WEBVTT
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-- and lift-off on
differential equations.
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So, this section is
about how to solve
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a system of first order,
first derivative, constant
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coefficient linear equations.
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And if we do it right, it turns
directly into linear algebra.
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The key idea is the solutions
to constant coefficient
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linear equations
are exponentials.
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So if you look for
an exponential,
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then all you have to find is
what's up there in the exponent
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and what multiplies
the exponential
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and that's the linear algebra.
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So -- and the result --
one thing we will fine --
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it's completely parallel
to powers of a matrix.
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So the last lecture was
about how would you compute
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A to the K or A to the 100?
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How do you compute high
powers of a matrix?
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Now it's not powers anymore,
but it's exponentials.
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That's the natural thing
for differential equation.
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Okay.
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But can I begin with an example?
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And I'll just go
through the mechanics.
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How would I solve
the differential --
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two differential equations?
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So I'm going to make it --
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I'll have a two by two
matrix and the coefficients
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are minus one two, one minus
two and I'd better give you
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some initial condition.
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So suppose it starts u at
times zero -- this is u1, u2 --
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let it -- let it --
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suppose everything is
in u1 at times zero.
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So -- at -- at the
start, it's all in u1.
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But what happens as time
goes on, du2/dt will --
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will be positive,
because of that u1 term,
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so flow will move into the u2
component and it will go out
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of the u1 component.
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So we'll just follow
that movement as time
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goes forward by looking at the
eigenvalues and eigenvectors
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of that matrix.
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That's a first job.
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Before you do anything
else, find the --
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find the matrix and its
eigenvalues and eigenvectors.
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So let me do that.
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Okay.
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So here's our matrix.
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Maybe you can tell
me right away what --
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what are the eigenvalues
and -- eigenvalues anyway.
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And then we can check.
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But can you spot any of the
eigenvalues of that matrix?
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We're looking for
two eigenvalues.
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Do you see --
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I mean, if I just wrote
that matrix down, what --
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what do you notice about it?
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It's singular, right.
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That -- that's a
singular matrix.
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That tells me right away
that one of the eigenvalues
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is lambda equals zero.
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I can -- that's a
singular matrix,
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the second column is minus
two times the first column,
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the determinant is zero,
it's -- it's singular,
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so zero is an eigenvalue and
the other eigenvalue will be --
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from the trace.
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I look at the
trace, the sum down
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the diagonal is minus three.
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That has to agree with
the sum of the eigenvalue,
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so that second eigenvalue
better be minus three.
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I could, of course --
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I could compute -- why
don't I over here --
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compute the determinant
of A minus lambda I,
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the determinant of this minus
one minus lambda two one minus
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two minus lambda matrix.
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But we know what's coming.
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When I do that multiplication,
I get a lambda squared.
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I get a two lambda and a one
lambda, that's a three lambda.
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And then -- now I'm going
to get the determinant,
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which is two minus
two which is zero.
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So there's my characteristic
polynomial, this determinant.
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And of course I factor that into
lambda times lambda plus three
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and I get the two eigenvalues
that we saw coming.
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What else do I need?
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The eigenvectors.
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So before I even think about the
differential equation or what
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-- how to solve it, let me
find the eigenvectors for this
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matrix.
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Okay.
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So take lambda equals zero --
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so that -- that's
the first eigenvalue.
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Lambda one equals zero
and the second eigenvalue
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will be lambda two
equals minus three.
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By the way, I --
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I already know something
important about this.
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The eigenvalues are
telling me something.
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You'll see how it comes
out, but let me point to --
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these numbers are
-- this eigenvalue,
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a negative eigenvalue,
is going to disappear.
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There's going to be an e to the
minus three t in the answer.
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That e to the minus
three t as times goes on
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is going to be very, very small.
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The other part of the
answer will involve an e
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to the zero t.
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But e to the zero t is
one and that's a constant.
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So I'm expecting that this
solution'll have two parts,
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an e to the zero t part and an
e to the minus three t part,
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and that -- and as time goes
on, the second part'll disappear
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and the first part
will be a steady
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It won't move. state.
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It will be -- at the end of
-- as t approaches infinity,
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this part disappears
and this is the --
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the e to the zero t
part is what I get.
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And I'm very interested in these
steady states, so that's --
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I get a steady state
when I have a zero
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eigenvalue.
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Okay.
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What about those eigenvectors?
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So what's the eigenvector that
goes with eigenvalue zero?
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Okay.
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The matrix is singular as
it is, the eigenvector is --
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is the guy in the null space,
so what vector is in the null
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space of that matrix?
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Let's see.
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I guess I probably give the
free variable the value one
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and I realize that if I want to
get zero I need a two up here.
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Okay?
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So Ax1 is zero x1.
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A x1 is zero x1.
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Fine.
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Okay.
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What about the other eigenvalue?
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Lambda two is minus three.
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Okay.
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How do I get the other
eigenvalue, then?
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For the moment --
can I mentally do it?
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I subtract minus three
along the diagonal,
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which means I add three --
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can I -- I'll just do it with an
erase -- erase for the moment.
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So I'm going to add
three to the diagonal.
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So this minus one will
become a two and --
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I'll make it in big
loopy letters --
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and when I add three to this
guy, the minus two becomes --
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well, I can't make one
very loopy, but how's that?
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Okay.
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Now that's A minus three I --
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A plus three I, sorry.
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That's A plus three I.
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It's supposed to
be singular, right?
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I-- if things --
if I did it right,
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this matrix should be
singular and the x2,
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the eigenvector should
be in its null space.
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Okay.
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What do I get for the
null space of this?
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Maybe minus one one,
or one minus one.
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Doesn't matter.
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Those are both perfectly good.
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Right?
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Because that's in the
null space of this.
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Now I'll -- because A times
that vector is three times that
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vector.
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Ax2 is minus three x2.
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Good.
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Okay.
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Can I get A again so
we see that correctly?
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That was a minus one and
that was a minus two.
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Good.
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Okay.
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That -- that's the first job.
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eigenvalues and eigenvectors.
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And already the
eigenvalues are telling me
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the most important
information about the answer.
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But now, what is the answer?
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The answer is -- the
solution will be U of T --
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okay.
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Now, wh- now I use those
eigenvalues and eigenvectors.
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The solution is some --
there are two eigenvalues.
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So I -- it -- so there're going
to be two special solutions
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here.
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Two pure exponential solutions.
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The first one is going to
be either the lambda one tx1
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and the -- so that solves the
equation, and so does this one.
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They both are solutions to
the differential equation.
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That's the general solution.
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The general solution
is a combination
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of that pure
exponential solution
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and that pure
exponential solution.
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Can I just see that those
guys do solve the equation?
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So let me just check -- check
on this one, for example.
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I -- I want to check that
the -- my equation --
00:09:53.360 --> 00:09:53.860
let's
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Check. remember, the
equation -- du/dt is Au.
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I plug in e to the
lambda one t x1
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and let's just see that
the equation's okay.
00:10:08.470 --> 00:10:12.370
I believe this is a
solution to that equation.
00:10:12.370 --> 00:10:13.980
So just plug it in.
00:10:13.980 --> 00:10:17.870
On the left-hand side, I
take the time derivative --
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so the left-hand side will be
lambda one, e to the lambda one
00:10:22.700 --> 00:10:25.540
t x1, right?
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The time derivative -- this is
the term that depends on time,
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it's just ordinary exponential,
its derivative brings down
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a lambda one.
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On the other side of the
equation it's A times this
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thing.
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A times either the lambda one t
x one, and does that check out?
