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PROFESSOR STRANG: OK, so
this is a review session

00:00:24.080 --> 00:00:27.170
open to questions on homework.

00:00:27.170 --> 00:00:31.390
Open to questions on
topics in the exam

00:00:31.390 --> 00:00:34.110
that's coming tomorrow.

00:00:34.110 --> 00:00:39.010
This morning I wrote down
what the four questions would

00:00:39.010 --> 00:00:42.790
be about, and I'm glad I did.

00:00:42.790 --> 00:00:46.450
I never-- should have done
this many times before.

00:00:46.450 --> 00:00:51.280
So you would know
exactly and get down

00:00:51.280 --> 00:00:54.300
to seeing what
those problems are.

00:00:54.300 --> 00:00:57.660
And of course the
matrices called K,

00:00:57.660 --> 00:01:05.590
and A transpose C A are going to
appear probably more than once.

00:01:05.590 --> 00:01:08.650
So, open for any questions.

00:01:08.650 --> 00:01:11.590
About any topic whatsoever.

00:01:11.590 --> 00:01:13.640
Please.

00:01:13.640 --> 00:01:14.450
Yes, thank you.

00:01:14.450 --> 00:01:16.360
AUDIENCE: [INAUDIBLE]

00:01:16.360 --> 00:01:18.460
PROFESSOR STRANG: The
fourth question on the exam?

00:01:18.460 --> 00:01:24.050
AUDIENCE: [INAUDIBLE]

00:01:24.050 --> 00:01:27.110
PROFESSOR STRANG: I'm glad
you used that word, fun.

00:01:27.110 --> 00:01:27.970
Yes.

00:01:27.970 --> 00:01:29.480
That's exactly what I mean.

00:01:29.480 --> 00:01:32.580
Section 2.4, and
they are fun, yeah.

00:01:32.580 --> 00:01:36.140
So I drew by hand a little
graph with nodes and edges.

00:01:36.140 --> 00:01:43.910
And you want to be able to
take that first basic step.

00:01:43.910 --> 00:01:46.510
So the first step,
which is as far as we

00:01:46.510 --> 00:01:51.690
got by last Wednesday, the
first lecture on Section 2.4,

00:01:51.690 --> 00:01:59.450
was just creating the matrix
A, understanding A transpose A,

00:01:59.450 --> 00:02:04.940
and of course there's more to
understand about A transpose A.

00:02:04.940 --> 00:02:07.030
Actually, why don't
we take one second.

00:02:07.030 --> 00:02:14.210
Suppose I have a graph
with six nodes, let's say.

00:02:14.210 --> 00:02:16.760
Can you imagine a
graph with six nodes?

00:02:16.760 --> 00:02:19.980
And every node connected
to every other node.

00:02:19.980 --> 00:02:25.460
So however many
edges that would be.

00:02:25.460 --> 00:02:32.560
Actually, my grandson just
got that question on his exam.

00:02:32.560 --> 00:02:37.410
He was told there were l islands
with a flight from every island

00:02:37.410 --> 00:02:39.370
to every other island,
and he was asked

00:02:39.370 --> 00:02:43.500
how many flights that makes.

00:02:43.500 --> 00:02:45.130
So I sent him the answer.

00:02:45.130 --> 00:02:51.310
But I was very happy
with his reply.

00:02:51.310 --> 00:02:53.770
He said "that's
exactly what I got."

00:02:53.770 --> 00:02:58.240
So, what do you know.

00:02:58.240 --> 00:02:59.820
It seems to work.

00:02:59.820 --> 00:03:01.080
So anyway.

00:03:01.080 --> 00:03:03.390
Suppose we had, how
many nodes did I say?

00:03:03.390 --> 00:03:04.410
Six?

00:03:04.410 --> 00:03:05.250
OK.

00:03:05.250 --> 00:03:11.150
So we have like a
six-node-- So n is six,

00:03:11.150 --> 00:03:15.070
and it's a complete
graph, this is really just

00:03:15.070 --> 00:03:19.720
to start us off talking
about some of these problems.

00:03:19.720 --> 00:03:23.990
So the matrix A, so I
think it would be 15,

00:03:23.990 --> 00:03:27.410
where did I come up
with that number 15?

00:03:27.410 --> 00:03:29.850
And is it right, actually?

00:03:29.850 --> 00:03:31.330
Yes.

00:03:31.330 --> 00:03:34.960
This is one way, would be the
first node has five edges going

00:03:34.960 --> 00:03:38.390
out and then the
second node would

00:03:38.390 --> 00:03:42.650
have four additional edges,
and three and two and one.

00:03:42.650 --> 00:03:45.370
And five, four, three,
two, one add to 15.

00:03:45.370 --> 00:03:51.210
What would be the shape
of A in that case?

00:03:51.210 --> 00:03:54.640
So it has a row for every edge.

00:03:54.640 --> 00:03:57.040
So 15 by 6, I think.

00:03:57.040 --> 00:04:06.030
OK, and I could create
A transpose A just

00:04:06.030 --> 00:04:08.710
to have a look at it.

00:04:08.710 --> 00:04:12.120
So it would be, what shape
would A transpose A be?

00:04:12.120 --> 00:04:13.800
Six by six.

00:04:13.800 --> 00:04:16.070
Symmetric, of course.

00:04:16.070 --> 00:04:19.780
Will it be singular
or non-singular?

00:04:19.780 --> 00:04:20.500
Singular.

00:04:20.500 --> 00:04:22.710
Singular, because we
haven't grounded any nodes.

00:04:22.710 --> 00:04:25.400
We've got all these nodes,
all these edges, nothing.

