done to show how
00:06:52.000 --> 00:06:58.000
that leads to the solution.
And then we will go back and
00:06:57.000 --> 00:07:03.000
see how to do it in general --
-- because you will see,
00:07:02.000 --> 00:07:08.000
as I do the decoupling in this
particular example,
00:07:05.000 --> 00:07:11.000
that that particular method,
though it is suggested,
00:07:09.000 --> 00:07:15.000
will not work in general.
I would need a more general
00:07:13.000 --> 00:07:19.000
method.
But let's first go to the
00:07:15.000 --> 00:07:21.000
example.
It is a slight modification of
00:07:17.000 --> 00:07:23.000
one you should have done in
recitation.
00:07:20.000 --> 00:07:26.000
I don't think I worked one of
these in the lecture,
00:07:23.000 --> 00:07:29.000
but to describe it I have to
draw two views of it to make
00:07:27.000 --> 00:07:33.000
sure you know exactly what I am
talking about.
00:07:32.000 --> 00:07:38.000
Sometimes it is called the two
compartment ice cube tray
00:07:35.000 --> 00:07:41.000
problem, a very old-fashion type
of ice cube tray.
00:07:38.000 --> 00:07:44.000
Not a modern one that is all
plastic where there is no
00:07:42.000 --> 00:07:48.000
leaking from one compartment to
another.
00:07:44.000 --> 00:07:50.000
The old kind of ice cube trays,
there were compartments and
00:07:48.000 --> 00:07:54.000
these were metal separated and
you leveled the liquid because
00:07:52.000 --> 00:07:58.000
it could leak through the bottom
that didn't go right to the
00:07:56.000 --> 00:08:02.000
bottom.
If you don't know what I am
00:07:58.000 --> 00:08:04.000
talking about it makes no
difference.
00:08:02.000 --> 00:08:08.000
This is the side view.
This is meant to be twice as
00:08:06.000 --> 00:08:12.000
long.
But, to make it quite clear,
00:08:09.000 --> 00:08:15.000
I will draw the top view of
this thing.
00:08:12.000 --> 00:08:18.000
You have to imagine this is a
rectangle, all the sides are
00:08:17.000 --> 00:08:23.000
parallel and everything.
This is one and this is two.
00:08:21.000 --> 00:08:27.000
All I am trying to say is that
the cross-sectional area of
00:08:26.000 --> 00:08:32.000
these two chambers,
this one has twice the
00:08:29.000 --> 00:08:35.000
cross-sectional area of this
one.
00:08:34.000 --> 00:08:40.000
So I will write a two here and
I will write a one there.
00:08:37.000 --> 00:08:43.000
Of course, it is this hole here
through which everything leaks.
00:08:42.000 --> 00:08:48.000
I am going to let x be the
height of this liquid,
00:08:45.000 --> 00:08:51.000
the water here,
and y the height of the water
00:08:48.000 --> 00:08:54.000
in that chamber.
Obviously, as time goes by,
00:08:51.000 --> 00:08:57.000
they both reach the same height
because of somebody's law.
00:08:55.000 --> 00:09:01.000
Now, what is the system of
differential equations that
00:08:59.000 --> 00:09:05.000
controls this?
Well, the essential thing is
00:09:04.000 --> 00:09:10.000
the flow rate through here.
That flow rate through the hole
00:09:11.000 --> 00:09:17.000
in units, let's say,
in liters per second.
00:09:16.000 --> 00:09:22.000
Just so you understand,
I am talking about the volume
00:09:22.000 --> 00:09:28.000
of liquid.
I am not talking about the
00:09:27.000 --> 00:09:33.000
velocity.
That is proportional to the
00:09:31.000 --> 00:09:37.000
area of the hole.
So the cross-sectional area of
00:09:37.000 --> 00:09:43.000
the hole.
And it is also times the
00:09:40.000 --> 00:09:46.000
velocity of the flow,
but the velocity of the flow
00:09:44.000 --> 00:09:50.000
depends upon the pressure
difference.
00:09:47.000 --> 00:09:53.000
And that pressure difference
depends upon the difference in
00:09:52.000 --> 00:09:58.000
height.
All those are various people's
00:09:55.000 --> 00:10:01.000
laws.
So times the height difference.
00:10:00.000 --> 00:10:06.000
Of course, you have to get the
sign right.
00:10:03.000 --> 00:10:09.000
I have just pointed out the
height difference is
00:10:06.000 --> 00:10:12.000
proportional to the pressure at
the hole.
00:10:10.000 --> 00:10:16.000
And it is that pressure at the
hole that determines the
00:10:14.000 --> 00:10:20.000
velocity with which the fluid
flows through.
00:10:17.000 --> 00:10:23.000
Where does this all produce our
equations?
00:10:21.000 --> 00:10:27.000
Well, x prime is equal to,
therefore, some constant,
00:10:25.000 --> 00:10:31.000
depending on the area of the
hole and this constant of
00:10:29.000 --> 00:10:35.000
proportionality with the
pressure and the units and
00:10:33.000 --> 00:10:39.000
everything else times the
pressure difference I am talking
00:10:37.000 --> 00:10:43.000
about.
Well, if fluid is going to flow
00:10:43.000 --> 00:10:49.000
in this direction that must mean
the y height is higher than the
00:10:49.000 --> 00:10:55.000
x height.
So, to make x prime positive,
00:10:52.000 --> 00:10:58.000
it should be y minus x here.
00:10:56.000 --> 00:11:02.000
Now, the y prime is different.
Because, again,
00:11:01.000 --> 00:11:07.000
the rate of fluid flow is
determined.
00:11:03.000 --> 00:11:09.000
This time, if y prime is
positive, if this is rising,
00:11:08.000 --> 00:11:14.000
as it will be in this case,
it's because the fluid is
00:11:12.000 --> 00:11:18.000
flowing in that direction.
It is because x is higher than
00:11:16.000 --> 00:11:22.000
y.
So this should be the same
00:11:18.000 --> 00:11:24.000
constant x minus y.