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Do we have equality there?
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Yes, because either the lambda
one t appears on both sides
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and the other one is Ax1
equal lambda one x1 -- check.
00:10:58.030 --> 00:11:02.590
Do you -- so, the -- we've come
to the first point to remember.
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These pure solutions.
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Those pure solutions are the
-- those pure exponentials are
00:11:10.760 --> 00:11:14.280
the differential
equations analogue of --
00:11:14.280 --> 00:11:17.040
last time we had pure powers.
00:11:17.040 --> 00:11:19.540
Last time -- so --
00:11:19.540 --> 00:11:24.200
so last time, the
analog was lambda --
00:11:24.200 --> 00:11:29.020
lambda one to the K-th power
x1, some amount of that,
00:11:29.020 --> 00:11:35.130
plus some amount of lambda
two to the K-th power x2.
00:11:35.130 --> 00:11:37.590
That was our formula
from last time.
00:11:37.590 --> 00:11:41.690
I put it up just to -- so
your eye compares those two
00:11:41.690 --> 00:11:42.880
formulas.
00:11:42.880 --> 00:11:45.550
Powers of lambda in the --
00:11:45.550 --> 00:11:48.610
in the difference equation
-- that -- this was in the --
00:11:48.610 --> 00:11:55.480
this was for the equation
uk plus one equals A uk.
00:11:55.480 --> 00:11:59.890
That was for the finite
step -- stepping by one.
00:11:59.890 --> 00:12:02.330
And we got powers,
now this is the one
00:12:02.330 --> 00:12:04.920
we're interested in,
the exponentials.
00:12:04.920 --> 00:12:08.750
So -- so that's --
that's the solution --
00:12:08.750 --> 00:12:10.660
what are c1 and c2?
00:12:10.660 --> 00:12:12.300
Then we're through.
00:12:12.300 --> 00:12:14.060
What are c1 and c2?
00:12:14.060 --> 00:12:17.340
Well, of course we know
these actual things.
00:12:17.340 --> 00:12:22.260
Let me just -- let
me come back to this.
00:12:22.260 --> 00:12:26.790
c1 is -- we haven't figured out
yet, but e to the lambda one t,
00:12:26.790 --> 00:12:32.890
the lambda one is zero so that's
just a one times x1 which is
00:12:32.890 --> 00:12:34.260
two one.
00:12:34.260 --> 00:12:39.590
So it's c1 times this one that's
not moving times the vector,
00:12:39.590 --> 00:12:44.660
the eigenvector two
one and c2 times --
00:12:44.660 --> 00:12:47.730
what's e to the lambda two t?
00:12:47.730 --> 00:12:51.830
Lambda two is minus three.
00:12:51.830 --> 00:12:54.010
So this is the term
that has the minus
00:12:54.010 --> 00:12:58.360
three t and its eigenvector
is this one minus one.
00:13:01.520 --> 00:13:06.640
So this vector
solves the equation
00:13:06.640 --> 00:13:08.580
and any multiple of it.
00:13:08.580 --> 00:13:12.190
This vector solves the equation
if it's got that factor
00:13:12.190 --> 00:13:14.620
e to the minus three t.
00:13:14.620 --> 00:13:17.980
We've got the answer
except for c1 and c2.
00:13:17.980 --> 00:13:22.970
So -- so everything I've done
is immediate as soon as you know
00:13:22.970 --> 00:13:25.570
the eigenvalues
and eigenvectors.
00:13:25.570 --> 00:13:27.640
So how do we get c1 and c2?
00:13:27.640 --> 00:13:30.930
That has to come from
the initial condition.
00:13:30.930 --> 00:13:38.740
So now I -- now I use -- u
of zero is given as one zero.
00:13:41.780 --> 00:13:46.346
So this is the initial condition
that will find c1 and c2.
00:13:46.346 --> 00:13:48.095
So let me do that on
the board underneath.
00:13:51.080 --> 00:13:52.935
At t equals zero, then --
00:13:56.500 --> 00:14:05.280
I get c1 times this guy plus
c2 and now I'm at times zero.
00:14:05.280 --> 00:14:08.320
So that's a one and
this is a one minus one
00:14:08.320 --> 00:14:12.920
and that's supposed to agree
with u of zero one zero.
00:14:19.550 --> 00:14:20.850
Okay.
00:14:20.850 --> 00:14:23.090
That should be two equations.
00:14:23.090 --> 00:14:26.500
That should give me c1 and
c2 and then I'm through.
00:14:26.500 --> 00:14:28.540
So what are c1 and c2?
00:14:28.540 --> 00:14:30.430
Let's see.
00:14:30.430 --> 00:14:33.000
I guess we could
actually spot them by eye
00:14:33.000 --> 00:14:36.800
or we could solve two
equations in two unknowns.
00:14:36.800 --> 00:14:38.090
Let's see.
00:14:38.090 --> 00:14:40.940
If these were both ones
-- so I'm just adding --
00:14:40.940 --> 00:14:43.630
then I would get three zero.
00:14:43.630 --> 00:14:46.740
So what's the -- what's
the solution, then?
00:14:49.970 --> 00:14:53.910
If -- if c1 and c2 are both
ones, I get three zero,
00:14:53.910 --> 00:14:55.990
so I want, like,
one third of that,
00:14:55.990 --> 00:14:57.750
because I want to get one zero.
00:14:57.750 --> 00:15:02.220
So I think it's c1 equals
a third, c2 equals a third.
00:15:05.460 --> 00:15:08.030
So finally I have the answer.
00:15:08.030 --> 00:15:11.000
Let me keep it in the
-- in this board here.
00:15:11.000 --> 00:15:20.530
Finally the answer is one third
of this plus one third of this.
00:15:24.450 --> 00:15:27.990
Do you see what -- what's
actually happening with this
00:15:27.990 --> 00:15:28.880
flow?
00:15:28.880 --> 00:15:32.630
This flow started out at --
the solution started out at one
00:15:32.630 --> 00:15:34.140
zero.
00:15:34.140 --> 00:15:36.790
Started at one zero.
00:15:36.790 --> 00:15:41.290
Then as time went on,
people moved, essentially.
00:15:41.290 --> 00:15:46.190
Some fraction of
this one moved here.
00:15:46.190 --> 00:15:52.030
And -- and in the limit, there's
-- there's the limit, as --
00:15:52.030 --> 00:15:52.530
right?
00:15:52.530 --> 00:15:55.540
As t goes to infinity,
as t gets very large,
00:15:55.540 --> 00:15:59.110
this disappears and this
is the steady state.
00:15:59.110 --> 00:16:02.550
So the steady state is --
00:16:02.550 --> 00:16:04.272
so the steady state --
00:16:08.700 --> 00:16:14.190
u -- we could call it u at
infinity is one third of two
00:16:14.190 --> 00:16:15.040
and one.
00:16:15.040 --> 00:16:17.110
It's -- it's two
thirds of one third.
00:16:19.970 --> 00:16:22.790
So that's the -- we really --
00:16:22.790 --> 00:16:25.280
I mean, you're
getting, like, total,
00:16:25.280 --> 00:16:29.790
insight into the
behavior of the solution,
00:16:29.790 --> 00:16:32.050
what the differential
equation does.
00:16:32.050 --> 00:16:37.480
Of course, we don't -- wouldn't
always have a steady state.
00:16:37.480 --> 00:16:40.580
Sometimes we would
approach zero.
00:16:40.580 --> 00:16:42.320
Sometimes we would blow up.