00:04:25.400 --> 00:04:27.910
We haven't taken
out that column;

00:04:27.910 --> 00:04:31.840
when I reduce it to five by
five, then it'll be invertible.

00:04:31.840 --> 00:04:37.110
But six by six, so what will
be the diagonal of this?

00:04:37.110 --> 00:04:43.870
This'll be now six by six,
the size will be six by six.

00:04:43.870 --> 00:04:49.190
And what will go on the diagonal
is the degrees of every node.

00:04:49.190 --> 00:04:51.140
That means how many
edges are coming in,

00:04:51.140 --> 00:04:52.800
and what number is that?

00:04:52.800 --> 00:04:53.570
Five.

00:04:53.570 --> 00:04:58.510
So I'll have five down the
diagonal, and what else,

00:04:58.510 --> 00:05:01.000
what will be off the diagonal?

00:05:01.000 --> 00:05:04.280
Minus, a whole lot of
minus ones, a minus one

00:05:04.280 --> 00:05:06.750
above and below for every edge.

00:05:06.750 --> 00:05:10.510
And since we have a
complete graph, how many

00:05:10.510 --> 00:05:12.690
minus ones have we got?

00:05:12.690 --> 00:05:13.850
All of them.

00:05:13.850 --> 00:05:14.950
All minus ones.

00:05:14.950 --> 00:05:27.980
So all minus ones
and all minus ones.

00:05:27.980 --> 00:05:33.390
That's fine to cross over
if you need to, sure.

00:05:33.390 --> 00:05:36.370
I'm not sure what else
to say about that matrix.

00:05:36.370 --> 00:05:38.460
Well, it's not invertible.

00:05:38.460 --> 00:05:43.530
Now let's take the next step
which, I'm now going probably

00:05:43.530 --> 00:05:47.730
beyond the exam part.

00:05:47.730 --> 00:05:52.120
Just really to get us started,
I ground the sixth node.

00:05:52.120 --> 00:05:54.830
Suppose I ground
node number six,

00:05:54.830 --> 00:05:57.750
that wipes out a row and
a column, is that right?

00:05:57.750 --> 00:06:01.120
So I'm now left with
a five by five matrix.

00:06:01.120 --> 00:06:04.070
It still has all minus
ones there and there,

00:06:04.070 --> 00:06:06.010
but now it's five
by five, now it

00:06:06.010 --> 00:06:09.280
is what kind of a matrix,
what are its properties?

00:06:09.280 --> 00:06:14.780
Square, obviously, symmetric
obviously, and now invertible.

00:06:14.780 --> 00:06:17.080
Positive definite, OK.

00:06:17.080 --> 00:06:20.760
So it's fives there and now
I would have four minus ones.

00:06:20.760 --> 00:06:23.600
Let's just write them in here.

00:06:23.600 --> 00:06:26.620
Typical row, now, in
this five by five matrix

00:06:26.620 --> 00:06:30.670
would have four minus ones and
of course more here, and more

00:06:30.670 --> 00:06:33.140
here, and one there.

00:06:33.140 --> 00:06:38.440
And symmetric.

00:06:38.440 --> 00:06:40.740
All I want to say
is that that matrix,

00:06:40.740 --> 00:06:43.690
we don't often write down
the inverses of matrices,

00:06:43.690 --> 00:06:46.230
but that one I think we could.

00:06:46.230 --> 00:06:49.380
I think we could actually, and
it's a little bit interesting

00:06:49.380 --> 00:06:54.740
to know, for that special
matrix, everything about it.

00:06:54.740 --> 00:06:58.600
We could find its eigenvalues,
its determinant, its pivots,

00:06:58.600 --> 00:07:01.780
the whole works for that matrix.

00:07:01.780 --> 00:07:04.780
And that's one page
of the book, maybe

00:07:04.780 --> 00:07:07.450
at the end of
Section 2.4, I think,

00:07:07.450 --> 00:07:11.290
comes more detail
about that matrix.

00:07:11.290 --> 00:07:18.890
So in a way that special guy
is like our special K matrix,

00:07:18.890 --> 00:07:22.010
-1, 2, -1, for
second differences.

00:07:22.010 --> 00:07:27.860
Somehow this is taking,
all nodes are connected.

00:07:27.860 --> 00:07:31.870
Instead of in a line, springs
in a line, points in a line,

00:07:31.870 --> 00:07:35.040
we now have everybody
connected to everybody.

00:07:35.040 --> 00:07:38.330
So this is sort of the
special matrix when

00:07:38.330 --> 00:07:40.200
everybody is
connected to everybody

00:07:40.200 --> 00:07:43.750
and we could learn all
about that particular one.

00:07:43.750 --> 00:07:48.340
But then, of course, if
some edges are not in then

00:07:48.340 --> 00:07:51.390
some zeroes will
appear off the diagonal

00:07:51.390 --> 00:07:54.220
in the adjacency matrix part.

00:07:54.220 --> 00:07:58.010
The degrees will drop a little
if we're missing some edges

00:07:58.010 --> 00:08:03.960
and the inverse will be
not some simple expression.

00:08:03.960 --> 00:08:05.470
Anyway, that's to
get us started.

00:08:05.470 --> 00:08:14.740
So that's really where the last
lecture, Friday, brought us

00:08:14.740 --> 00:08:16.780
to this point.

00:08:16.780 --> 00:08:21.290
And I'll take this chance
to add in the block matrix

00:08:21.290 --> 00:08:24.010
just because I think of
it as quite important.

00:08:24.010 --> 00:08:26.950
So for this case,
C is the identity.