But notice that right-hand side
00:11:23.000 --> 00:11:29.000
is the rate at which fluid is
flowing into this tank.
00:11:27.000 --> 00:11:33.000
That is not the rate at which y
is changing.
00:11:32.000 --> 00:11:38.000
It is the rate at which 2y is
changing.
00:11:34.000 --> 00:11:40.000
Why isn't there a constant
here?
00:11:36.000 --> 00:11:42.000
There is.
It's one.
00:11:38.000 --> 00:11:44.000
That is the one,
this one cross-section.
00:11:41.000 --> 00:11:47.000
The area here is one and the
cross-sectional area here is
00:11:45.000 --> 00:11:51.000
two.
And that is the reason for the
00:11:47.000 --> 00:11:53.000
one here and the two here,
because we are interested in
00:11:51.000 --> 00:11:57.000
the rate at which fluid is being
added to this,
00:11:54.000 --> 00:12:00.000
which is only related to the
height, the rate at which the
00:11:58.000 --> 00:12:04.000
height is rising if you take
into account the cross-sectional
00:12:03.000 --> 00:12:09.000
area.
So there is the system.
00:12:07.000 --> 00:12:13.000
In order to use nothing but
integers here,
00:12:11.000 --> 00:12:17.000
I am going to take c equals to
two, so I don't have to put in
00:12:17.000 --> 00:12:23.000
halves.
The final system is x prime
00:12:21.000 --> 00:12:27.000
equals minus 2x,
you have to write them in the
00:12:25.000 --> 00:12:31.000
correct order,
and y prime equals,
00:12:29.000 --> 00:12:35.000
the twos cancel because c is
two, is x minus y.
00:12:35.000 --> 00:12:41.000
So there is our system.
Now the problem is I want to
00:12:38.000 --> 00:12:44.000
solve it by decoupling it.
I want, in other words,
00:12:42.000 --> 00:12:48.000
to find new variables,
u and v, which are more natural
00:12:46.000 --> 00:12:52.000
to the problem than the x and y
that are so natural to the
00:12:50.000 --> 00:12:56.000
problem that the new system will
just consistently be two
00:12:54.000 --> 00:13:00.000
side-by-side equations instead
of the single equation.
00:12:59.000 --> 00:13:05.000
The question is,
what should u and v be?
00:13:01.000 --> 00:13:07.000
Now, the difference between
what I am going to do now and
00:13:05.000 --> 00:13:11.000
what I am going to do later in
the period is later in the
00:13:08.000 --> 00:13:14.000
period I will give you a
systematic way of finding what u
00:13:12.000 --> 00:13:18.000
and v should be.
Now we are going to psyche out
00:13:15.000 --> 00:13:21.000
what they should be in the way
in which people who solve
00:13:18.000 --> 00:13:24.000
systems often do.
I am going to use the fact that
00:13:21.000 --> 00:13:27.000
this is not just an abstract
system of equations.
00:13:24.000 --> 00:13:30.000
It comes from some physical
problem.
00:13:28.000 --> 00:13:34.000
And I ask, is there some system
of variables,
00:13:31.000 --> 00:13:37.000
which somehow go more deeply
into the structure of what's
00:13:36.000 --> 00:13:42.000
going on here than the naīve
variables, which simply tell me
00:13:41.000 --> 00:13:47.000
how high the two tank levels
are?
00:13:44.000 --> 00:13:50.000
That is the obvious thing I can
see, but there are some
00:13:48.000 --> 00:13:54.000
variables that go more deeply.
Now, one of them is sort of
00:13:53.000 --> 00:13:59.000
obvious and suggested both the
form of the equation and by
00:13:58.000 --> 00:14:04.000
this.
Simply, the difference in
00:14:01.000 --> 00:14:07.000
heights is, in some ways,
a more natural variable because
00:14:05.000 --> 00:14:11.000
that is directly related to the
pressure difference,
00:14:09.000 --> 00:14:15.000
which is directly related to
the velocity of flow.
00:14:12.000 --> 00:14:18.000
They will differ by just
constant.
00:14:14.000 --> 00:14:20.000
I am going to call that the
second variable,
00:14:17.000 --> 00:14:23.000
or the difference in height
let's call it.
00:14:20.000 --> 00:14:26.000
That's x minus y.
That is a very natural variable
00:14:23.000 --> 00:14:29.000
for the problem.
The question is,
00:14:25.000 --> 00:14:31.000
what should the other one be?
Now you sort of stare at that
00:14:31.000 --> 00:14:37.000
for a while until it occurs to
you that something is constant.
00:14:35.000 --> 00:14:41.000
What is constant in this
problem?
00:14:38.000 --> 00:14:44.000
Well, the tank is sitting
there, that is constant.
00:14:42.000 --> 00:14:48.000
But what thing,
which might be a variable,
00:14:45.000 --> 00:14:51.000
clearly must be a constant?
It will be the total amount of
00:14:49.000 --> 00:14:55.000
water in the two tanks.
These things vary,
00:14:52.000 --> 00:14:58.000
but the total amount of water
stays the same because it is a
00:14:57.000 --> 00:15:03.000
homogenous problem.
No water is coming in from the
00:15:02.000 --> 00:15:08.000
outside, and none is leaving the
tanks through a little hole.
00:15:08.000 --> 00:15:14.000
Okay.
What is the expression for the
00:15:11.000 --> 00:15:17.000
total amount of water in the
tanks?
00:15:14.000 --> 00:15:20.000
x plus 2y.
Therefore, that is a natural
00:15:19.000 --> 00:15:25.000
variable also.
It is independent of this one.
00:15:23.000 --> 00:15:29.000
It is not a simple multiply of
it.
00:15:26.000 --> 00:15:32.000
It is a really different
variable.
00:15:31.000 --> 00:15:37.000
This variable represents the
total amount of liquid in the
00:15:36.000 --> 00:15:42.000
two tanks.
This represents the pressure up
00:15:40.000 --> 00:15:46.000
to a constant factor.
It is proportional to the
00:15:45.000 --> 00:15:51.000
pressure at the hole.
Okay.