00:16:42.320 --> 00:16:45.250
Can we straighten
out those cases?
00:16:45.250 --> 00:16:47.510
The eigenvalue should tell us.
00:16:47.510 --> 00:16:50.170
So when do we get --
00:16:50.170 --> 00:16:54.440
so -- so let me ask first,
when do we get stability?
00:16:57.220 --> 00:17:00.250
That's u of t going to zero.
00:17:03.070 --> 00:17:05.660
When would the solution
go to zero no matter
00:17:05.660 --> 00:17:09.230
what the initial condition is?
00:17:09.230 --> 00:17:11.140
Negative eigenvalues, right.
00:17:11.140 --> 00:17:12.609
Negative eigenvalues.
00:17:12.609 --> 00:17:13.630
But now I have to --
00:17:13.630 --> 00:17:16.950
I have to ask you
for one more step.
00:17:16.950 --> 00:17:20.420
Suppose the eigenvalues
are complex numbers?
00:17:20.420 --> 00:17:22.680
Because we know they could be.
00:17:22.680 --> 00:17:27.760
Then we want stability --
this -- this -- we want --
00:17:27.760 --> 00:17:35.260
we need all these e to the
lambda t-s all going to zero
00:17:35.260 --> 00:17:40.920
and somehow that asks us
to have lambda negative.
00:17:40.920 --> 00:17:43.470
But suppose lambda
is a complex number?
00:17:43.470 --> 00:17:45.690
Then what's the test?
00:17:45.690 --> 00:17:50.340
What -- if lambda's a
complex number like, oh,
00:17:50.340 --> 00:17:54.730
suppose lambda is negative
plus an imaginary part?
00:17:54.730 --> 00:17:59.810
Say lambda is minus
three plus six i?
00:17:59.810 --> 00:18:01.120
What -- what happens then?
00:18:01.120 --> 00:18:03.530
Can we just, like,
do a -- a case here?
00:18:03.530 --> 00:18:11.550
If -- if this lambda is
minus three plus six it,
00:18:11.550 --> 00:18:14.170
how big is that number?
00:18:14.170 --> 00:18:18.450
Does this -- does this imaginary
part play a -- play a --
00:18:18.450 --> 00:18:20.840
play a role here or not?
00:18:20.840 --> 00:18:22.850
Or how big is --
00:18:22.850 --> 00:18:25.700
what's the absolute value
of that -- of that quantity?
00:18:28.530 --> 00:18:32.670
It's just e to the
minus three t, right?
00:18:32.670 --> 00:18:36.880
Because this other part, this --
the -- the magnitude -- the --
00:18:36.880 --> 00:18:41.795
this -- e to the six it -- what
-- that has absolute value one.
00:18:44.680 --> 00:18:45.180
Right?
00:18:45.180 --> 00:18:50.540
That's just this cosine of
six t plus i, sine of six t.
00:18:50.540 --> 00:18:53.060
And the absolute
value squared will
00:18:53.060 --> 00:18:56.230
be cos squared plus sine
squared will be one.
00:18:56.230 --> 00:18:59.680
This is -- this complex number
is running around the unit
00:18:59.680 --> 00:19:00.660
circle.
00:19:00.660 --> 00:19:04.770
This com- this -- the -- it's
the real part that matters.
00:19:04.770 --> 00:19:07.020
This is what I'm trying to do.
00:19:07.020 --> 00:19:10.980
Real part of lambda
has to be negative.
00:19:10.980 --> 00:19:14.880
If lambda's a complex
number, it's the real part,
00:19:14.880 --> 00:19:19.200
the minus three, that
either makes us go to zero
00:19:19.200 --> 00:19:24.940
or doesn't, or let
-- or blows up.
00:19:24.940 --> 00:19:27.380
The imaginary part won't
-- will just, like,
00:19:27.380 --> 00:19:30.690
oscillate between
the two components.
00:19:30.690 --> 00:19:31.360
Okay.
00:19:31.360 --> 00:19:33.230
So that's stability.
00:19:33.230 --> 00:19:36.040
And what about --
00:19:36.040 --> 00:19:37.305
what about a steady state?
00:19:42.130 --> 00:19:45.490
When would we have,
a steady state,
00:19:45.490 --> 00:19:47.390
always in the same direction?
00:19:47.390 --> 00:19:48.160
So let me --
00:19:48.160 --> 00:19:51.280
I'll take this part away --
00:19:51.280 --> 00:19:54.280
when -- so that was, like,
checking that it's --
00:19:54.280 --> 00:19:57.830
that it's the real part
that we care about.
00:19:57.830 --> 00:20:01.530
Now, we have a
steady state when --
00:20:01.530 --> 00:20:12.500
when lambda one is zero and the
other eigenvalues have what?
00:20:12.500 --> 00:20:14.990
So I'm looking -- like,
that example was, like,
00:20:14.990 --> 00:20:18.910
perfect for a steady state.
00:20:18.910 --> 00:20:22.760
We have a zero eigenvalue
and the other eigenvalues,
00:20:22.760 --> 00:20:25.070
we want those to disappear.
00:20:25.070 --> 00:20:28.975
So the other eigenvalues
have real part negative.
00:20:31.880 --> 00:20:35.700
And we blow up, for sure --
00:20:35.700 --> 00:20:45.380
we blow up if any real
part of lambda is positive.
00:20:49.090 --> 00:20:54.240
So if I -- if I reverse the
sign of A -- of the matrix A --
00:20:54.240 --> 00:20:57.420
suppose instead of the matrix
I had, the A that I had,
00:20:57.420 --> 00:20:58.390
I changed it --
00:20:58.390 --> 00:21:00.770
I changed all its sines.
00:21:00.770 --> 00:21:04.950
What would that do to the
eigenvalues and eigenvectors?
00:21:04.950 --> 00:21:08.090
If I -- if -- if I know the
eigenvalues and eigenvectors
00:21:08.090 --> 00:21:11.520
of A, tell me about minus A.
00:21:11.520 --> 00:21:14.780
What happens to the eigenvalues?
00:21:14.780 --> 00:21:18.410
For minus A, they'll
all change sine.
00:21:18.410 --> 00:21:20.660
So I'll have blow up.
00:21:20.660 --> 00:21:23.020
This -- instead of the
e to the minus three t,
00:21:23.020 --> 00:21:26.460
if I change that to minus --
if I change the sines in that
00:21:26.460 --> 00:21:30.810
matrix, I would change
the trace to plus three,
00:21:30.810 --> 00:21:34.020
I would have an e to the plus
three t and I would have blow
00:21:34.020 --> 00:21:36.150
up.
00:21:36.150 --> 00:21:39.430
Of course the zero eigenvalue
would stay at zero,
00:21:39.430 --> 00:21:42.490
but the other guy
is taking off in --
00:21:42.490 --> 00:21:45.091
if I reversed all the sines.
00:21:45.091 --> 00:21:45.590
Okay.
00:21:45.590 --> 00:21:51.090
So this is -- this is
the stability picture.
00:21:51.090 --> 00:21:56.680
And for a two by two
matrix, we can actually
00:21:56.680 --> 00:22:01.220
pin down even more
closely what that means.
00:22:01.220 --> 00:22:02.710
Can I -- let -- can I do that?
00:22:02.710 --> 00:22:04.410
Let me do that --
00:22:04.410 --> 00:22:05.810
I want to --
00:22:05.810 --> 00:22:11.230
for a two by two matrix, I
can tell whether the real part
00:22:11.230 --> 00:22:14.740
of the eigenvalues is
negative, I -- well, let me --
00:22:14.740 --> 00:22:18.480
let me tell you what I
have in mind for that.