00:08:26.950 --> 00:08:31.240
So I would have the identity up
in that block, A in that block,

00:08:31.240 --> 00:08:33.850
A transpose in this block.

00:08:33.850 --> 00:08:39.590
That would be my mixed
method matrix, you could say.

00:08:39.590 --> 00:08:42.700
My saddle point matrix.

00:08:42.700 --> 00:08:45.810
It starts out very
positive definite.

00:08:45.810 --> 00:08:48.010
But it ends up
negative definite.

00:08:48.010 --> 00:08:50.450
And that's typical of
mixed methods, when

00:08:50.450 --> 00:08:57.070
both unknowns, the currents
as well as the potentials,

00:08:57.070 --> 00:08:59.920
are included in the system.

00:08:59.920 --> 00:09:04.330
So A transpose w, that was
f, I think, and this is b.

00:09:04.330 --> 00:09:07.810
I just mentioned that again,
it was in Friday's lecture

00:09:07.810 --> 00:09:10.750
and it's in the book
but I would just

00:09:10.750 --> 00:09:14.010
want to say I
often refer to this

00:09:14.010 --> 00:09:17.480
as the fundamental problem
of numerical analysis,

00:09:17.480 --> 00:09:20.190
is how do you solve that system.

00:09:20.190 --> 00:09:23.120
And of course elimination
is one way to do it.

00:09:23.120 --> 00:09:26.990
When I eliminate w,
that will lead me

00:09:26.990 --> 00:09:32.080
to the equation A
transpose Au equals,

00:09:32.080 --> 00:09:39.020
I think it'll be, there'll be an
A transpose b minus f, I think.

00:09:39.020 --> 00:09:41.220
C being the identity there.

00:09:41.220 --> 00:09:44.770
So that's the mixed method,
this is the displacement method,

00:09:44.770 --> 00:09:47.970
and this is the popular one.

00:09:47.970 --> 00:09:49.960
Because it's all at once.

00:09:49.960 --> 00:09:55.240
But people think
about this one, too.

00:09:55.240 --> 00:09:59.860
So that's like saying again
what was in Friday's lecture

00:09:59.860 --> 00:10:04.100
and will be used going forward.

00:10:04.100 --> 00:10:06.130
OK, that was just
to get us started.

00:10:06.130 --> 00:10:12.080
Now, please let's
have some questions.

00:10:12.080 --> 00:10:14.180
We need another question.

00:10:14.180 --> 00:10:15.250
Who else?

00:10:15.250 --> 00:10:16.010
Yes, thank you.

00:10:16.010 --> 00:10:17.527
AUDIENCE: [INAUDIBLE]

00:10:17.527 --> 00:10:19.360
PROFESSOR STRANG: The
first on the homework.

00:10:19.360 --> 00:10:23.590
What number was that?

00:10:23.590 --> 00:10:27.540
Section 2.2, number six?

00:10:27.540 --> 00:10:30.250
About the trapezoidal rule?

00:10:30.250 --> 00:10:31.470
Yes.

00:10:31.470 --> 00:10:38.040
OK, now I did speak about that
a little in the last review

00:10:38.040 --> 00:10:42.501
session, but can I just say
a couple words more about it

00:10:42.501 --> 00:10:43.000
here?

00:10:43.000 --> 00:10:44.432
AUDIENCE: [INAUDIBLE]

00:10:44.432 --> 00:10:45.890
PROFESSOR STRANG:
What's it asking?

00:10:45.890 --> 00:10:48.560
Yes.

00:10:48.560 --> 00:10:50.920
People often ask me
that about my problems.

00:10:50.920 --> 00:10:54.030
I don't know.

00:10:54.030 --> 00:10:55.320
You can't read my mind?

00:10:55.320 --> 00:10:56.550
You should.

00:10:56.550 --> 00:11:03.930
OK, so the point is that
for special differential

00:11:03.930 --> 00:11:07.090
equations-- So let me just
summarize what we did there.

00:11:07.090 --> 00:11:10.600
So this we did before, but
I didn't do everything.

00:11:10.600 --> 00:11:16.030
So what we did before was point
out that the system du/dt =

00:11:16.030 --> 00:11:24.100
Au conserves
energy. u squared, u

00:11:24.100 --> 00:11:27.910
of time-- for all
time, u of time

00:11:27.910 --> 00:11:40.220
squared stays constant if A
transpose equals minus A. OK,

00:11:40.220 --> 00:11:42.780
essentially you
take the derivative,

00:11:42.780 --> 00:11:45.900
it's got two terms because
we've got a product there,

00:11:45.900 --> 00:11:47.520
a product rule.

00:11:47.520 --> 00:11:52.170
The derivative will be,
one term will involve A,

00:11:52.170 --> 00:11:54.730
the other term will
involve A transpose,

00:11:54.730 --> 00:11:57.730
and if our matrix has this
antisymmetric property,

00:11:57.730 --> 00:12:01.010
those terms will cancel;
the derivative will be zero,

00:12:01.010 --> 00:12:03.590
and that'll mean that
this is a constant.

00:12:03.590 --> 00:12:06.010
OK, so that's the
differential equation.

00:12:06.010 --> 00:12:09.090
Now, the question was about
the difference equation.

00:12:09.090 --> 00:12:13.540
So we're taking the
trapezoidal rule

00:12:13.540 --> 00:12:20.410
and we want to show that
u_n squared stays constant

00:12:20.410 --> 00:12:22.390
for the trapezoidal rule.