00:15:48.000 --> 00:15:54.000
Now what I am going to do is
say this is my change of
00:15:53.000 --> 00:15:59.000
variable.
Now let's plug in and see what
00:15:57.000 --> 00:16:03.000
happens to the system when I
plug in these two variables.
00:16:04.000 --> 00:16:10.000
And how do I do that?
Well, I want to substitute and
00:16:09.000 --> 00:16:15.000
get the new system.
The new system,
00:16:13.000 --> 00:16:19.000
or rather the old system,
but what makes it new is in
00:16:19.000 --> 00:16:25.000
terms of u and v.
What will that be?
00:16:22.000 --> 00:16:28.000
Well, u prime is x prime plus
2y prime.
00:16:30.000 --> 00:16:36.000
But I know what x prime plus 2y
prime is because I
00:16:35.000 --> 00:16:41.000
can calculate it for this.
What will it be?
00:16:40.000 --> 00:16:46.000
x prime plus 2y prime is
negative 2x plus twice y prime,
00:16:45.000 --> 00:16:51.000
so it's plus 2x,
which is zero.
00:16:48.000 --> 00:16:54.000
And how about these two?
2y minus twice this,
00:16:52.000 --> 00:16:58.000
because I want this plus twice
that, so it 2y minus 2y,
00:16:57.000 --> 00:17:03.000
again, zero.
The right-hand side becomes
00:17:02.000 --> 00:17:08.000
zero after I calculate x prime
plus 2y.
00:17:06.000 --> 00:17:12.000
So that is zero.
That would just,
00:17:09.000 --> 00:17:15.000
of course, clear.
Now, that makes sense,
00:17:12.000 --> 00:17:18.000
of course.
Since the total amount is
00:17:15.000 --> 00:17:21.000
constant, that says that u prime
is zero.
00:17:19.000 --> 00:17:25.000
Okay.
What is v prime?
00:17:21.000 --> 00:17:27.000
v prime is x prime minus y
prime.
00:17:25.000 --> 00:17:31.000
What is that?
Well, once again we have to
00:17:31.000 --> 00:17:37.000
calculate.
x prime minus y prime is minus
00:17:36.000 --> 00:17:42.000
2x minus x, which is minus 3x,
and 2y minus negative y,
00:17:43.000 --> 00:17:49.000
which makes plus 3y.
All right.
00:17:46.000 --> 00:17:52.000
What is the system?
The system is u prime equals
00:17:52.000 --> 00:17:58.000
zero and v prime
equals minus three times x minus
00:18:00.000 --> 00:18:06.000
y. But x minus y is v.
00:18:07.000 --> 00:18:13.000
In other words,
00:18:09.000 --> 00:18:15.000
these new two variables
decouple the system.
00:18:14.000 --> 00:18:20.000
And we got them,
as scientists often do,
00:18:18.000 --> 00:18:24.000
by physical considerations.
These variables go more deeply
00:18:24.000 --> 00:18:30.000
into what is going on in that
system of two tanks than simply
00:18:31.000 --> 00:18:37.000
the two heights,
which are too obvious as
00:18:35.000 --> 00:18:41.000
variables.
All right.
00:18:38.000 --> 00:18:44.000
What is the solution?
Well, the solution is,
00:18:42.000 --> 00:18:48.000
u equals a constant and v is
equal to?
00:18:45.000 --> 00:18:51.000
Well, the solution to this
equation is a different
00:18:50.000 --> 00:18:56.000
arbitrary constant from that
one.
00:18:53.000 --> 00:18:59.000
These are side-by-side
equations that have nothing
00:18:57.000 --> 00:19:03.000
whatever to do with each other,
remember?
00:19:02.000 --> 00:19:08.000
Times e to the minus 3t.
00:19:05.000 --> 00:19:11.000
Now, there are two options.
Either one leaves the solution
00:19:10.000 --> 00:19:16.000
in terms of those new variables,
saying they are more natural to
00:19:15.000 --> 00:19:21.000
the problem, but sometimes,
of course, one wants the answer
00:19:20.000 --> 00:19:26.000
in terms of the old one.
But, if you do that,
00:19:24.000 --> 00:19:30.000
then you have to solve that.
In order to save a little time,
00:19:29.000 --> 00:19:35.000
since this is purely linear
algebra, I am going to write --
00:19:36.000 --> 00:19:42.000
Instead of taking two minutes
to actually do the calculation
00:19:39.000 --> 00:19:45.000
in front of you,
I will just write down what the
00:19:42.000 --> 00:19:48.000
answer is --
00:19:53.000 --> 00:19:59.000
-- in terms of u and x and y.
In other words,
00:19:56.000 --> 00:20:02.000
this is a perfectly good way to
leave the answer if you are
00:20:02.000 --> 00:20:08.000
allowed to do it.
But if somebody says they want
00:20:06.000 --> 00:20:12.000
the answer in terms of x and y,
well, you have to give them
00:20:10.000 --> 00:20:16.000
what they are paying for.
In terms of x and y,
00:20:13.000 --> 00:20:19.000
you have first to solve those
equations backwards for x and y
00:20:17.000 --> 00:20:23.000
in terms of u and v in which
case you will get x equals
00:20:21.000 --> 00:20:27.000
one-third of u plus 2v.
00:20:24.000 --> 00:20:30.000
Use the inverse matrix or just
do elimination,
00:20:27.000 --> 00:20:33.000
whatever you usually like to
do.
00:20:31.000 --> 00:20:37.000
And the other one will be
one-third of u minus v.
00:20:41.000 --> 00:20:47.000
And then, if you substitute in,
00:20:47.000 --> 00:20:53.000
you will see what you will get
is one-third of c1.
00:20:56.000 --> 00:21:02.000
Sorry.
u is c1.
00:21:15.000 --> 00:21:21.000
c1 plus 2 c2 e to the negative
3t.
00:21:19.000 --> 00:21:25.000
And this is one-third of c1
minus c2 e to the minus 3t.
