WEBVTT
00:00:00.500 --> 00:00:01.320
GILBERT STRANG: OK.
00:00:01.320 --> 00:00:07.920
A third video about
stability for second order,
00:00:07.920 --> 00:00:10.010
constant coefficient equations.
00:00:10.010 --> 00:00:13.820
But we'll move on
to matrices here.
00:00:13.820 --> 00:00:17.020
So this is a rather
special video.
00:00:17.020 --> 00:00:19.700
So this is our
familiar equation.
00:00:19.700 --> 00:00:23.070
And I took a to b1,
I just divided out a.
00:00:23.070 --> 00:00:25.240
No problem.
00:00:25.240 --> 00:00:29.860
So that's one second
order equation.
00:00:29.860 --> 00:00:36.030
But we know how to convert it
to two first order equations.
00:00:36.030 --> 00:00:37.480
And here they are.
00:00:37.480 --> 00:00:39.460
So this is two equations.
00:00:39.460 --> 00:00:42.590
That's a 2 by 2 matrix there.
00:00:42.590 --> 00:00:46.860
And so let me read
the top equation.
00:00:46.860 --> 00:00:54.190
It says that dy dt
is 0y plus 1dy dt.
00:00:54.190 --> 00:00:56.810
So that equation
is a triviality.
00:00:56.810 --> 00:01:00.950
dy dt equals dy dt.
00:01:00.950 --> 00:01:03.520
The second equation
is the real one.
00:01:03.520 --> 00:01:07.320
The derivative of y
prime is y double prime.
00:01:07.320 --> 00:01:13.380
So this is second derivative
here, equals minus cy and minus
00:01:13.380 --> 00:01:14.940
b y prime.
00:01:14.940 --> 00:01:17.590
And that's my equation
y double prime,
00:01:17.590 --> 00:01:23.000
when I bring the minus
cy over as plus cy,
00:01:23.000 --> 00:01:28.130
and I bring the minus b y
prime over as plus b y prime.
00:01:28.130 --> 00:01:29.420
I have my equation.
00:01:29.420 --> 00:01:33.950
So that equation is
the same as that one.
00:01:33.950 --> 00:01:37.420
It's just written
with a vector unknown.
00:01:37.420 --> 00:01:41.170
It's a system, system
of two equations.
00:01:41.170 --> 00:01:43.850
And it's got a 2 by 2 matrix.
00:01:43.850 --> 00:01:48.360
And it's called, this
particular matrix with a 0
00:01:48.360 --> 00:01:52.400
and a 1 is called
the companion matrix.
00:01:52.400 --> 00:01:59.040
Companion, so this is the
companion equation to that one.
00:01:59.040 --> 00:02:00.210
OK.
00:02:00.210 --> 00:02:03.590
So whatever we know
about this equation,
00:02:03.590 --> 00:02:09.229
from the exponents
s1 and s2, we're
00:02:09.229 --> 00:02:11.220
going to have the
same information out
00:02:11.220 --> 00:02:13.040
of this equation.
00:02:13.040 --> 00:02:15.250
But the language changes.
00:02:15.250 --> 00:02:17.780
And that's really the
point of this video,
00:02:17.780 --> 00:02:20.520
just to tell you the
change in language.
00:02:20.520 --> 00:02:22.260
So here it is.
00:02:22.260 --> 00:02:28.170
The old exponents, s1 and s2,
for that problem, and everybody
00:02:28.170 --> 00:02:30.050
watching this video
is remembering
00:02:30.050 --> 00:02:38.440
that the s's solve s squared
plus Bs plus C equals 0.
00:02:38.440 --> 00:02:40.750
So that's always
what are s's are.
00:02:40.750 --> 00:02:45.580
So that has two roots,
s1 and s2 that control
00:02:45.580 --> 00:02:49.250
everything, control stability.
00:02:49.250 --> 00:02:55.210
Now if I do it in this
language, I no longer
00:02:55.210 --> 00:02:56.850
call them s1 and s2.
00:02:56.850 --> 00:02:59.380
But they're the
same two numbers.
00:02:59.380 --> 00:03:03.770
What I call them is
eigenvalues, a cool word,
00:03:03.770 --> 00:03:08.510
half German half English
maybe, kind of a crazy word.
00:03:08.510 --> 00:03:10.810
But it's well established.
00:03:10.810 --> 00:03:13.840
Those same numbers
would be called
00:03:13.840 --> 00:03:17.360
the eigenvalues of the matrix.
00:03:17.360 --> 00:03:21.210
You see, the matrix in
this problem is the same.
00:03:21.210 --> 00:03:25.210
We've got the same information
as the equation here.
00:03:25.210 --> 00:03:26.790
So those are the eigenvalues.