00:12:22.390 --> 00:12:30.640
And so what that means, in
other words, is step by step,

00:12:30.640 --> 00:12:38.330
u_(n+1), and I could
write it u_(n+1) squared,

00:12:38.330 --> 00:12:43.500
but the other way to write that
and the way we have to work

00:12:43.500 --> 00:12:47.410
with it is that, is the same.

00:12:47.410 --> 00:12:51.010
Now, that was just an identity,
that's just the meaning.

00:12:51.010 --> 00:12:56.570
Now, I want to show
that that's the key.

00:12:56.570 --> 00:12:59.280
That's what we
would want to prove.

00:12:59.280 --> 00:13:05.950
That the trapezoidal rule copies
the property of constant energy

00:13:05.950 --> 00:13:08.030
of the differential equations.

00:13:08.030 --> 00:13:11.880
And of course, you know
that in oscillating springs

00:13:11.880 --> 00:13:16.600
when there's no source, no
forces coming from outside,

00:13:16.600 --> 00:13:19.070
the total energy
will stay constant.

00:13:19.070 --> 00:13:22.300
And you could think of
many other situations.

00:13:22.300 --> 00:13:26.320
You have a spacecraft, where
you've turned off the engines.

00:13:26.320 --> 00:13:29.220
It's just going there,
it's possibly rotating.

00:13:29.220 --> 00:13:35.390
So there you've got
angular velocity

00:13:35.390 --> 00:13:37.540
included in the total energy.

00:13:37.540 --> 00:13:41.180
Important fact, if energy stays
constant you want to know it.

00:13:41.180 --> 00:13:44.960
And you're very happy
if the finite difference

00:13:44.960 --> 00:13:46.610
method copies it.

00:13:46.610 --> 00:13:53.330
OK, so then it was just
a question of-- Here

00:13:53.330 --> 00:13:56.360
we took derivatives
to do that one.

00:13:56.360 --> 00:13:58.540
Here we're going to be
playing with differences,

00:13:58.540 --> 00:14:04.080
and my suggestion
was that the good way

00:14:04.080 --> 00:14:08.600
to get it was to
take that vector

00:14:08.600 --> 00:14:17.040
times the trapezoidal equation
and show that this turned out

00:14:17.040 --> 00:14:21.410
to-- The trapezoidal equation
is something equals zero,

00:14:21.410 --> 00:14:26.860
and you hope, and it takes a
few lines of jiggling around,

00:14:26.860 --> 00:14:31.480
that when you do that
you'll get the difference,

00:14:31.480 --> 00:14:32.920
you get exactly this.

00:14:32.920 --> 00:14:40.220
You get u_(n+1) transpose
u_(n+1) minus u_n transpose

00:14:40.220 --> 00:14:41.160
u_n.

00:14:41.160 --> 00:14:44.100
That's the goal.

00:14:44.100 --> 00:14:47.000
We know that the
trapezoidal equation--

00:14:47.000 --> 00:14:49.730
maybe I move everything onto
one side so I have something

00:14:49.730 --> 00:14:51.420
equals zero.

00:14:51.420 --> 00:14:57.340
Then my trick is take
that vector equation,

00:14:57.340 --> 00:15:02.030
multiply by that, play
around with those terms

00:15:02.030 --> 00:15:03.370
and you'll get this.

00:15:03.370 --> 00:15:07.380
So, since that is
zero, this is zero.

00:15:07.380 --> 00:15:11.060
And that's exactly what
our goal was to prove.

00:15:11.060 --> 00:15:13.570
So it's just in
the jiggling around

00:15:13.570 --> 00:15:17.840
and maybe we don't want to
take the full time because I'll

00:15:17.840 --> 00:15:18.700
post that.

00:15:18.700 --> 00:15:22.680
Actually, I may post some
of these solutions even

00:15:22.680 --> 00:15:24.040
before the quiz.

00:15:24.040 --> 00:15:26.890
And therefore before
the homework is due,

00:15:26.890 --> 00:15:29.600
just because this particular
homework, as I say,

00:15:29.600 --> 00:15:32.500
is not-- The graders
are just going

00:15:32.500 --> 00:15:39.020
to be so busy with
all the quizzes.

00:15:39.020 --> 00:15:40.680
This is for learning.

00:15:40.680 --> 00:15:43.620
Now, here's the one thing
you want to learn out

00:15:43.620 --> 00:15:46.940
of this messy computation.

00:15:46.940 --> 00:15:55.070
You also have a term,
you'll also find a term U_n,

00:15:55.070 --> 00:15:57.310
when you just do
this mechanically,

00:15:57.310 --> 00:16:00.830
you'll find a u_(n+1)
transpose u_n,

00:16:00.830 --> 00:16:06.410
and you'll find a u_n
transpose u_(n+1),

00:16:06.410 --> 00:16:10.270
and they'll come in
with opposite signs.

00:16:10.270 --> 00:16:13.130
That'll be when you've
plugged in the fact

00:16:13.130 --> 00:16:17.580
that A transpose equals minus
A, and all I wanted to do

00:16:17.580 --> 00:16:20.900
is ask you, what does
that term amount to?

00:16:20.900 --> 00:16:23.510
Because that term will show up.

00:16:23.510 --> 00:16:24.620
One way or another.

00:16:24.620 --> 00:16:26.620
And what does it equal?

00:16:26.620 --> 00:16:27.480
Zero.

00:16:27.480 --> 00:16:28.800
Everybody should know that.

00:16:28.800 --> 00:16:31.350
That's the one
thing, that you have

00:16:31.350 --> 00:16:34.230
to add to just
mechanically computing,

00:16:34.230 --> 00:16:39.950
is the fact that the dot product
of that vector with that,

00:16:39.950 --> 00:16:46.870
v transpose w is the same
as w transpose v. So,

00:16:46.870 --> 00:16:50.550
it's good to just call attention
to the easy things that

00:16:50.550 --> 00:16:51.510
are like that.