00:21:24.000 --> 00:21:30.000
And so, the final solution is,
00:21:28.000 --> 00:21:34.000
in terms of the way we usually
write out the answer,
00:21:33.000 --> 00:21:39.000
x will be what?
Well, it will be one-third c1
00:21:38.000 --> 00:21:44.000
times the eigenvector one,
one plus one-third times c2
00:21:43.000 --> 00:21:49.000
times the eigenvector two,
negative one times e to the
00:21:49.000 --> 00:21:55.000
minus 3t.
That is the solution written
00:21:54.000 --> 00:22:00.000
out in terms of x and y either
as a vector in the usual way or
00:22:00.000 --> 00:22:06.000
separately in terms of x and y.
But, notice,
00:22:05.000 --> 00:22:11.000
in order to do that you have to
have these backwards equations.
00:22:10.000 --> 00:22:16.000
In other words,
I need the equations in that
00:22:13.000 --> 00:22:19.000
form.
I need the equations because
00:22:16.000 --> 00:22:22.000
they tell me what the new
variables are.
00:22:19.000 --> 00:22:25.000
But I also have to have the
equations the other way in order
00:22:24.000 --> 00:22:30.000
to get the solution in terms of
x and y, finally.
00:22:27.000 --> 00:22:33.000
Okay.
That was all an example.
00:22:31.000 --> 00:22:37.000
For the rest of the period,
I would like to show you the
00:22:35.000 --> 00:22:41.000
general method of doing the same
thing which does not depend upon
00:22:40.000 --> 00:22:46.000
being clever about the choice of
the new variables.
00:22:43.000 --> 00:22:49.000
And then, at the very end of
the period, I will apply the
00:22:48.000 --> 00:22:54.000
general method to this problem
to see whether we get the same
00:22:52.000 --> 00:22:58.000
answer or not.
What is the general method?
00:22:55.000 --> 00:23:01.000
Our problem is the decouple.
Now, the first thing is you
00:23:00.000 --> 00:23:06.000
cannot always decouple.
To decouple the eigenvalues
00:23:05.000 --> 00:23:11.000
must all be real and
non-defective.
00:23:09.000 --> 00:23:15.000
In other words,
if they are repeated they must
00:23:13.000 --> 00:23:19.000
be complete.
You must have enough
00:23:16.000 --> 00:23:22.000
independent eigenvectors.
So they must be real and
00:23:21.000 --> 00:23:27.000
complete.
If repeated,
00:23:23.000 --> 00:23:29.000
they must be complete.
They must not be defective.
00:23:30.000 --> 00:23:36.000
As I told you at the time when
we studied complete and
00:23:34.000 --> 00:23:40.000
incomplete, the most common case
in which this occurs is when the
00:23:40.000 --> 00:23:46.000
matrix is symmetric.
If the matrix is real and
00:23:44.000 --> 00:23:50.000
symmetric then you can always
decouple the system.
00:23:49.000 --> 00:23:55.000
That is a very important
theorem, particularly since many
00:23:54.000 --> 00:24:00.000
of the equilibrium problems
normally lead to symmetric
00:23:59.000 --> 00:24:05.000
matrices and are solved by
decoupling.
00:24:04.000 --> 00:24:10.000
Okay.
So what are we looking for?
00:24:17.000 --> 00:24:23.000
We are assuming this and we
need it.
00:24:19.000 --> 00:24:25.000
In general, otherwise,
you cannot decouple if you have
00:24:23.000 --> 00:24:29.000
complex eigenvalues and you
cannot decouple if you have
00:24:27.000 --> 00:24:33.000
defective eigenvalues.
00:24:34.000 --> 00:24:40.000
Well, what are we looking for?
We are looking for new
00:24:39.000 --> 00:24:45.000
variables.
u, v equals a1,
00:24:42.000 --> 00:24:48.000
b1, a2, b2 times the x,
y.
00:24:49.000 --> 00:24:55.000
And this matrix is called D,
the decoupling matrix and is
00:24:56.000 --> 00:25:02.000
what we are looking for.
How do I choose those new
00:25:01.000 --> 00:25:07.000
variables u and v when I don't
have any physical considerations
00:25:06.000 --> 00:25:12.000
to guide me as I did before?
Now, the key is to look instead
00:25:11.000 --> 00:25:17.000
at what you are going to need.
Remember, we are changing
00:25:15.000 --> 00:25:21.000
variables.
And, as I told you from the
00:25:18.000 --> 00:25:24.000
first days of the term,
when you change variables look
00:25:22.000 --> 00:25:28.000
at what you are going to need to
substitute in to make the change
00:25:27.000 --> 00:25:33.000
of variables.
Don't just start writing
00:25:32.000 --> 00:25:38.000
equations.
What we are going to need to
00:25:35.000 --> 00:25:41.000
plug into that system and change
it to the (u,
00:25:39.000 --> 00:25:45.000
v) coordinates is not u and v
in terms of x and y.
00:25:44.000 --> 00:25:50.000
What we need is x and y in
terms of u and v to do the
00:25:49.000 --> 00:25:55.000
substitution.
What we need is the inverse of
00:25:52.000 --> 00:25:58.000
this.
So, in order to do the
00:25:55.000 --> 00:26:01.000
substitution,
what we need is (x,
00:25:58.000 --> 00:26:04.000
y).
Oops.
00:26:00.000 --> 00:26:06.000
Let's call them prime.
Let's call these a1,
00:26:04.000 --> 00:26:10.000
b1, a2, b2 because these are
going to be much more important
00:26:09.000 --> 00:26:15.000
to the problem than the other
ones.
00:26:12.000 --> 00:26:18.000
Okay.
I am going to,
00:26:13.000 --> 00:26:19.000
I should call this matrix D
inverse, that would be a
00:26:18.000 --> 00:26:24.000
sensible thing to call it.
Since this is the important
00:26:22.000 --> 00:26:28.000
matrix, this is the one we are
going to need to do the
00:26:27.000 --> 00:26:33.000
substitution,
I am going to give it another
00:26:31.000 --> 00:26:37.000
letter instead.
And the letter that comes after
00:26:37.000 --> 00:26:43.000
D is E.
