GILBERT STRANG: OK.
A third video about
stability for second order,
constant coefficient equations.
But we'll move on
to matrices here.
So this is a rather
special video.
So this is our
familiar equation.
And I took a to b1,
I just divided out a.
No problem.
So that's one second
order equation.
But we know how to convert it
to two first order equations.
And here they are.
So this is two equations.
That's a 2 by 2 matrix there.
And so let me read
the top equation.
It says that dy dt
is 0y plus 1dy dt.
So that equation
is a triviality.
dy dt equals dy dt.
The second equation
is the real one.
The derivative of y
prime is y double prime.
So this is second derivative
here, equals minus cy and minus
b y prime.
And that's my equation
y double prime,
when I bring the minus
cy over as plus cy,
and I bring the minus b y
prime over as plus b y prime.
I have my equation.
So that equation is
the same as that one.
It's just written
with a vector unknown.
It's a system, system
of two equations.
And it's got a 2 by 2 matrix.
And it's called, this
particular matrix with a 0
and a 1 is called
the companion matrix.
Companion, so this is the
companion equation to that one.
OK.
So whatever we know
about this equation,
from the exponents
s1 and s2, we're
going to have the
same information out
of this equation.
But the language changes.
And that's really the
point of this video,
just to tell you the
change in language.
So here it is.
The old exponents, s1 and s2,
for that problem, and everybody
watching this video
is remembering
that the s's solve s squared
plus Bs plus C equals 0.
So that's always
what are s's are.
So that has two roots,
s1 and s2 that control
everything, control stability.
Now if I do it in this
language, I no longer
call them s1 and s2.
But they're the
same two numbers.
What I call them is
eigenvalues, a cool word,
half German half English
maybe, kind of a crazy word.
But it's well established.
Those same numbers
would be called
the eigenvalues of the matrix.
You see, the matrix in
this problem is the same.
We've got the same information
as the equation here.
So those are the eigenvalues.
And may I just tell you
what you may know already?
That everybody writes lambda,
a Greek lambda, for eigenvalue.
So where I had two exponents,
here I have two eigenvalues.
And those numbers are the
same as those numbers.
And they satisfy
the same equation.
And when we meet matrices and
eigenvalues properly and soon,
we'll see about eigenvalues
of other matrices.
And we'll see that for these
particular companion matrices,
the eigenvalues solve
the same equation
that the exponents solve,
this quadratic s squared
and Bs and C equals 0.
OK.
And stability,
remember that stability
has been real part of those
roots of those exponents
less than zero, because
then the exponential
has that negative real
part, and goes to zero.
Now we're just using, so
that was our old language.
And our new language would
be real part of lambda,
less than zero.
Stable matrix is real
part of the eigenvalues,
lambda less than zero.
So we're just really
exchanging the letters s
and the single high order
equation for the letter lambda,
and two first order equations.
OK.
I'm doing this without-- just
connecting the lambda to the s,
but without telling you what
the lambda is on its own.
OK.
So let me remember.
So, here I've taken
a further step.
Because basically
I've said everything
about a second order equation.
We know the condition
for stability.
The condition is that the
damping should be positive,
B should be positive.
And the frequency squared
better come out positive.
So C should be positive.
So B positive and C
positive were the case
when this was our matrix.
Now I just have a
few minutes more.
So why don't I allow
any 2 by 2 matrix.
I'm not going to give you the
theory of eigenvalues here.
But just make the connection.
OK.
So I want to make
the connection.
And you remember that
the companion matrix
had a special form 0.
a was zero, b was 1, c
was the minus the big C,
and d was minus the B.
That was the companion.
So what am I going to say
at this early, almost too
early moment about eigenvalues?
Because I'll have to
do those properly.
Eigenvalues and
eigenvectors are the key
to a system of equations.
And you understand
what I mean by system?
It means that the unknown-- that
I have more than one equation.
My matrix is 2 by 2,
or 3 by 3, or n by n.
My unknown z has 2 or 3
or n different components.
It's a vector.
So z is a vector.
A matrix multiplies a vector.
That's what matrices do.
They multiply vectors.
So that's the general picture.
And this was an
especially important case.
So we can decide
on the stability.
So I'll just summarize the
stability for that system.
The stability will
be-- well I have
to tell you something about
the solutions to that system.
Remember z is a vector.
So here are solutions.
z is-- it turns out
this is the key.
That there is an e--
you expect exponentials.
And you expect now eigenvalues
instead of s there.
And now we need a vector.
And let me just
call that vector x1.
And this will be
the eigenvector.
And this is the eigenvalue.
And if I look for a
solution of that form,
put it into my equation,
out pops the key equation
for eigenvectors.
So again, I put this, hope for
solution, into the equation.
And I'll discover that
a times this vector x1
should be lambda 1 times x1.
Oh well, I have a lot
to say about that.
But if it holds, if a times
x1 is lambda 1 times x1,
then when I put this
in, the equation works.
I've got a solution.
Well I've got one solution.
And of course for
second order things,
I'm looking for two solutions.
So the complete
solution would also
be-- so I could
have it's linear.
So I can always
multiply by a constant.
And then I would expect a
second one, of the same form,
e to some other eigenvalue,
like some other exponent
times some other eigenvector.
Here's my look-ahead message
that solutions look like that.
So we're looking
for an eigenvalue,
and looking for an eigenvector.
And there is the key equation
they have to satisfy.
And that equation
comes when we put this
into the differential equation
and make the two sides agree.
So that's what's coming.
Eigenvalues and eigenvectors
control the stability
for systems of equations.
And that's what
the world is mostly
looking at, single equation,
once in awhile but very,
very often a system.
And it'll be the
eigenvalues that tell us.
So are the eigenvalues positive?
In that case we
blow up, unstable.
Are the eigenvalues negative,
or at least the real part
is negative?
That's the stable case
that we live with.
Good, thanks.