00:16:51.510 --> 00:16:53.780
Why is that?

00:16:53.780 --> 00:16:59.280
That's because both sides,
this is equal to what? v_1*w_1,

00:16:59.280 --> 00:17:03.440
v_2*w_2, v_3*w_3,
component by component.

00:17:03.440 --> 00:17:05.360
And this is w_1*v_1.

00:17:05.360 --> 00:17:08.250
But we're just talking
numbers at that point.

00:17:08.250 --> 00:17:11.070
So the numbers of v_1 times
w_1 are certainly the same

00:17:11.070 --> 00:17:12.850
as w_1 times v_1.

00:17:12.850 --> 00:17:15.880
Every component by component,
they're exactly the same

00:17:15.880 --> 00:17:19.760
and of course then the
dot products are the same.

00:17:19.760 --> 00:17:26.770
So that's the fact which
for this v and that w,

00:17:26.770 --> 00:17:31.130
make the term go away,
that's still sitting there.

00:17:31.130 --> 00:17:35.870
Other terms go away
because of this property.

00:17:35.870 --> 00:17:38.740
Having written that
and recognizing

00:17:38.740 --> 00:17:42.100
that we have Fourier
stuff coming up

00:17:42.100 --> 00:17:48.860
in the last third of the course,
where we have complex numbers.

00:17:48.860 --> 00:17:57.220
I have to say that if when
I have complex vectors,

00:17:57.220 --> 00:18:01.080
do you know about those?

00:18:01.080 --> 00:18:04.610
The dot product, or
the length squared,

00:18:04.610 --> 00:18:09.450
if this was a vector of complex,
with possibly complex numbers,

00:18:09.450 --> 00:18:12.060
I wouldn't take the
length squared just

00:18:12.060 --> 00:18:14.390
by adding up these squares.

00:18:14.390 --> 00:18:19.410
Suppose my-- Yes, I'm really
anticipating weeks ahead,

00:18:19.410 --> 00:18:23.230
but suppose my
vector was [1,  i].

00:18:23.230 --> 00:18:27.790
What's the length of
that particular vector v?

00:18:27.790 --> 00:18:31.620
Well, if I do v transpose
v, what do I get?

00:18:31.620 --> 00:18:36.770
For v equals [1, i], what does
v transpose v turn out to be?

00:18:36.770 --> 00:18:37.880
Zero.

00:18:37.880 --> 00:18:40.300
One squared plus
i squared is zero.

00:18:40.300 --> 00:18:41.570
No good.

00:18:41.570 --> 00:18:45.060
So obviously some rule
has to change a little bit

00:18:45.060 --> 00:18:46.780
to get the correct number.

00:18:46.780 --> 00:18:51.660
The correct length squared,
I would rather expect two.

00:18:51.660 --> 00:18:54.610
The size of that squared plus
the size of that squared.

00:18:54.610 --> 00:18:56.315
So I don't want to
square i, I want

00:18:56.315 --> 00:18:58.410
to square its absolute value.

00:18:58.410 --> 00:19:01.930
And the way to do
that is conjugate one

00:19:01.930 --> 00:19:03.920
of the two things.

00:19:03.920 --> 00:19:08.710
Now I'm taking [1, i], and on
the other side I have [1, -i]

00:19:08.710 --> 00:19:11.660
and that gives me
the two that I want.

00:19:11.660 --> 00:19:17.740
So what I'm doing, when
I've got complex vectors

00:19:17.740 --> 00:19:21.000
then I would really
do that, and now that

00:19:21.000 --> 00:19:23.190
is not the same as that.

00:19:23.190 --> 00:19:24.241
Right.

00:19:24.241 --> 00:19:24.740
Yeah.

00:19:24.740 --> 00:19:30.720
If in one case if I'm
doing the conjugate of v

00:19:30.720 --> 00:19:32.730
and the other case it's
the conjugate of w,

00:19:32.730 --> 00:19:35.880
then I've got a
complex conjugate.

00:19:35.880 --> 00:19:40.830
OK, that's a throwaway
comment that just is relevant

00:19:40.830 --> 00:19:47.260
because it keeps us focused
for a moment on the real case,

00:19:47.260 --> 00:19:52.640
where we do have equals.

00:19:52.640 --> 00:19:55.960
Now, I don't know if that
was sufficient answer?

00:19:55.960 --> 00:19:58.520
It wasn't a complete
answer because I

00:19:58.520 --> 00:20:03.740
didn't do the manipulations,
but the solutions

00:20:03.740 --> 00:20:06.150
posted should show you those.

00:20:06.150 --> 00:20:09.360
And, of course, you can organize
them a little different.

00:20:09.360 --> 00:20:11.760
OK, good for that one.

00:20:11.760 --> 00:20:12.410
Yes, please.

00:20:12.410 --> 00:20:16.450
AUDIENCE: [INAUDIBLE]

00:20:16.450 --> 00:20:18.410
PROFESSOR STRANG: The
other two terms here?

00:20:18.410 --> 00:20:22.870
AUDIENCE: [INAUDIBLE]

00:20:22.870 --> 00:20:25.970
PROFESSOR STRANG: Yes.

00:20:25.970 --> 00:20:28.810
You couldn't cancel them.

00:20:28.810 --> 00:20:34.710
Well, I recommend just, that's
how the best mathematics is

00:20:34.710 --> 00:20:35.320
done, right?

00:20:35.320 --> 00:20:39.200
You want zero,
you just X it out.