Now, E is an excellent choice
00:26:40.000 --> 00:26:46.000
because it is also the first
letter of the word eigenvector.
00:26:46.000 --> 00:26:52.000
And the point is the matrix E,
which is going to work,
00:26:51.000 --> 00:26:57.000
is the matrix whose columns are
the two eigenvectors.
00:27:08.000 --> 00:27:14.000
The columns are the two
eigenvectors.
00:27:11.000 --> 00:27:17.000
Now, even if you didn't know
anything that would be
00:27:16.000 --> 00:27:22.000
practically the only reasonable
choice anybody could make.
00:27:21.000 --> 00:27:27.000
What are we looking for?
To make a linear change of
00:27:26.000 --> 00:27:32.000
variables like this really means
to pick new i and j vectors.
00:27:33.000 --> 00:27:39.000
You know, from the first days
of 18.02, what you want is a new
00:27:37.000 --> 00:27:43.000
coordinate system in the plane.
And the coordinate system in
00:27:41.000 --> 00:27:47.000
the plane is determined as soon
as you tell what the new i is
00:27:46.000 --> 00:27:52.000
and what the new j is in the new
system.
00:27:48.000 --> 00:27:54.000
To establish a linear change of
coordinates amounts to picking
00:27:53.000 --> 00:27:59.000
two new vectors that are going
to play the role of i and j
00:27:57.000 --> 00:28:03.000
instead of the old i and the old
j.
00:28:01.000 --> 00:28:07.000
Okay, so pick two vectors which
somehow are important to this
00:28:07.000 --> 00:28:13.000
matrix.
Well, there are only two,
00:28:11.000 --> 00:28:17.000
the eigenvectors.
What else could they possibly
00:28:17.000 --> 00:28:23.000
be? Now, what is the relation?
00:28:20.000 --> 00:28:26.000
I say with this,
what happens is I say that
00:28:25.000 --> 00:28:31.000
alpha one corresponds,
and alpha two,
00:28:29.000 --> 00:28:35.000
these are vectors in the
xy-system.
00:28:35.000 --> 00:28:41.000
Well, if I change the
coordinates to u and v,
00:28:38.000 --> 00:28:44.000
in the uv-system they will
correspond to the vectors one,
00:28:42.000 --> 00:28:48.000
zero. In other words,
00:28:45.000 --> 00:28:51.000
the vector that we would
normally call i in the u,
00:28:48.000 --> 00:28:54.000
v system.
And this one will correspond to
00:28:52.000 --> 00:28:58.000
the vector zero, one.
00:28:54.000 --> 00:29:00.000
Now, if you don't believe that
I will calculate it for you.
00:29:00.000 --> 00:29:06.000
The calculation is trivial.
Look.
00:29:04.000 --> 00:29:10.000
What have we got?
(x, y) equals a1,
00:29:09.000 --> 00:29:15.000
b1, a2, b2.
00:29:13.000 --> 00:29:19.000
This is the column vector alpha
one.
00:29:18.000 --> 00:29:24.000
This is the column vector alpha
two.
00:29:22.000 --> 00:29:28.000
Now, here is u and v.
Suppose I make u and v equal to
00:29:30.000 --> 00:29:36.000
one, zero, what happens to x and
y?
00:29:35.000 --> 00:29:41.000
Your matrix multiply.
One, zero.
00:29:38.000 --> 00:29:44.000
So a1 plus zero,
b1 plus zero.
00:29:45.000 --> 00:29:51.000
It corresponds to the column
vector (a1, b1).
00:29:50.000 --> 00:29:56.000
And in the same way zero,
one corresponds to
00:29:57.000 --> 00:30:03.000
(a2, b2).
00:30:05.000 --> 00:30:11.000
Just by matrix multiplication.
And that shows that these
00:30:12.000 --> 00:30:18.000
correspond.
In the uv-system the two
00:30:16.000 --> 00:30:22.000
eigenvectors are now called i
and j.
00:30:21.000 --> 00:30:27.000
Well, that looks very
promising, but the program now
00:30:27.000 --> 00:30:33.000
is to do the substitution to
substitute into the system x
00:30:34.000 --> 00:30:40.000
prime equals Ax and
see if it is decoupled in the
00:30:42.000 --> 00:30:48.000
uv-coordinates.
Now, I don't dare let you do
00:30:47.000 --> 00:30:53.000
this by yourself because you
will run into trouble.
00:30:51.000 --> 00:30:57.000
Nothing is going to happen.
You will just get a mess and
00:30:54.000 --> 00:31:00.000
will say I must be missing
something.
00:30:57.000 --> 00:31:03.000
And that is because you are
missing something.
00:31:01.000 --> 00:31:07.000
What you are missing,
and this is a good occasion to
00:31:05.000 --> 00:31:11.000
tell you, is that,
in general, three-quarters of
00:31:09.000 --> 00:31:15.000
the civilized world does not
introduce eigenvalues and
00:31:14.000 --> 00:31:20.000
eigenvectors the way you learn
them in 18.03.
00:31:18.000 --> 00:31:24.000
They use a different definition
that is identical.
00:31:22.000 --> 00:31:28.000
I mean it is equivalent.
The concept is the same,
00:31:27.000 --> 00:31:33.000
but it looks a little
different.
00:31:31.000 --> 00:31:37.000
Our definition is what?
Well, what is an eigenvalue and
00:31:35.000 --> 00:31:41.000
eigenvector?
The basic thing is this
00:31:37.000 --> 00:31:43.000
equation.
00:31:46.000 --> 00:31:52.000
This is a two-by-two matrix,
right?
00:31:48.000 --> 00:31:54.000
This is a column vector with
two entries.
00:31:51.000 --> 00:31:57.000
The product has to be a column
vector with two entries,
00:31:55.000 --> 00:32:01.000
but both entries are supposed
to be zero so I will write it
00:31:59.000 --> 00:32:05.000
this way.
This way first defines what an
00:32:03.000 --> 00:32:09.000
eigenvalue is.