00:22:18.480 --> 00:22:21.040
So two by two stability --
00:22:21.040 --> 00:22:25.750
let me -- this is
just a little comment.
00:22:25.750 --> 00:22:27.506
Two by two stability.
00:22:31.240 --> 00:22:35.930
So my matrix, now,
is just a b c d
00:22:35.930 --> 00:22:41.770
and I'm looking for the real
parts of both eigenvalues
00:22:41.770 --> 00:22:42.910
to be negative.
00:22:47.480 --> 00:22:47.980
Okay.
00:22:52.330 --> 00:22:55.300
What -- how can I tell
from looking at the matrix,
00:22:55.300 --> 00:22:58.230
without computing
its eigenvalues,
00:22:58.230 --> 00:23:02.150
whether the two
eigenvalues are negative,
00:23:02.150 --> 00:23:04.930
or at least their real
parts are negative?
00:23:04.930 --> 00:23:07.260
What would that tell
me about the trace?
00:23:07.260 --> 00:23:10.830
So -- so the trace --
00:23:10.830 --> 00:23:14.930
that's this a plus d --
00:23:14.930 --> 00:23:19.470
what can you tell me about
the trace in the case of a two
00:23:19.470 --> 00:23:21.760
by two stable matrix?
00:23:21.760 --> 00:23:25.320
That means the eigenvalues
have -- are negative,
00:23:25.320 --> 00:23:28.660
or at least the real parts of
those eigenvalues are negative
00:23:28.660 --> 00:23:33.140
-- then, when I take the -- when
I look at the matrix and find
00:23:33.140 --> 00:23:36.930
its trace, what -- what
do I know about that?
00:23:36.930 --> 00:23:38.360
It's negative, right.
00:23:38.360 --> 00:23:40.940
This is the sum of
the -- this equals --
00:23:40.940 --> 00:23:47.010
this equals lambda one plus
lambda two, so it's negative.
00:23:47.010 --> 00:23:49.590
The two eigenvalues, by
the way, will have --
00:23:49.590 --> 00:23:54.990
if they're complex -- will have
plus six i and minus six i.
00:23:54.990 --> 00:23:59.860
The complex parts will -- will
be conjugates of each other
00:23:59.860 --> 00:24:04.720
and disappear when we add and
the trace will be negative.
00:24:04.720 --> 00:24:06.710
Okay, the trace
has to be negative.
00:24:06.710 --> 00:24:09.030
Is that enough --
00:24:09.030 --> 00:24:14.670
is a negative trace enough
to make the matrix stable?
00:24:14.670 --> 00:24:16.180
Shouldn't be enough, right?
00:24:16.180 --> 00:24:19.270
Can I -- can you make -- what's
a matrix that has a negative
00:24:19.270 --> 00:24:24.040
trace but still it's not stable?
00:24:24.040 --> 00:24:27.500
So it -- it has a blow -- it
still has a blow-up factor
00:24:27.500 --> 00:24:30.900
and a -- and a --
and a decaying one.
00:24:30.900 --> 00:24:33.820
So what would be a -- so
just -- just to see --
00:24:33.820 --> 00:24:35.920
maybe I just put that here.
00:24:35.920 --> 00:24:40.790
This -- now I'm looking for an
example where the trace could
00:24:40.790 --> 00:24:48.080
be negative but still blow up.
00:24:48.080 --> 00:24:52.830
Of course -- yeah,
let's just take one.
00:24:52.830 --> 00:24:57.990
Oh, look, let me -- let me make
it minus two zero zero one.
00:24:57.990 --> 00:25:00.140
Okay.
00:25:00.140 --> 00:25:04.810
There's a case where that
matrix has negative trace --
00:25:04.810 --> 00:25:06.390
I know its
eigenvalues of course.
00:25:06.390 --> 00:25:09.750
They're minus two and
one and it blows up.
00:25:09.750 --> 00:25:12.780
It's got -- it's got a
plus one eigenvalue here,
00:25:12.780 --> 00:25:17.280
so there would be an e to
the plus t in the solution
00:25:17.280 --> 00:25:21.170
and it'll blow up if it has
any second component at all.
00:25:21.170 --> 00:25:23.900
I need another condition.
00:25:23.900 --> 00:25:25.615
And it's a condition
on the determinant.
00:25:28.240 --> 00:25:29.560
And what's that condition?
00:25:29.560 --> 00:25:32.730
If I know that the
two eigenvalues --
00:25:32.730 --> 00:25:36.090
suppose I know they're
negative, both negative.
00:25:36.090 --> 00:25:39.790
What does that tell me
about the determinant?
00:25:39.790 --> 00:25:41.260
Let me ask again.
00:25:41.260 --> 00:25:44.990
If I know both the
eigenvalues are negative,
00:25:44.990 --> 00:25:47.760
then I know the
trace is negative
00:25:47.760 --> 00:25:53.170
but the determinant is
positive, because it's
00:25:53.170 --> 00:25:56.450
the product of the
two eigenvalues.
00:25:56.450 --> 00:26:00.580
So this determinant is
lambda one times lambda two.
00:26:00.580 --> 00:26:04.540
This is -- this is lambda
one times lambda two
00:26:04.540 --> 00:26:08.060
and if they're both negative,
the product is positive.
00:26:08.060 --> 00:26:11.860
So positive determinant,
negative trace.
00:26:11.860 --> 00:26:17.200
I can easily track down the --
this condition also for the --
00:26:17.200 --> 00:26:20.380
if -- if there's an imaginary
part hanging around.
00:26:20.380 --> 00:26:20.880
Okay.
00:26:20.880 --> 00:26:25.550
So that's a -- like a
small but quite useful,
00:26:25.550 --> 00:26:29.820
because two by two is
always important --
00:26:29.820 --> 00:26:33.100
comment on stability.
00:26:33.100 --> 00:26:33.750
Okay.
00:26:33.750 --> 00:26:40.170
So let's just look
at the picture again.
00:26:40.170 --> 00:26:41.290
Okay.
00:26:41.290 --> 00:26:43.980
The main part of my
lecture, the one thing
00:26:43.980 --> 00:26:46.700
you want to be able to,
like, just do automatically
00:26:46.700 --> 00:26:51.750
is this step of
solving the system.
00:26:51.750 --> 00:26:54.190
Find the eigenvalues,
find the eigenvectors,
00:26:54.190 --> 00:26:56.000
find the coefficients.
00:26:56.000 --> 00:27:01.090
And notice -- what's the matrix
-- in this linear system here,
00:27:01.090 --> 00:27:05.590
I can't help -- if I -- if I
have equations like that --
00:27:05.590 --> 00:27:08.900
that's my equations
column at a time --
00:27:08.900 --> 00:27:11.710
what's the matrix
form of that equation?
00:27:11.710 --> 00:27:18.580
So -- so this -- this
system of equations is --
00:27:18.580 --> 00:27:26.710
is some matrix multiplying
c1, c2 to give u of zero.
00:27:26.710 --> 00:27:29.370
One zero.
00:27:29.370 --> 00:27:30.510
What's the matrix?
00:27:30.510 --> 00:27:35.400
Well, it's obviously
two one, one minus one,
00:27:35.400 --> 00:27:37.760
but we have a name, or
at least a letter --
00:27:37.760 --> 00:27:40.090
actually a name for that matrix.
00:27:40.090 --> 00:27:42.670
Wh- what matrix
are we s- are we --
00:27:42.670 --> 00:27:47.390
are we dealing with here in
this step of finding the c-s?