00:20:39.200 --> 00:20:40.700
Anyway.

00:20:40.700 --> 00:20:44.160
Let me leave the posted
solutions to be a hint

00:20:44.160 --> 00:20:46.140
and come back to it.

00:20:46.140 --> 00:20:47.390
Yeah.

00:20:47.390 --> 00:20:52.150
AUDIENCE: [INAUDIBLE]

00:20:52.150 --> 00:20:54.000
PROFESSOR STRANG:
The next problem was?

00:20:54.000 --> 00:20:56.850
AUDIENCE: [INAUDIBLE]

00:20:56.850 --> 00:20:58.200
PROFESSOR STRANG: Oh, yes.

00:20:58.200 --> 00:21:01.010
OK, that one I spoke a
little bit about, but now

00:21:01.010 --> 00:21:08.850
let me read from the
problem set that I got.

00:21:08.850 --> 00:21:12.870
I noticed that was quite brief.

00:21:12.870 --> 00:21:16.510
Oh, to find that actual angle?

00:21:16.510 --> 00:21:20.170
Somehow that's a little
interesting, isn't it?

00:21:20.170 --> 00:21:24.660
AUDIENCE: [INAUDIBLE]

00:21:24.660 --> 00:21:28.160
PROFESSOR STRANG: To tell the
truth, I meant numerically.

00:21:28.160 --> 00:21:33.500
I meant, what's the
point of that question.

00:21:33.500 --> 00:21:38.750
The point is we're trying to
solve, this isn't a big deal.

00:21:38.750 --> 00:21:44.530
But it was just if we're
using this trapezoidal method,

00:21:44.530 --> 00:21:48.620
the beauty of that, exactly
what our thing proves,

00:21:48.620 --> 00:21:54.170
is-- Here the constant
energy surface is the circle.

00:21:54.170 --> 00:21:58.310
The point of the trapezoidal
method for this simple equation

00:21:58.310 --> 00:22:07.560
u''+u=0, which amounted to the
equation uv' equals something

00:22:07.560 --> 00:22:17.280
like, our a matrix
was antisymmetric.

00:22:17.280 --> 00:22:19.750
So it fit perfectly
in that problem,

00:22:19.750 --> 00:22:23.010
and if we started on the
circle we stay on the circle.

00:22:23.010 --> 00:22:28.400
And if I take 32 steps I come
back pretty closely to here,

00:22:28.400 --> 00:22:33.600
and I just thought it might be
fun to figure out numerically,

00:22:33.600 --> 00:22:37.570
with MATLAB or a
calculator or something,

00:22:37.570 --> 00:22:42.260
we take an angle, theta, is that
what the problem asks, and then

00:22:42.260 --> 00:22:46.800
come around here to
32 theta, and 32 theta

00:22:46.800 --> 00:22:49.160
will not be exactly 2pi.

00:22:49.160 --> 00:22:51.000
But darned close.

00:22:51.000 --> 00:22:53.750
Because you could see in
the figure in the book

00:22:53.750 --> 00:22:56.230
it wasn't too far off.

00:22:56.230 --> 00:22:59.030
So the question was,
what is that theta?

00:22:59.030 --> 00:23:02.870
So I think that the
formula turned out

00:23:02.870 --> 00:23:09.070
to be that each step multiplied
by this one plus i delta t,

00:23:09.070 --> 00:23:15.330
or h, on two divided by one
minus i delta t over two.

00:23:15.330 --> 00:23:21.590
And when you plug in
delta t to be 2pi over 32,

00:23:21.590 --> 00:23:26.100
so that's the, what
did I say, that's

00:23:26.100 --> 00:23:28.260
the tangent of
theta or something?

00:23:28.260 --> 00:23:32.330
Sorry, I've forgotten the
way the problem was put.

00:23:32.330 --> 00:23:38.220
Oh, it's e to the i theta, yeah.

00:23:38.220 --> 00:23:41.350
What's the main point
about that complex number?

00:23:41.350 --> 00:23:42.970
When you look at
that complex number

00:23:42.970 --> 00:23:47.950
what's the most
essential thing you see?

00:23:47.950 --> 00:23:49.296
That it, yeah, tell me again.

00:23:49.296 --> 00:23:50.170
AUDIENCE: [INAUDIBLE]

00:23:50.170 --> 00:23:52.320
PROFESSOR STRANG:
Magnitude one, great.

00:23:52.320 --> 00:23:55.230
It's a number divided by
its complex conjugate,

00:23:55.230 --> 00:23:57.270
so it's a number
of magnitude one.

00:23:57.270 --> 00:24:01.180
And now tell me, if you see
a complex number of magnitude

00:24:01.180 --> 00:24:03.380
one, what jumps to mind?

00:24:03.380 --> 00:24:06.950
What form do you
naturally put it in?

00:24:06.950 --> 00:24:07.450
e^(i*theta).

00:24:10.540 --> 00:24:15.117
Every complex number of absolute
value one is just beautifully

00:24:15.117 --> 00:24:16.450
written in the form e^(i*theta).

00:24:19.780 --> 00:24:24.070
That complex number is some
point on the unit circle,

00:24:24.070 --> 00:24:24.790
so there it is.

00:24:24.790 --> 00:24:28.360
Right there, there
it is, e^(i*theta).

00:24:28.360 --> 00:24:31.870
With that-- theta
is negative there,

00:24:31.870 --> 00:24:33.670
because we're going
the wrong way.

00:24:33.670 --> 00:24:35.940
No big deal.

00:24:35.940 --> 00:24:38.510
Maybe here theta's positive.