It is something that makes the
00:32:06.000 --> 00:32:12.000
determinant zero.
And then it defines what an
00:32:09.000 --> 00:32:15.000
eigenvector is.
It is, then,
00:32:11.000 --> 00:32:17.000
a solution to the system that
you can get because the
00:32:15.000 --> 00:32:21.000
determinant is zero.
Now, that is not what most
00:32:19.000 --> 00:32:25.000
people do.
What most people do is the
00:32:21.000 --> 00:32:27.000
following.
They write this equation
00:32:24.000 --> 00:32:30.000
differently by having something
on both sides.
00:32:29.000 --> 00:32:35.000
Using the distributive law,
what goes on the left side is A
00:32:33.000 --> 00:32:39.000
alpha one.
What is that?
00:32:35.000 --> 00:32:41.000
That is a column vector with
two entries.
00:32:38.000 --> 00:32:44.000
What goes on the right?
Well, lambda one times the
00:32:42.000 --> 00:32:48.000
identity times alpha one.
Now, the identity matrix times
00:32:47.000 --> 00:32:53.000
anything just reproduces what
was there.
00:32:50.000 --> 00:32:56.000
There is no difference between
writing the identity times alpha
00:32:55.000 --> 00:33:01.000
one and just alpha one all by
itself.
00:33:00.000 --> 00:33:06.000
So that is what I am going to
do.
00:33:02.000 --> 00:33:08.000
This is the definition of
eigenvalue and eigenvector that
00:33:07.000 --> 00:33:13.000
all the other people use.
Most linear algebra books use
00:33:12.000 --> 00:33:18.000
this definition,
or most books use a different
00:33:16.000 --> 00:33:22.000
approach and say,
here is an eigenvalue and an
00:33:20.000 --> 00:33:26.000
eigenvector.
And it requires them to define
00:33:23.000 --> 00:33:29.000
them in the opposite order.
First what alpha one is and
00:33:28.000 --> 00:33:34.000
then what lambda one is.
See, I don't have any
00:33:33.000 --> 00:33:39.000
determinant now.
So what is the definition?
00:33:36.000 --> 00:33:42.000
And they like it because it has
a certain geometric flavor that
00:33:40.000 --> 00:33:46.000
this one lacks entirely.
This is good for solving
00:33:43.000 --> 00:33:49.000
differential equations,
which is why we are using it in
00:33:47.000 --> 00:33:53.000
18.03, but this has a certain
geometric content.
00:33:50.000 --> 00:33:56.000
This way thinks of A as a
linear transformation of the
00:33:54.000 --> 00:34:00.000
plane, a shearing of the plane.
You take the plane and do
00:33:57.000 --> 00:34:03.000
something to it.
Or, you squish it like that.
00:34:02.000 --> 00:34:08.000
Or, you rotate it.
That's okay,
00:34:04.000 --> 00:34:10.000
too.
And the matrix defines a linear
00:34:07.000 --> 00:34:13.000
transformation to the plane,
every vector goes to another
00:34:11.000 --> 00:34:17.000
vector.
The question it asks is,
00:34:14.000 --> 00:34:20.000
is there a vector which is
taken by this linear
00:34:18.000 --> 00:34:24.000
transformation and just left
alone or stretched,
00:34:21.000 --> 00:34:27.000
is kept in the same direction
but stretched?
00:34:25.000 --> 00:34:31.000
Or, maybe its direction is
reversed and it is stretched or
00:34:29.000 --> 00:34:35.000
it shrunk.
But, in general,
00:34:33.000 --> 00:34:39.000
if there are real eigenvalues
there will be such vectors that
00:34:38.000 --> 00:34:44.000
are just left in the same
direction but just stretched or
00:34:43.000 --> 00:34:49.000
shrunk.
And what is the lambda?
00:34:46.000 --> 00:34:52.000
The lambda then is the amount
by which they are stretched or
00:34:51.000 --> 00:34:57.000
shrunk, the factor.
This way, first we have to find
00:34:55.000 --> 00:35:01.000
the vector, which is left
essentially unchanged,
00:34:59.000 --> 00:35:05.000
and then the number here that
goes with it is the stretching
00:35:04.000 --> 00:35:10.000
factor or the shrinking factor.
But the end result is the pair
00:35:11.000 --> 00:35:17.000
alpha one and lambda one,
regardless of which order you
00:35:15.000 --> 00:35:21.000
find them, satisfied the same
equation.
00:35:18.000 --> 00:35:24.000
Now, a consequence of this
definition we are going to need
00:35:22.000 --> 00:35:28.000
in the calculation that I am
going to do in just a moment.
00:35:26.000 --> 00:35:32.000
Let me calculate that out.
What I want to do is calculate
00:35:31.000 --> 00:35:37.000
the matrix A times E.
I am going to need to calculate
00:35:36.000 --> 00:35:42.000
that.
Now, what is that?
00:35:38.000 --> 00:35:44.000
Remember, E is the matrix whose
columns are the eigenvectors.
00:35:43.000 --> 00:35:49.000
That is the matrix alpha one,
alpha two.
00:35:47.000 --> 00:35:53.000
Now, what is this?
Well, in both Friday's lecture
00:35:51.000 --> 00:35:57.000
and Monday's lecture,
I used the fact that if you do
00:35:55.000 --> 00:36:01.000
a multiplication like that it is
the same thing as doing the
00:36:01.000 --> 00:36:07.000
multiplication A alpha one and
putting it in the first column.
00:36:08.000 --> 00:36:14.000
And then A alpha two is the
column vector that goes in the
00:36:13.000 --> 00:36:19.000
second column.
But what is this?
00:36:16.000 --> 00:36:22.000
This is lambda one alpha one.
And this is lambda two alpha
00:36:22.000 --> 00:36:28.000
two by this other definition of
eigenvalue and eigenvector.
00:36:28.000 --> 00:36:34.000
And what is this?
Can I write this in terms of
00:36:34.000 --> 00:36:40.000
matrices?
Yes indeed I can.