00:27:47.390 --> 00:27:50.690
Its letter is S --
00:27:50.690 --> 00:27:52.340
it's the eigenvector matrix.
00:27:52.340 --> 00:27:52.970
Of course.
00:27:52.970 --> 00:27:55.230
These are the
eigenvectors, there
00:27:55.230 --> 00:27:57.150
in the columns of our matrix.
00:27:57.150 --> 00:28:02.520
So this is S c
equals u of zero --
00:28:02.520 --> 00:28:09.640
is the -- that step where you
find the actual coefficients,
00:28:09.640 --> 00:28:14.690
you find out how much of
each pure exponential is
00:28:14.690 --> 00:28:16.910
in the solution.
00:28:16.910 --> 00:28:20.660
By getting it right at the
start, then it stays right
00:28:20.660 --> 00:28:21.320
forever.
00:28:21.320 --> 00:28:24.820
I -- you're seeing this
picture that each --
00:28:24.820 --> 00:28:29.090
each pure exponential goes on
its own way once you start it
00:28:29.090 --> 00:28:29.620
from u of
00:28:29.620 --> 00:28:30.490
zero.
00:28:30.490 --> 00:28:33.810
So you start it by
figuring out how much
00:28:33.810 --> 00:28:37.880
each one is present in u of
zero and then off they go.
00:28:37.880 --> 00:28:38.740
Okay.
00:28:38.740 --> 00:28:45.110
So -- so that's a system of
two equations in two unknowns
00:28:45.110 --> 00:28:48.070
coupled --
00:28:48.070 --> 00:28:53.630
the matrix sort of couples
u1 and u2 and the eigenvalues
00:28:53.630 --> 00:28:57.770
and eigenvectors uncouple
it, diagonalize it.
00:28:57.770 --> 00:29:00.830
Actually -- okay, now can I --
00:29:00.830 --> 00:29:04.600
can I think in terms
of S and lambda?
00:29:04.600 --> 00:29:07.330
So I want to write
the solution down,
00:29:07.330 --> 00:29:10.840
again in terms of S and lambda.
00:29:10.840 --> 00:29:11.340
Okay.
00:29:11.340 --> 00:29:14.890
I'll do that on this far board.
00:29:14.890 --> 00:29:15.890
Okay.
00:29:15.890 --> 00:29:20.540
So I'm coming back to --
00:29:20.540 --> 00:29:26.470
I'm coming back to our
equation du/dt equals Au.
00:29:26.470 --> 00:29:35.890
Now this matrix A couples them.
00:29:35.890 --> 00:29:39.010
The whole point of
eigenvectors is to uncouple.
00:29:39.010 --> 00:29:46.510
So one way to see that is
introduce set u equal A --
00:29:46.510 --> 00:29:47.010
not
00:29:47.010 --> 00:29:54.580
A. It's S, the eigenvector
matrix that uncouples it.
00:29:54.580 --> 00:29:58.890
So I'm -- I'm taking this
equation as I'm given,
00:29:58.890 --> 00:30:04.150
coupled with -- with A has --
is probably full of non-zeroes,
00:30:04.150 --> 00:30:08.130
but I'm -- by uncoupling it,
I mean I'm diagonalizing it.
00:30:08.130 --> 00:30:11.630
If I can get a
diagonal matrix, I'm --
00:30:11.630 --> 00:30:12.470
I'm in.
00:30:12.470 --> 00:30:13.120
Okay.
00:30:13.120 --> 00:30:14.950
So I plug that in.
00:30:14.950 --> 00:30:18.720
This is A S v.
00:30:18.720 --> 00:30:20.813
And this is S dv/dt.
00:30:25.810 --> 00:30:26.890
S is a constant.
00:30:26.890 --> 00:30:30.050
It's -- this it the
eigenvector matrix.
00:30:30.050 --> 00:30:31.635
This is the eigenvector matrix.
00:30:34.820 --> 00:30:35.330
Okay.
00:30:35.330 --> 00:30:37.550
Now I'm going to bring
S inverse over here.
00:30:40.390 --> 00:30:41.285
And what have I got?
00:30:45.160 --> 00:30:53.470
That combination S inverse A S
is lambda, the diagonal matrix.
00:30:53.470 --> 00:30:56.940
That's -- that's the
point, that in --
00:30:56.940 --> 00:31:01.400
if I'm using the
eigenvectors as my basis,
00:31:01.400 --> 00:31:06.700
then my system of
equations is just diagonal.
00:31:06.700 --> 00:31:10.420
I -- each -- there's
no coupling anymore --
00:31:10.420 --> 00:31:13.130
dv1/dt is lambda one v1.
00:31:13.130 --> 00:31:20.890
So let's just write that
down. dv1/ dt is lambda one v1
00:31:20.890 --> 00:31:26.470
and so on for all
n of the equations.
00:31:26.470 --> 00:31:29.610
It's a system of equations
but they're not connected,
00:31:29.610 --> 00:31:32.865
so they're easy to solve
and why don't I just
00:31:32.865 --> 00:31:33.865
write down the solution?
00:31:36.880 --> 00:31:47.160
v -- well, v is now some
e to the lambda one t --
00:31:47.160 --> 00:31:49.600
well, okay --
00:31:49.600 --> 00:31:56.330
I guess my idea here now is to
use, the natural notation --
00:31:56.330 --> 00:32:04.740
v of T is e to the
lambda tv of zero.
00:32:04.740 --> 00:32:16.150
And u of t will be Se to the
lambda t S inverse, u of zero.
00:32:16.150 --> 00:32:20.990
This is the -- this is the,
formula I'm headed for.
00:32:25.220 --> 00:32:29.530
This -- this matrix, S e
to the lambda t S inverse,
00:32:29.530 --> 00:32:32.040
that's my exponential.
00:32:32.040 --> 00:32:40.483
That's my e to the A t, is this
S e to the lambda t S inverse.
00:32:43.600 --> 00:32:47.310
So my -- my job really now is
to explain what's going on with
00:32:47.310 --> 00:32:49.620
this matrix up in
the exponential.
00:32:49.620 --> 00:32:51.610
What does that mean?
00:32:51.610 --> 00:32:54.110
What does it mean to
say e to a matrix?
00:32:58.100 --> 00:33:01.680
This ought to be easier because
this is e to a diagonal matrix,
00:33:01.680 --> 00:33:03.970
but still it's a matrix.
00:33:03.970 --> 00:33:07.350
What do we mean by e to the A t?
00:33:07.350 --> 00:33:13.120
Because really e to the
A t is my answer here.
00:33:13.120 --> 00:33:18.620
My -- my answer to
this equation is --
00:33:18.620 --> 00:33:26.170
this u of t, this is my -- this
is my e to the A t u of zero.
00:33:26.170 --> 00:33:30.907
So it's -- my job is
really now to say what's --
00:33:30.907 --> 00:33:31.740
what does that mean?
00:33:31.740 --> 00:33:33.780
What's the exponential
of a matrix
00:33:33.780 --> 00:33:38.390
and why is that formula correct?
00:33:38.390 --> 00:33:38.950
Okay.
00:33:38.950 --> 00:33:42.190
So I'll put that on
the board underneath.
00:33:42.190 --> 00:33:45.210
What's the exponential
of a matrix?
00:33:45.210 --> 00:33:47.430
Let me come back here.
00:33:47.430 --> 00:33:49.460
So I'm talking about
matrix exponentials.
00:33:55.040 --> 00:33:57.540
e to the At.