00:24:38.510 --> 00:24:43.660
I've forgotten, so
I won't try to go

00:24:43.660 --> 00:24:47.450
either the clockwise or the
counterclockwise way around.

00:24:47.450 --> 00:24:51.070
So, if I wanted to figure out
what theta was and plugged

00:24:51.070 --> 00:24:54.260
in these things, let's see.

00:24:54.260 --> 00:25:03.590
So that's pi over 32, delta t
over two would be pi over 32,

00:25:03.590 --> 00:25:09.640
and this guy would be its
conjugate. pi over 32,

00:25:09.640 --> 00:25:12.670
and then in this
solution that'll

00:25:12.670 --> 00:25:23.580
be plugged on the homework
this will be, I think maybe,

00:25:23.580 --> 00:25:30.040
maybe the theta comes out to
be, it's kind of cool actually,

00:25:30.040 --> 00:25:37.000
twice the arc tangent of
pi over 32 or something.

00:25:37.000 --> 00:25:38.130
I didn't know that.

00:25:38.130 --> 00:25:44.200
But that'll be in the
solutions for you to check.

00:25:44.200 --> 00:25:50.440
So now, why do I like
e^(i*theta) so much?

00:25:50.440 --> 00:25:53.470
Because now I could tell
you what this point is,

00:25:53.470 --> 00:25:56.870
after you've done it 32 times.

00:25:56.870 --> 00:25:59.330
What angle have you reached?

00:25:59.330 --> 00:26:05.520
This is the crunch line of using
complex numbers, e^(i*theta),

00:26:05.520 --> 00:26:09.950
is that they're absolutely
great for taking powers.

00:26:09.950 --> 00:26:15.450
If I take the 32nd power of
x plus iy, I'm lost, right.

00:26:15.450 --> 00:26:19.890
x plus iy to the 32nd
power starts out x^32,

00:26:19.890 --> 00:26:24.600
ends up i^32 y^32, with
horrible stuff in between.

00:26:24.600 --> 00:26:30.187
But what is the 32nd
power of e^(i*theta)?

00:26:30.187 --> 00:26:30.770
e^(i*32theta).

00:26:33.740 --> 00:26:37.160
Just that angle 32 times
exactly as we've drawn it.

00:26:37.160 --> 00:26:39.300
So that's the point
e^(i*32theta).

00:26:42.250 --> 00:26:49.790
OK, and therefore if we now
know what theta is, so yeah.

00:26:49.790 --> 00:26:55.900
So it must be pretty near
2pi, but not exactly.

00:26:55.900 --> 00:26:59.750
I guess that's about right.

00:26:59.750 --> 00:27:03.200
In fact, having got
this far, the tangent

00:27:03.200 --> 00:27:09.210
of a very small angle
is approximately what?

00:27:09.210 --> 00:27:11.470
It's approximately
the angle, right?

00:27:11.470 --> 00:27:14.760
The sine of a very small angle
is approximately the angle.

00:27:14.760 --> 00:27:16.910
The cosine is approximately one.

00:27:16.910 --> 00:27:19.090
The tangent is
approximately the angle.

00:27:19.090 --> 00:27:25.380
So this, 32 theta,
is 32 times two times

00:27:25.380 --> 00:27:28.180
approximately the angle.

00:27:28.180 --> 00:27:33.760
And what answer do you get?

00:27:33.760 --> 00:27:35.250
2pi.

00:27:35.250 --> 00:27:38.820
Which makes sense.

00:27:38.820 --> 00:27:47.430
So you could say what the
trapezoidal method has done

00:27:47.430 --> 00:27:52.390
is to replace the exact
angle by the inverse tangent

00:27:52.390 --> 00:27:53.810
approximately.

00:27:53.810 --> 00:27:56.090
That's sort of nice.

00:27:56.090 --> 00:27:59.090
In this example you
can get as far as that

00:27:59.090 --> 00:28:03.220
and you could actually
find out how near that is.

00:28:03.220 --> 00:28:06.540
And, by the way, how near
would I expect it to be?

00:28:06.540 --> 00:28:11.450
I would expect it,
so what do we know

00:28:11.450 --> 00:28:16.190
about the trapezoidal method
without having proved it?

00:28:16.190 --> 00:28:19.530
It's second order
accurate, right?

00:28:19.530 --> 00:28:21.340
If it was first
order accurate, I

00:28:21.340 --> 00:28:26.120
would expect it to miss by
something of the size of theta.

00:28:26.120 --> 00:28:28.380
Maybe a fraction of theta.

00:28:28.380 --> 00:28:30.150
But being second
order accurate, I'm

00:28:30.150 --> 00:28:31.890
expecting it to
miss by something

00:28:31.890 --> 00:28:34.820
of size theta squared.

00:28:34.820 --> 00:28:40.190
So it would be pretty
near zero, right.

00:28:40.190 --> 00:28:47.080
And actually, another way I
know it's around-- So the error

00:28:47.080 --> 00:28:49.870
would be something
like, it would have a 32

00:28:49.870 --> 00:28:51.740
squared in the denominator.

00:28:51.740 --> 00:28:59.660
And I've just thought of
another way to see that.

00:28:59.660 --> 00:29:02.220
We just said that the
first term in the arc

00:29:02.220 --> 00:29:05.750
tangent of a small angle,
theta, of a small angle,

00:29:05.750 --> 00:29:10.150
alpha, whatever that is, pi over
32, the first term in the arc

00:29:10.150 --> 00:29:11.490
tangent is?

00:29:11.490 --> 00:29:12.560
The angle.

00:29:12.560 --> 00:29:14.310
That's what we just said.

