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GILBERT STRANG: OK.

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A third video about
stability for second order,

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constant coefficient equations.

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But we'll move on
to matrices here.

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So this is a rather
special video.

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So this is our
familiar equation.

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And I took a to b1,
I just divided out a.

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No problem.

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So that's one second
order equation.

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But we know how to convert it
to two first order equations.

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And here they are.

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So this is two equations.

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That's a 2 by 2 matrix there.

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And so let me read
the top equation.

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It says that dy dt
is 0y plus 1dy dt.

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So that equation
is a triviality.

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dy dt equals dy dt.

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The second equation
is the real one.

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The derivative of y
prime is y double prime.

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So this is second derivative
here, equals minus cy and minus

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b y prime.

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And that's my equation
y double prime,

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when I bring the minus
cy over as plus cy,

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and I bring the minus b y
prime over as plus b y prime.

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I have my equation.

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So that equation is
the same as that one.

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It's just written
with a vector unknown.

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It's a system, system
of two equations.

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And it's got a 2 by 2 matrix.

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And it's called, this
particular matrix with a 0

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and a 1 is called
the companion matrix.

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Companion, so this is the
companion equation to that one.

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OK.

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So whatever we know
about this equation,

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from the exponents
s1 and s2, we're

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going to have the
same information out

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of this equation.

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But the language changes.

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And that's really the
point of this video,

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just to tell you the
change in language.

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So here it is.

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The old exponents, s1 and s2,
for that problem, and everybody

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watching this video
is remembering

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that the s's solve s squared
plus Bs plus C equals 0.

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So that's always
what are s's are.

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So that has two roots,
s1 and s2 that control

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everything, control stability.

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Now if I do it in this
language, I no longer

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call them s1 and s2.

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But they're the
same two numbers.

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What I call them is
eigenvalues, a cool word,

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half German half English
maybe, kind of a crazy word.

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But it's well established.

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Those same numbers
would be called

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the eigenvalues of the matrix.

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You see, the matrix in
this problem is the same.

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We've got the same information
as the equation here.

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So those are the eigenvalues.

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And may I just tell you
what you may know already?

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That everybody writes lambda,
a Greek lambda, for eigenvalue.

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So where I had two exponents,
here I have two eigenvalues.

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And those numbers are the
same as those numbers.

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And they satisfy
the same equation.

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And when we meet matrices and
eigenvalues properly and soon,

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we'll see about eigenvalues
of other matrices.

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And we'll see that for these
particular companion matrices,

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the eigenvalues solve
the same equation

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that the exponents solve,
this quadratic s squared

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and Bs and C equals 0.

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OK.

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And stability,
remember that stability

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has been real part of those
roots of those exponents

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less than zero, because
then the exponential

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has that negative real
part, and goes to zero.

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Now we're just using, so
that was our old language.

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And our new language would
be real part of lambda,

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less than zero.

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Stable matrix is real
part of the eigenvalues,

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lambda less than zero.

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So we're just really
exchanging the letters s

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and the single high order
equation for the letter lambda,

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and two first order equations.

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OK.

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I'm doing this without-- just
connecting the lambda to the s,

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but without telling you what
the lambda is on its own.

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OK.

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So let me remember.

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So, here I've taken
a further step.

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Because basically
I've said everything

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about a second order equation.

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We know the condition
for stability.

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The condition is that the
damping should be positive,

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B should be positive.

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And the frequency squared
better come out positive.

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So C should be positive.

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So B positive and C
positive were the case

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when this was our matrix.

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Now I just have a
few minutes more.

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So why don't I allow
any 2 by 2 matrix.

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I'm not going to give you the
theory of eigenvalues here.

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But just make the connection.

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OK.

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So I want to make
the connection.

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And you remember that
the companion matrix

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had a special form 0.

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a was zero, b was 1, c
was the minus the big C,

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and d was minus the B.
That was the companion.

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So what am I going to say
at this early, almost too

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early moment about eigenvalues?

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Because I'll have to
do those properly.

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Eigenvalues and
eigenvectors are the key

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to a system of equations.

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And you understand
what I mean by system?

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It means that the unknown-- that
I have more than one equation.

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My matrix is 2 by 2,
or 3 by 3, or n by n.

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My unknown z has 2 or 3
or n different components.

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It's a vector.

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So z is a vector.

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A matrix multiplies a vector.

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That's what matrices do.

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They multiply vectors.

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So that's the general picture.

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And this was an
especially important case.

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So we can decide
on the stability.

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So I'll just summarize the
stability for that system.

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The stability will
be-- well I have

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to tell you something about
the solutions to that system.

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Remember z is a vector.

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So here are solutions.

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z is-- it turns out
this is the key.

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That there is an e--
you expect exponentials.

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And you expect now eigenvalues
instead of s there.

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And now we need a vector.

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And let me just
call that vector x1.

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And this will be
the eigenvector.

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And this is the eigenvalue.

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And if I look for a
solution of that form,

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put it into my equation,
out pops the key equation

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for eigenvectors.

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So again, I put this, hope for
solution, into the equation.

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And I'll discover that
a times this vector x1

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should be lambda 1 times x1.

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Oh well, I have a lot
to say about that.

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But if it holds, if a times
x1 is lambda 1 times x1,

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then when I put this
in, the equation works.

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I've got a solution.

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Well I've got one solution.

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And of course for
second order things,

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I'm looking for two solutions.

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So the complete
solution would also

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be-- so I could
have it's linear.

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So I can always
multiply by a constant.

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And then I would expect a
second one, of the same form,

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e to some other eigenvalue,
like some other exponent

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times some other eigenvector.

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Here's my look-ahead message
that solutions look like that.

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So we're looking
for an eigenvalue,

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and looking for an eigenvector.

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And there is the key equation
they have to satisfy.

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And that equation
comes when we put this

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into the differential equation
and make the two sides agree.

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So that's what's coming.

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Eigenvalues and eigenvectors
control the stability

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for systems of equations.

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And that's what
the world is mostly

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looking at, single equation,
once in awhile but very,

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very often a system.

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And it'll be the
eigenvalues that tell us.

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So are the eigenvalues positive?

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In that case we
blow up, unstable.

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Are the eigenvalues negative,
or at least the real part

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is negative?

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That's the stable case
that we live with.

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Good, thanks.