00:33:57.540 --> 00:33:58.350
Okay.
00:33:58.350 --> 00:34:01.090
How are we going to define
the exponential of a --
00:34:01.090 --> 00:34:01.735
of something?
00:34:04.490 --> 00:34:09.940
The trick -- the idea is --
the thing to go back to is
00:34:09.940 --> 00:34:15.860
exponential -- there's a
power series for exponentials.
00:34:15.860 --> 00:34:19.690
That's how you -- you -- the
good way to define e to the x
00:34:19.690 --> 00:34:25.469
is the power series one plus
x plus one half x squared,
00:34:25.469 --> 00:34:29.489
one six x cubed and we'll
do it now when the --
00:34:29.489 --> 00:34:30.770
when we have a matrix.
00:34:30.770 --> 00:34:34.739
So the one becomes the
identity, the x is At,
00:34:34.739 --> 00:34:42.620
the x squared is At squared
and I divide by two.
00:34:42.620 --> 00:34:47.600
The cube, the x cube
is At cubed over six,
00:34:47.600 --> 00:34:50.880
and what's the
general term in here?
00:34:50.880 --> 00:34:55.929
At to the n-th
power divided by --
00:34:55.929 --> 00:34:57.710
and it goes on.
00:34:57.710 --> 00:35:01.430
But what do I divide by?
00:35:01.430 --> 00:35:05.350
So, you see the pattern
-- here I divided by one,
00:35:05.350 --> 00:35:10.080
here I divided by one by two by
six, those are the factorials.
00:35:10.080 --> 00:35:10.965
It's n factorial.
00:35:14.110 --> 00:35:17.445
That was, like, the one
beautiful Taylor series.
00:35:20.300 --> 00:35:23.180
The one beautiful Taylor series
-- well, there are two --
00:35:23.180 --> 00:35:25.960
there are two beautiful
Taylor series in this world.
00:35:25.960 --> 00:35:29.550
The Taylor series
for e to the x is
00:35:29.550 --> 00:35:35.090
the s with x to the
n-th over n factorial.
00:35:35.090 --> 00:35:38.680
And all I'm doing is doing
the same thing for matrixes.
00:35:38.680 --> 00:35:40.850
The other beautiful
Taylor series
00:35:40.850 --> 00:35:47.920
is just the sum of x to the
n-th not divided by n factorial.
00:35:47.920 --> 00:35:51.300
Can you -- do you know
what function that one is?
00:35:51.300 --> 00:35:53.990
So if I take --
this is the series,
00:35:53.990 --> 00:35:58.070
all these sums are going
from zero to infinity.
00:35:58.070 --> 00:35:59.960
What's -- what
function have I got --
00:35:59.960 --> 00:36:02.770
this is like a side comment --
00:36:02.770 --> 00:36:06.750
this is one plus x plus x
squared plus x cubed plus x
00:36:06.750 --> 00:36:09.430
to the fourth not divided
by anything, what's --
00:36:09.430 --> 00:36:11.800
what's that function?
00:36:11.800 --> 00:36:15.380
One plus x plus x squared plus
x cubed plus x fourth forever
00:36:15.380 --> 00:36:18.700
is one over one minus x.
00:36:18.700 --> 00:36:24.120
It's the geometric series, the
nicest power series of all.
00:36:24.120 --> 00:36:28.850
So, actually, of course, there
would be a similar thing here.
00:36:28.850 --> 00:36:36.810
If -- if I wanted, I minus
A t inverse would be --
00:36:36.810 --> 00:36:39.060
now I've got matrixes.
00:36:39.060 --> 00:36:43.150
I've got matrixes everywhere,
but it's just like that series
00:36:43.150 --> 00:36:46.690
with -- and just like this
one without the divisions.
00:36:46.690 --> 00:36:56.310
It's I plus At plus At squared
plus At cubed and forever.
00:36:59.200 --> 00:37:02.460
So that's actually a --
a reasonable way to find
00:37:02.460 --> 00:37:04.660
the inverse of a matrix.
00:37:04.660 --> 00:37:07.500
If I chop it off --
00:37:07.500 --> 00:37:10.120
well, it's reasonable
if t is small.
00:37:10.120 --> 00:37:13.330
If t is a small number, then --
00:37:13.330 --> 00:37:15.940
then t squared is
extremely small,
00:37:15.940 --> 00:37:19.350
t cubed is even smaller,
so approximately
00:37:19.350 --> 00:37:22.480
that inverse is I plus At.
00:37:22.480 --> 00:37:24.660
I can keep more terms if I like.
00:37:24.660 --> 00:37:25.830
Do you see what I'm doing?
00:37:25.830 --> 00:37:31.440
I'm just saying we can do the
same thing for matrixes that we
00:37:31.440 --> 00:37:35.600
do for ordinary functions
and the good thing about
00:37:35.600 --> 00:37:38.470
the exponential
series -- so in a way,
00:37:38.470 --> 00:37:41.990
this series is
better than this one.
00:37:41.990 --> 00:37:43.290
Why?
00:37:43.290 --> 00:37:45.410
Because this one
always converges.
00:37:45.410 --> 00:37:48.440
I'm dividing by these
bigger and bigger numbers,
00:37:48.440 --> 00:37:54.770
so whatever matrix A and however
large t is, that series --
00:37:54.770 --> 00:37:57.430
these terms go to zero.
00:37:57.430 --> 00:38:01.960
The series adds up to a finite
sum, e to the At is a -- is --
00:38:01.960 --> 00:38:04.390
is completely defined.
00:38:04.390 --> 00:38:08.710
Whereas this second
guy could fail, right?
00:38:08.710 --> 00:38:11.730
If At is big --
00:38:11.730 --> 00:38:15.190
somehow if At has an
eigenvalue larger than one,
00:38:15.190 --> 00:38:18.690
then when I square it it'll
have that eigenvalue squared,
00:38:18.690 --> 00:38:21.820
when I cube it the
eigenvalue will be cubed --
00:38:21.820 --> 00:38:26.560
that series will blow up
unless the eigenvalues of At
00:38:26.560 --> 00:38:28.500
are smaller than one.
00:38:28.500 --> 00:38:32.150
So when the eigenvalues of
At are smaller than one --
00:38:32.150 --> 00:38:33.610
so I'd better put that in.
00:38:33.610 --> 00:38:38.260
The -- all eigenvalues
of At below one --
00:38:38.260 --> 00:38:42.230
then that series converges
and gives me the inverse.
00:38:42.230 --> 00:38:42.970
Okay.
00:38:42.970 --> 00:38:47.590
So this is the guy I'm chiefly
interested in, and I wanted
00:38:47.590 --> 00:38:51.820
to connect it to --
00:38:51.820 --> 00:38:52.400
oh, okay.
00:38:52.400 --> 00:38:55.810
What's -- how do I -- how do
I get -- this is my, like,
00:38:55.810 --> 00:38:58.160
main thing now to do --
00:38:58.160 --> 00:39:02.270
how do I get e to the At --
00:39:02.270 --> 00:39:05.760
how do I see that e to the
At is the same as this?
00:39:10.450 --> 00:39:16.410
In other words, I can find e to
the At by finding S and lambda,
00:39:16.410 --> 00:39:18.710
because now e to the lambda t
00:39:18.710 --> 00:39:21.350
is easy.
00:39:21.350 --> 00:39:24.820
Lambda's a diagonal matrix
and we can write down either
00:39:24.820 --> 00:39:27.250
the lambda t -- and will
right -- in a minute.