00:03:26.790 --> 00:03:30.240
And may I just tell you
what you may know already?
00:03:30.240 --> 00:03:35.870
That everybody writes lambda,
a Greek lambda, for eigenvalue.
00:03:35.870 --> 00:03:40.850
So where I had two exponents,
here I have two eigenvalues.
00:03:40.850 --> 00:03:44.320
And those numbers are the
same as those numbers.
00:03:44.320 --> 00:03:48.160
And they satisfy
the same equation.
00:03:48.160 --> 00:03:54.390
And when we meet matrices and
eigenvalues properly and soon,
00:03:54.390 --> 00:03:58.420
we'll see about eigenvalues
of other matrices.
00:03:58.420 --> 00:04:02.990
And we'll see that for these
particular companion matrices,
00:04:02.990 --> 00:04:06.110
the eigenvalues solve
the same equation
00:04:06.110 --> 00:04:09.510
that the exponents solve,
this quadratic s squared
00:04:09.510 --> 00:04:12.220
and Bs and C equals 0.
00:04:12.220 --> 00:04:14.380
OK.
00:04:14.380 --> 00:04:18.640
And stability,
remember that stability
00:04:18.640 --> 00:04:24.450
has been real part of those
roots of those exponents
00:04:24.450 --> 00:04:29.270
less than zero, because
then the exponential
00:04:29.270 --> 00:04:33.340
has that negative real
part, and goes to zero.
00:04:33.340 --> 00:04:37.470
Now we're just using, so
that was our old language.
00:04:37.470 --> 00:04:41.760
And our new language would
be real part of lambda,
00:04:41.760 --> 00:04:44.790
less than zero.
00:04:44.790 --> 00:04:50.080
Stable matrix is real
part of the eigenvalues,
00:04:50.080 --> 00:04:52.190
lambda less than zero.
00:04:52.190 --> 00:04:56.580
So we're just really
exchanging the letters s
00:04:56.580 --> 00:05:02.840
and the single high order
equation for the letter lambda,
00:05:02.840 --> 00:05:06.240
and two first order equations.
00:05:06.240 --> 00:05:07.220
OK.
00:05:07.220 --> 00:05:12.362
I'm doing this without-- just
connecting the lambda to the s,
00:05:12.362 --> 00:05:16.000
but without telling you what
the lambda is on its own.
00:05:16.000 --> 00:05:16.870
OK.
00:05:16.870 --> 00:05:21.570
So let me remember.
00:05:21.570 --> 00:05:24.880
So, here I've taken
a further step.
00:05:24.880 --> 00:05:31.150
Because basically
I've said everything
00:05:31.150 --> 00:05:34.680
about a second order equation.
00:05:34.680 --> 00:05:39.330
We know the condition
for stability.
00:05:39.330 --> 00:05:42.580
The condition is that the
damping should be positive,
00:05:42.580 --> 00:05:44.590
B should be positive.
00:05:44.590 --> 00:05:48.480
And the frequency squared
better come out positive.
00:05:48.480 --> 00:05:50.850
So C should be positive.
00:05:50.850 --> 00:05:55.380
So B positive and C
positive were the case
00:05:55.380 --> 00:05:58.950
when this was our matrix.
00:05:58.950 --> 00:06:00.570
Now I just have a
few minutes more.
00:06:00.570 --> 00:06:07.050
So why don't I allow
any 2 by 2 matrix.
00:06:07.050 --> 00:06:10.710
I'm not going to give you the
theory of eigenvalues here.
00:06:10.710 --> 00:06:13.080
But just make the connection.
00:06:13.080 --> 00:06:13.700
OK.
00:06:13.700 --> 00:06:16.340
So I want to make
the connection.
00:06:16.340 --> 00:06:19.400
And you remember that
the companion matrix
00:06:19.400 --> 00:06:21.650
had a special form 0.
00:06:21.650 --> 00:06:27.400
a was zero, b was 1, c
was the minus the big C,
00:06:27.400 --> 00:06:30.555
and d was minus the B.
That was the companion.
00:06:36.380 --> 00:06:40.770
So what am I going to say
at this early, almost too
00:06:40.770 --> 00:06:42.790
early moment about eigenvalues?
00:06:42.790 --> 00:06:45.880
Because I'll have to
do those properly.
00:06:45.880 --> 00:06:49.950
Eigenvalues and
eigenvectors are the key
00:06:49.950 --> 00:06:52.070
to a system of equations.
00:06:52.070 --> 00:06:54.290
And you understand
what I mean by system?
00:06:54.290 --> 00:06:59.020
It means that the unknown-- that
I have more than one equation.
00:06:59.020 --> 00:07:03.530
My matrix is 2 by 2,
or 3 by 3, or n by n.