00:36:36.000 --> 00:36:42.000
This is the matrix alpha one,
alpha two times this matrix
00:36:42.000 --> 00:36:48.000
lambda one, lambda two,
zero, zero.
00:36:46.000 --> 00:36:52.000
Check it out.
Lambda one plus zero,
00:36:50.000 --> 00:36:56.000
lambda one times this thing
plus zero, the first entry is
00:36:56.000 --> 00:37:02.000
exactly that.
And the same way the second
00:37:01.000 --> 00:37:07.000
column doing the same
calculation is exactly this.
00:37:06.000 --> 00:37:12.000
What is that?
That is e times this matrix
00:37:10.000 --> 00:37:16.000
lambda one, zero,
zero, lambda two.
00:37:14.000 --> 00:37:20.000
Okay.
We are almost finished now.
00:37:17.000 --> 00:37:23.000
Now we can carry out our work.
We are going to do the
00:37:22.000 --> 00:37:28.000
substitution.
I start with a system.
00:37:26.000 --> 00:37:32.000
Remember where we are.
I am starting with this system.
00:37:33.000 --> 00:37:39.000
I am going to make the
substitution x equal to this
00:37:39.000 --> 00:37:45.000
matrix E, whose columns are the
eigenvectors.
00:37:45.000 --> 00:37:51.000
I am in introducing,
in other words,
00:37:49.000 --> 00:37:55.000
new variables u and v according
to that thing.
00:37:55.000 --> 00:38:01.000
u is the column vector,
u and v.
00:38:00.000 --> 00:38:06.000
And x, as usual,
00:38:03.000 --> 00:38:09.000
is the column vector x and y.
So I am going to plug it in.
00:38:08.000 --> 00:38:14.000
Okay.
Let's plug it in.
00:38:10.000 --> 00:38:16.000
What do I get?
I take the derivative.
00:38:13.000 --> 00:38:19.000
E is a constant matrix so that
makes E times u prime,
00:38:17.000 --> 00:38:23.000
is equal to A times,
x is E times u again.
00:38:21.000 --> 00:38:27.000
Now, at this point,
00:38:24.000 --> 00:38:30.000
you would be stuck,
except I calculated for you A
00:38:28.000 --> 00:38:34.000
times E is E times that funny
diagonal matrix with the
00:38:32.000 --> 00:38:38.000
lambdas.
So this is E times that funny
00:38:38.000 --> 00:38:44.000
matrix of the lambdas,
the eigenvalues,
00:38:41.000 --> 00:38:47.000
and still the u at the end of
it.
00:38:45.000 --> 00:38:51.000
So where are we?
E times u prime equals E times
00:38:50.000 --> 00:38:56.000
this thing.
Well, multiply both sides by E
00:38:54.000 --> 00:39:00.000
inverse and you can cancel them
out.
00:38:59.000 --> 00:39:05.000
And so the end result is that
after you have made the
00:39:04.000 --> 00:39:10.000
substitution in terms of the new
variables u, what you get is u
00:39:10.000 --> 00:39:16.000
prime equals lambda one,
lambda two, zero,
00:39:14.000 --> 00:39:20.000
zero times u.
Let's write that out in terms
00:39:18.000 --> 00:39:24.000
of a system.
This is u prime is equal to,
00:39:22.000 --> 00:39:28.000
well, this is u,
v here.
00:39:24.000 --> 00:39:30.000
It is lambda one times u plus
zero times v.
00:39:30.000 --> 00:39:36.000
And the other one is v prime
equals zero times u plus lambda
00:39:40.000 --> 00:39:46.000
two times v.
We are decoupled.
00:39:45.000 --> 00:39:51.000
In just one sentence you would
say --
00:39:52.000 --> 00:39:58.000
In other words,
if you were reading a book that
00:39:55.000 --> 00:40:01.000
sort of assumed you knew what
was going on,
00:39:59.000 --> 00:40:05.000
all it would say is as usual.
That is to make you feel bad.
00:40:04.000 --> 00:40:10.000
Or, as is well-known to make
you feel even worse.
00:40:08.000 --> 00:40:14.000
Or, the system is decoupled by
choosing as the new basis for
00:40:13.000 --> 00:40:19.000
the system the eigenvectors of
the matrix and in terms of the
00:40:18.000 --> 00:40:24.000
resulting new coordinates,
the decoupled system will be
00:40:23.000 --> 00:40:29.000
the following where the
constants are the eigenvalues.
00:40:29.000 --> 00:40:35.000
And so the solution will be u
equals c1 times e to the lambda1
00:40:33.000 --> 00:40:39.000
t and v
is equal to c2 times e to the
00:40:38.000 --> 00:40:44.000
lambda2 t.
00:40:40.000 --> 00:40:46.000
Of course, if you want it back
in terms now of x and y,
00:40:44.000 --> 00:40:50.000
you will have to go back to
here, to these equations and
00:40:48.000 --> 00:40:54.000
then plug in for u and v what
they are.
00:40:51.000 --> 00:40:57.000
And then you will get the
answer in terms of x and y.
00:40:55.000 --> 00:41:01.000
Okay.
We have just enough time to
00:40:59.000 --> 00:41:05.000
actually carry out this little
program.
00:41:04.000 --> 00:41:10.000
It takes a lot longer to derive
than it does actually to do,
00:41:10.000 --> 00:41:16.000
so let's do it for this system
that we were talking about
00:41:16.000 --> 00:41:22.000
before.
Decouple the system,
00:41:19.000 --> 00:41:25.000
x, y prime equals the matrixes
negative two,
00:41:24.000 --> 00:41:30.000
two, one, negative one.
00:41:36.000 --> 00:41:42.000
Okay.
What do I do?
00:41:37.000 --> 00:41:43.000
Well, I first have to calculate
the eigenvalues in the
00:41:42.000 --> 00:41:48.000
eigenvectors,
so the Ev's and Ev's.
00:41:45.000 --> 00:41:51.000
The characteristic equation is
lambda squared.
00:41:49.000 --> 00:41:55.000
The trace is negative three,
but you have to change the
00:41:54.000 --> 00:42:00.000
sign.