00:39:27.250 --> 00:39:29.810
But how -- do you see what --
00:39:29.810 --> 00:39:33.290
do you see that
we're hoping for a --
00:39:33.290 --> 00:39:38.540
we're hoping that we can
compute either the A T from S
00:39:38.540 --> 00:39:41.450
and lambda --
00:39:41.450 --> 00:39:44.980
and I have to look at that
definition and say, okay,
00:39:44.980 --> 00:39:48.660
how do -- how do I get an S and
the lambda to come out of that?
00:39:48.660 --> 00:39:50.670
Okay, can -- do you see how I --
00:39:50.670 --> 00:39:54.830
I want to connect that to
that, from that definition.
00:39:54.830 --> 00:39:59.090
So let me erase this --
the geometric series,
00:39:59.090 --> 00:40:08.250
which isn't part of the
differential equations case
00:40:08.250 --> 00:40:14.200
and get the S and the
lambda into this picture.
00:40:14.200 --> 00:40:15.900
Oh, okay.
00:40:15.900 --> 00:40:16.420
Here we go.
00:40:19.210 --> 00:40:22.930
So identity is fine.
00:40:22.930 --> 00:40:26.880
Now -- all right, you --
you -- you'll see how I'm --
00:40:26.880 --> 00:40:31.900
how I'm -- how I going to
get A replaced by S and S is
00:40:31.900 --> 00:40:32.550
in lambda's?
00:40:32.550 --> 00:40:36.520
Well I use the fundamental
formula of this whole chapter.
00:40:36.520 --> 00:40:43.540
A is S lambda S inverse
and then times t.
00:40:43.540 --> 00:40:45.451
That's At.
00:40:45.451 --> 00:40:45.950
Okay.
00:40:45.950 --> 00:40:48.910
What's A squared t?
00:40:48.910 --> 00:40:51.450
I can -- I've got
the divide by two,
00:40:51.450 --> 00:40:56.580
I've got the t squared
and I've got an A squared.
00:40:56.580 --> 00:41:02.580
All right, I -- so I've got
a -- there's A -- there's A.
00:41:02.580 --> 00:41:04.390
Now square it.
00:41:04.390 --> 00:41:05.880
So what happens
when I square it?
00:41:05.880 --> 00:41:08.050
We've seen that before.
00:41:08.050 --> 00:41:16.120
When I square it, I get S
lambda squared S inverse, right?
00:41:16.120 --> 00:41:20.070
When I square that thing,
the -- there's an S and a --
00:41:20.070 --> 00:41:23.840
an S cancels out an S inverse.
00:41:23.840 --> 00:41:25.900
I'm left with the S
on the left, the S
00:41:25.900 --> 00:41:29.140
inverse on the right and
lambda squared in the middle.
00:41:29.140 --> 00:41:31.660
And so on.
00:41:31.660 --> 00:41:36.680
The next one'll be S
lambda cubed, S inverse --
00:41:36.680 --> 00:41:39.590
times t cubed over
three factorial.
00:41:39.590 --> 00:41:45.190
And now -- what do I do now?
00:41:45.190 --> 00:41:48.440
I want to pull an S
out from everything.
00:41:48.440 --> 00:41:53.010
I want an S out of
the whole thing.
00:41:53.010 --> 00:41:57.020
Well, look, I'd better
write the identity how?
00:41:57.020 --> 00:42:01.250
I -- I want to be able to pull
an S out and an S inverse out
00:42:01.250 --> 00:42:04.840
from the other side, so I just
write the identity as S times S
00:42:04.840 --> 00:42:05.820
inverse.
00:42:05.820 --> 00:42:11.120
So I have an S there and
an S inverse from this side
00:42:11.120 --> 00:42:13.240
and what have I
got in the middle?
00:42:16.170 --> 00:42:18.241
If I pull out an S
and an S inverse,
00:42:18.241 --> 00:42:19.490
what have I got in the middle?
00:42:19.490 --> 00:42:23.260
I've got the
identity, a lambda t,
00:42:23.260 --> 00:42:26.650
a lambda squared t
squared over two --
00:42:26.650 --> 00:42:30.780
I've got e to the lambda t.
00:42:30.780 --> 00:42:32.600
That's what's in the middle.
00:42:32.600 --> 00:42:36.630
That's my formula
for e to the At.
00:42:36.630 --> 00:42:39.120
Oh, now I have to ask you.
00:42:39.120 --> 00:42:42.290
Does this formula always work?
00:42:42.290 --> 00:42:45.420
This formula always works --
00:42:45.420 --> 00:42:48.540
well, except it's
an infinite series.
00:42:48.540 --> 00:42:51.800
But what do I mean
by always work?
00:42:51.800 --> 00:42:55.300
And this one doesn't
always work and I just
00:42:55.300 --> 00:42:58.460
have to remind you
of what assumption
00:42:58.460 --> 00:43:00.660
is built into this
formula that's
00:43:00.660 --> 00:43:03.580
not built into the original.
00:43:03.580 --> 00:43:07.900
The assumption that A
can be diagonalized.
00:43:07.900 --> 00:43:11.660
You'll remember that
there are some small --
00:43:11.660 --> 00:43:14.770
sm- some subset of
matrixes that don't
00:43:14.770 --> 00:43:18.000
have n independent
eigenvectors, so we
00:43:18.000 --> 00:43:20.590
don't have an S inverse
for those matrixes
00:43:20.590 --> 00:43:24.780
and the whole
diagonalization breaks down.
00:43:24.780 --> 00:43:26.860
We could still
make it triangular.
00:43:26.860 --> 00:43:28.010
I'll tell you about that.
00:43:28.010 --> 00:43:32.930
But diagonal we can't do for
those particular degenerate
00:43:32.930 --> 00:43:37.310
matrixes that don't have enough
independent eigenvectors.
00:43:37.310 --> 00:43:40.240
But otherwise, this is golden.
00:43:40.240 --> 00:43:40.970
Okay.
00:43:40.970 --> 00:43:44.780
So that's the formula --
that's the matrix exponential.
00:43:44.780 --> 00:43:48.680
Now it just remains for me to
say what is e to the lambda t?
00:43:48.680 --> 00:43:50.460
Can I just do that?
00:43:50.460 --> 00:43:55.280
Let me just put that
in the corner here.
00:43:55.280 --> 00:44:02.180
What is the exponential
of a diagonal matrix?
00:44:02.180 --> 00:44:10.140
Remember lambda is diagonal,
lambda one down to lambda n.
00:44:10.140 --> 00:44:14.520
What's the exponential
of that diagonal matrix?
00:44:14.520 --> 00:44:19.550
Because our whole point is
that this ought to be simple.
00:44:19.550 --> 00:44:23.240
Our whole point is that to take
the exponential of a diagonal
00:44:23.240 --> 00:44:27.560
matrix ought to be
completely decoupled --
00:44:27.560 --> 00:44:30.300
it ought to be diagonal,
in other words, and it is.
00:44:30.300 --> 00:44:37.010
It's just e to the lambda
one t, e to the lambda two t,
00:44:37.010 --> 00:44:41.070
e to the lambda n t, all zeroes.
00:44:41.070 --> 00:44:47.620
So -- so if we have a diagonal
matrix and I plug it into this
00:44:47.620 --> 00:44:52.140
exponential formula,
everything's diagonal
00:44:52.140 --> 00:44:55.710
and the diagonal terms are
just the ordinary scaler
00:44:55.710 --> 00:44:58.930
exponentials e to
the lambda one t.