00:07:03.530 --> 00:07:08.740
My unknown z has 2 or 3
or n different components.
00:07:08.740 --> 00:07:09.710
It's a vector.
00:07:09.710 --> 00:07:11.340
So z is a vector.
00:07:11.340 --> 00:07:13.150
A matrix multiplies a vector.
00:07:13.150 --> 00:07:14.920
That's what matrices do.
00:07:14.920 --> 00:07:16.570
They multiply vectors.
00:07:16.570 --> 00:07:20.040
So that's the general picture.
00:07:20.040 --> 00:07:24.730
And this was an
especially important case.
00:07:24.730 --> 00:07:31.200
So we can decide
on the stability.
00:07:31.200 --> 00:07:36.300
So I'll just summarize the
stability for that system.
00:07:36.300 --> 00:07:38.500
The stability will
be-- well I have
00:07:38.500 --> 00:07:41.920
to tell you something about
the solutions to that system.
00:07:41.920 --> 00:07:43.940
Remember z is a vector.
00:07:43.940 --> 00:07:45.830
So here are solutions.
00:07:45.830 --> 00:07:52.500
z is-- it turns out
this is the key.
00:07:52.500 --> 00:07:56.890
That there is an e--
you expect exponentials.
00:07:56.890 --> 00:08:02.440
And you expect now eigenvalues
instead of s there.
00:08:02.440 --> 00:08:04.470
And now we need a vector.
00:08:04.470 --> 00:08:07.180
And let me just
call that vector x1.
00:08:07.180 --> 00:08:09.576
And this will be
the eigenvector.
00:08:14.360 --> 00:08:17.085
And this is the eigenvalue.
00:08:21.070 --> 00:08:24.940
And if I look for a
solution of that form,
00:08:24.940 --> 00:08:29.710
put it into my equation,
out pops the key equation
00:08:29.710 --> 00:08:30.640
for eigenvectors.
00:08:30.640 --> 00:08:36.860
So again, I put this, hope for
solution, into the equation.
00:08:36.860 --> 00:08:42.740
And I'll discover that
a times this vector x1
00:08:42.740 --> 00:08:45.250
should be lambda 1 times x1.
00:08:45.250 --> 00:08:47.620
Oh well, I have a lot
to say about that.
00:08:51.100 --> 00:08:55.400
But if it holds, if a times
x1 is lambda 1 times x1,
00:08:55.400 --> 00:08:59.680
then when I put this
in, the equation works.
00:08:59.680 --> 00:09:01.060
I've got a solution.
00:09:01.060 --> 00:09:02.540
Well I've got one solution.
00:09:02.540 --> 00:09:05.320
And of course for
second order things,
00:09:05.320 --> 00:09:07.430
I'm looking for two solutions.
00:09:07.430 --> 00:09:09.940
So the complete
solution would also
00:09:09.940 --> 00:09:12.760
be-- so I could
have it's linear.
00:09:12.760 --> 00:09:15.170
So I can always
multiply by a constant.
00:09:15.170 --> 00:09:20.440
And then I would expect a
second one, of the same form,
00:09:20.440 --> 00:09:25.410
e to some other eigenvalue,
like some other exponent
00:09:25.410 --> 00:09:29.070
times some other eigenvector.
00:09:29.070 --> 00:09:37.400
Here's my look-ahead message
that solutions look like that.
00:09:37.400 --> 00:09:39.790
So we're looking
for an eigenvalue,
00:09:39.790 --> 00:09:41.630
and looking for an eigenvector.
00:09:41.630 --> 00:09:44.950
And there is the key equation
they have to satisfy.
00:09:44.950 --> 00:09:47.650
And that equation
comes when we put this
00:09:47.650 --> 00:09:51.730
into the differential equation
and make the two sides agree.
00:09:51.730 --> 00:09:53.930
So that's what's coming.
00:09:53.930 --> 00:09:57.250
Eigenvalues and eigenvectors
control the stability
00:09:57.250 --> 00:10:00.530
for systems of equations.
00:10:00.530 --> 00:10:02.570
And that's what
the world is mostly
00:10:02.570 --> 00:10:06.770
looking at, single equation,
once in awhile but very,
00:10:06.770 --> 00:10:08.440
very often a system.
00:10:08.440 --> 00:10:11.270
And it'll be the
eigenvalues that tell us.
00:10:11.270 --> 00:10:14.540
So are the eigenvalues positive?
00:10:14.540 --> 00:10:17.440
In that case we
blow up, unstable.
00:10:17.440 --> 00:10:20.680
Are the eigenvalues negative,
or at least the real part
00:10:20.680 --> 00:10:21.880
is negative?
00:10:21.880 --> 00:10:25.000
That's the stable case
that we live with.
00:10:25.000 --> 00:10:26.950
Good, thanks.