The determinant is two minus
00:41:57.000 --> 00:42:03.000
two, so that is zero.
There is no constant term here.
00:42:02.000 --> 00:42:08.000
It is zero.
That is the characteristic
00:42:05.000 --> 00:42:11.000
equation.
The roots are obviously lambda
00:42:08.000 --> 00:42:14.000
equals zero, lambda equals
negative three.
00:42:12.000 --> 00:42:18.000
And what are the eigenvectors
that go with that?
00:42:15.000 --> 00:42:21.000
With lambda equals zero goes
the eigenvector,
00:42:19.000 --> 00:42:25.000
minus two.
Well, I subtract zero here,
00:42:22.000 --> 00:42:28.000
so the equation I have to solve
is minus 2 a1 plus --
00:42:28.000 --> 00:42:34.000
I am not going to write 2 a2,
which is what you have been
00:42:32.000 --> 00:42:38.000
writing up until now.
The reason is because I ran
00:42:36.000 --> 00:42:42.000
into trouble with the notation
and I had to use,
00:42:40.000 --> 00:42:46.000
as the eigenvector,
not a1, a2 but a1,
00:42:43.000 --> 00:42:49.000
b1.
So it should be a b1 here,
00:42:46.000 --> 00:42:52.000
not the a2 that you are used
to.
00:42:48.000 --> 00:42:54.000
The solution,
therefore, is alpha one equals
00:42:52.000 --> 00:42:58.000
one, one.
And for lambda equals negative
00:42:55.000 --> 00:43:01.000
three, the corresponding
eigenvector this time will be --
00:43:02.000 --> 00:43:08.000
Now I have to subtract negative
three from here,
00:43:06.000 --> 00:43:12.000
so negative two minus negative
three makes one.
00:43:10.000 --> 00:43:16.000
That is a1 plus 2 b1 equals
zero,
00:43:15.000 --> 00:43:21.000
a logical choice for the
eigenvector here.
00:43:19.000 --> 00:43:25.000
The second eigenvector would be
make b1 equal to one,
00:43:24.000 --> 00:43:30.000
let's say, and then a1 will be
negative two.
00:43:28.000 --> 00:43:34.000
Okay.
Now what do we have to do?
00:43:32.000 --> 00:43:38.000
Now, what we want is the matrix
E.
00:43:35.000 --> 00:43:41.000
The matrix E is the matrix of
eigenvectors,
00:43:38.000 --> 00:43:44.000
so it is the matrix one,
one, negative two,
00:43:42.000 --> 00:43:48.000
one.
00:43:44.000 --> 00:43:50.000
The next thing we want is what
the new variables u and v are.
00:43:50.000 --> 00:43:56.000
For that, we will need E
inverse.
00:43:52.000 --> 00:43:58.000
How do you calculate the
inverse of a two-by-two matrix?
00:43:59.000 --> 00:44:05.000
You switch the two diagonal
elements, there I have switched
00:44:04.000 --> 00:44:10.000
them, and you leave the other
two where they are but change
00:44:09.000 --> 00:44:15.000
their sign.
So it is two up here and
00:44:13.000 --> 00:44:19.000
negative one there.
Maybe I should make this one
00:44:17.000 --> 00:44:23.000
purple and then that one purple
to indicate that I have switched
00:44:23.000 --> 00:44:29.000
them.
I am not done yet.
00:44:25.000 --> 00:44:31.000
I have to divide by the
determinant.
00:44:30.000 --> 00:44:36.000
What is the determinant?
It is one minus negative two,
00:44:36.000 --> 00:44:42.000
which is three,
so I have to divide by three.
00:44:42.000 --> 00:44:48.000
I multiply everything here by
one-third.
00:44:47.000 --> 00:44:53.000
Okay.
And what is the decoupled
00:44:51.000 --> 00:44:57.000
system?
The new variables are u equals
00:44:56.000 --> 00:45:02.000
one-third.
00:45:08.000 --> 00:45:14.000
In other words,
the new variables are given by
00:45:11.000 --> 00:45:17.000
D.
It is u, v equals one,
00:45:13.000 --> 00:45:19.000
two, negative one, one
times one-third times x,y.
00:45:19.000 --> 00:45:25.000
That is the expression for u,
00:45:21.000 --> 00:45:27.000
v in terms of x and y.
It's this matrix D,
00:45:25.000 --> 00:45:31.000
the decoupling matrix which is
the one that is used.
00:45:30.000 --> 00:45:36.000
And that gives this system u
equals one-third of x plus 2y
00:45:37.000 --> 00:45:43.000
on top.
And what is the v entry?
00:45:44.000 --> 00:45:50.000
v is one-third of minus x plus
y.
00:45:51.000 --> 00:45:57.000
Now, are those the same
variables that I used before?
00:46:08.000 --> 00:46:14.000
Yes.
This is my new and better you,
00:46:10.000 --> 00:46:16.000
the one I got by just blindly
following the method instead of
00:46:14.000 --> 00:46:20.000
looking for physical things with
physical meaning.
00:46:17.000 --> 00:46:23.000
It differs from the old one
just by a constant factor.
00:46:21.000 --> 00:46:27.000
Now, that doesn't have any
effect on the resulting equation
00:46:25.000 --> 00:46:31.000
because if the old one is u
prime equals zero the new one is
00:46:29.000 --> 00:46:35.000
one-third u prime equals zero.
It is still the same equation,
00:46:34.000 --> 00:46:40.000
in other words.
And how about this one?
00:46:36.000 --> 00:46:42.000
This one differs from the other
one by the factor minus
00:46:40.000 --> 00:46:46.000
one-third.
If I multiply that v through by
00:46:43.000 --> 00:46:49.000
minus one-third,
I get this v.
00:46:45.000 --> 00:46:51.000
And, therefore,
that too does not affect the
00:46:47.000 --> 00:46:53.000
second equation.
I simply multiply both sides by
00:46:51.000 --> 00:46:57.000
minus one-third.
The new v still satisfies the
00:46:54.000 --> 00:47:00.000
equation minus three times v.