00:44:58.930 --> 00:45:01.840
Okay, so that's -- that's --
00:45:01.840 --> 00:45:06.050
in a sense, I'm doing here, on
this board, with -- with, like,
00:45:06.050 --> 00:45:11.100
formulas what I did on the --
00:45:11.100 --> 00:45:15.940
in the first half of the
lecture with specific matrix A
00:45:15.940 --> 00:45:19.270
and specific eigenvalues
and eigenvectors.
00:45:19.270 --> 00:45:21.990
The -- so let me show
you the formulas again.
00:45:21.990 --> 00:45:24.770
But the -- so you --
00:45:24.770 --> 00:45:27.300
I guess -- what should
you know from this
00:45:27.300 --> 00:45:28.300
lecture?
00:45:28.300 --> 00:45:34.180
You should know what this
matrix exponential is and, like,
00:45:34.180 --> 00:45:36.670
when does it go to zero?
00:45:36.670 --> 00:45:38.390
Tell me again, now,
the answer to that.
00:45:38.390 --> 00:45:41.350
When does e to
the At approach --
00:45:41.350 --> 00:45:45.510
get smaller and
smaller as t increases?
00:45:45.510 --> 00:45:49.070
Well, the S and the S
inverse aren't moving.
00:45:49.070 --> 00:45:51.780
It's this one that has to
get smaller and smaller
00:45:51.780 --> 00:45:57.160
and that one has this
simple diagonal form.
00:45:57.160 --> 00:46:01.740
And it goes to zero provided
every one of these lambdas --
00:46:01.740 --> 00:46:04.840
I -- I need to have each one
of these guys go to zero,
00:46:04.840 --> 00:46:09.930
so I need every real part of
every eigenvalue negative.
00:46:12.650 --> 00:46:13.230
Right?
00:46:13.230 --> 00:46:15.690
If the real part is
negative, that's --
00:46:15.690 --> 00:46:19.160
that takes the exponential
-- that forces --
00:46:19.160 --> 00:46:21.420
the exponential goes to zero.
00:46:21.420 --> 00:46:24.480
Okay, so that -- that's
really the difference.
00:46:24.480 --> 00:46:33.250
If I can just draw the -- here's
a picture of the -- of the --
00:46:33.250 --> 00:46:36.780
this is the complex plain.
00:46:36.780 --> 00:46:42.080
Here's the real axis and
here's the imaginary axis.
00:46:42.080 --> 00:46:43.960
And where do the
eigenvalues have
00:46:43.960 --> 00:46:47.570
to be for stability in
differential equations?
00:46:47.570 --> 00:46:52.470
They have to be over here,
in the left half plain.
00:46:52.470 --> 00:46:55.910
So the left half plain is this
plain, real part of lambda,
00:46:55.910 --> 00:46:58.820
less than zero.
00:46:58.820 --> 00:47:01.190
Where do the ma- where
do the eigenvalues have
00:47:01.190 --> 00:47:06.500
to be for powers of the
matrix to go to zero?
00:47:06.500 --> 00:47:11.960
Powers of the matrix go to zero
if the eigenvalues are in here.
00:47:11.960 --> 00:47:17.800
So this is the stability
region for powers --
00:47:17.800 --> 00:47:22.490
this is the region -- absolute
value of lambda, less than one.
00:47:22.490 --> 00:47:26.740
That's the stability for -- that
tells us that the powers of A
00:47:26.740 --> 00:47:30.260
go to zero, this tells us
that the exponential of A goes
00:47:30.260 --> 00:47:31.220
to zero.
00:47:31.220 --> 00:47:31.790
Okay.
00:47:31.790 --> 00:47:33.700
One final example.
00:47:33.700 --> 00:47:38.580
Let me just write down how
to deal with a final example.
00:47:38.580 --> 00:47:39.980
Let's see.
00:47:44.480 --> 00:47:48.930
So my final example will be a
single equation, u''+bu'+Ku=0.
00:47:57.040 --> 00:48:01.000
One equation, second order.
00:48:01.000 --> 00:48:03.490
How do I --
00:48:03.490 --> 00:48:05.290
and maybe I should have used --
00:48:05.290 --> 00:48:08.170
I'll use -- I prefer
to use y here,
00:48:08.170 --> 00:48:12.170
because that's what you see
in differential equations.
00:48:12.170 --> 00:48:14.790
And I want u to be a vector.
00:48:14.790 --> 00:48:23.730
So how do I change one second
order equation into a two
00:48:23.730 --> 00:48:28.250
by two first order system?
00:48:28.250 --> 00:48:30.540
Just the way I
did for Fibonacci.
00:48:30.540 --> 00:48:38.180
I'll let u be y prime and y.
00:48:38.180 --> 00:48:43.210
What I'm going to do is I'm
going to add an extra equation,
00:48:43.210 --> 00:48:46.620
y prime equals y prime.
00:48:46.620 --> 00:48:50.800
So I take this -- so by --
00:48:50.800 --> 00:48:55.110
so using this as
the vector unknown,
00:48:55.110 --> 00:48:58.830
now my equation is u prime.
00:48:58.830 --> 00:49:00.800
My first order
system is u prime,
00:49:00.800 --> 00:49:06.160
that'll be y double prime y
prime, the derivative of u,
00:49:06.160 --> 00:49:10.940
okay, now the differential
equation is telling me that y
00:49:10.940 --> 00:49:14.430
double prime is m- so
I'm just looking for --
00:49:14.430 --> 00:49:17.160
what's this matrix?
00:49:17.160 --> 00:49:19.410
y prime y.
00:49:19.410 --> 00:49:23.220
I'm looking for the matrix A.
00:49:23.220 --> 00:49:28.460
What's the matrix in case I have
a single first order equation
00:49:28.460 --> 00:49:31.140
and I want to make it
into a two by two system?
00:49:31.140 --> 00:49:32.270
Okay, simple.
00:49:32.270 --> 00:49:35.920
The first row of the matrix
is given by the equation.
00:49:35.920 --> 00:49:43.800
So y''-by'-ky -- no problem.
00:49:43.800 --> 00:49:47.240
And what's the second
row on the matrix?
00:49:47.240 --> 00:49:48.660
Then we're done.
00:49:48.660 --> 00:49:50.710
The second row of the
matrix I want just
00:49:50.710 --> 00:49:54.490
to be the trivial equation
y prime equals y prime,
00:49:54.490 --> 00:49:56.340
so I need a one
and a zero there.
00:49:59.240 --> 00:50:03.950
So matrixes like these,
the gen- the general case,
00:50:03.950 --> 00:50:09.050
if I had a five by five -- if
I had a fifth order equation
00:50:09.050 --> 00:50:11.590
and I wanted a five
by five matrix,
00:50:11.590 --> 00:50:15.850
I would see the coefficients of
the equation up there and then
00:50:15.850 --> 00:50:21.260
my four trivial equations
would put ones here.
00:50:21.260 --> 00:50:27.110
This is the kind of matrix
that goes from a fifth order
00:50:27.110 --> 00:50:32.060
to a five by five first order.
00:50:35.140 --> 00:50:40.010
So the -- and the eigenvalues
will come out in a natural way
00:50:40.010 --> 00:50:41.350
connected to the differential
00:50:41.350 --> 00:50:41.970
equation.
00:50:41.970 --> 00:50:45.840
Okay, that's
differential equations.
00:50:45.840 --> 00:50:49.890
The -- a parallel lecture
compared to powers of a matrix
00:50:49.890 --> 00:50:52.060
we can now do exponentials.
00:50:52.060 --> 00:50:53.610
Thanks.