00:29:14.310 --> 00:29:16.520
Then, do you know
what would come next?

00:29:16.520 --> 00:29:20.890
Now we're looking at the error.

00:29:20.890 --> 00:29:27.300
So that of a very small
angle will start theta,

00:29:27.300 --> 00:29:30.460
and I want to ask you about
how many theta squareds

00:29:30.460 --> 00:29:32.810
and theta cubes.

00:29:32.810 --> 00:29:39.300
You're seeing what you can do
with paper and pencils type

00:29:39.300 --> 00:29:41.520
stuff.

00:29:41.520 --> 00:29:46.999
Here's my main question, how
many theta squareds in there?

00:29:46.999 --> 00:29:48.040
You want to make a guess?

00:29:48.040 --> 00:29:51.960
A mathematician's
favorite number, zero.

00:29:51.960 --> 00:29:55.820
Right, there will be no
theta squared terms in.

00:29:55.820 --> 00:29:59.860
That's an odd function, so
I'm expecting only odd powers

00:29:59.860 --> 00:30:02.190
and therefore I won't
be surprised to see

00:30:02.190 --> 00:30:04.230
theta cubed come up first.

00:30:04.230 --> 00:30:07.170
And then when I
multiply by the 32,

00:30:07.170 --> 00:30:14.890
I get the theta squared that
I guessed we would have.

00:30:14.890 --> 00:30:22.040
OK, once again I'll stop there
because that's very narrow path

00:30:22.040 --> 00:30:26.070
to be following but
it shows you how.

00:30:26.070 --> 00:30:28.760
You know, there's
a lot of room still

00:30:28.760 --> 00:30:31.440
for what you can do
with paper and pencil

00:30:31.440 --> 00:30:35.320
to understand a model problem.

00:30:35.320 --> 00:30:38.030
And then the computer
would tell us

00:30:38.030 --> 00:30:42.510
about a serious problem of
following the solar system

00:30:42.510 --> 00:30:44.560
for a million years.

00:30:44.560 --> 00:30:48.740
OK, another totally
different question, if I can.

00:30:48.740 --> 00:30:49.510
Yes, thank you.

00:30:49.510 --> 00:30:50.760
AUDIENCE: [INAUDIBLE]

00:30:50.760 --> 00:30:51.968
PROFESSOR STRANG: Yeah, sure.

00:30:51.968 --> 00:30:53.020
AUDIENCE: [INAUDIBLE]

00:30:53.020 --> 00:30:56.000
PROFESSOR STRANG: 2.4.1, right.

00:30:56.000 --> 00:30:57.110
A mistake in the book.

00:30:57.110 --> 00:30:59.010
AUDIENCE: [INAUDIBLE]

00:30:59.010 --> 00:31:01.300
PROFESSOR STRANG:
Or in the, yeah.

00:31:01.300 --> 00:31:04.230
It's quite possible.

00:31:04.230 --> 00:31:12.680
OK, there's a printed
error in the graph.

00:31:12.680 --> 00:31:13.200
Yeah.

00:31:13.200 --> 00:31:18.380
So in numbering the
edges, well let's

00:31:18.380 --> 00:31:19.800
blame it on the printer, right?

00:31:19.800 --> 00:31:20.840
Not the author.

00:31:20.840 --> 00:31:26.190
OK, so the diagonal edge,
that five probably was

00:31:26.190 --> 00:31:28.950
intended to be a three, yeah.

00:31:28.950 --> 00:31:29.940
Thank you.

00:31:29.940 --> 00:31:35.840
So we'll catch that
in the next printing.

00:31:35.840 --> 00:31:42.000
And you recognize that always,
numbering the edges and nodes

00:31:42.000 --> 00:31:43.810
is a pretty
arbitrary thing, it's

00:31:43.810 --> 00:31:48.020
just if we number differently
that just reorders

00:31:48.020 --> 00:31:50.040
the rows of the matrix.

00:31:50.040 --> 00:31:51.580
If we number the
edge differently,

00:31:51.580 --> 00:31:55.950
it'll reorder the rows and
it'll reorder rows and columns

00:31:55.950 --> 00:31:59.470
of A transpose A. So it won't
make a serious difference

00:31:59.470 --> 00:32:00.250
in the matrix.

00:32:00.250 --> 00:32:00.750
Yeah.

00:32:00.750 --> 00:32:02.932
AUDIENCE: [INAUDIBLE]

00:32:02.932 --> 00:32:05.140
PROFESSOR STRANG: Do you
want to go back to this guy?

00:32:05.140 --> 00:32:06.400
OK.

00:32:06.400 --> 00:32:10.291
AUDIENCE: So if you have
an anti-symmetric matrix,

00:32:10.291 --> 00:32:12.166
does it follow that the
eigenvectors used are

00:32:12.166 --> 00:32:12.749
perpendicular?

00:32:12.749 --> 00:32:16.470
PROFESSOR STRANG: This
is a good question.

00:32:16.470 --> 00:32:20.910
This guy, way up here,
with this property,

00:32:20.910 --> 00:32:22.860
AUDIENCE: The eigenvectors
are perpendicular?

00:32:22.860 --> 00:32:25.068
PROFESSOR STRANG: The
eigenvectors are perpendicular.

00:32:25.068 --> 00:32:26.610
Yes, yeah.

00:32:26.610 --> 00:32:32.820
So we have, there's this, like,
the nobility among matrices

00:32:32.820 --> 00:32:35.610
are the ones with
perpendicular eigenvectors.

00:32:35.610 --> 00:32:39.100
So that includes
symmetric matrices,

00:32:39.100 --> 00:32:43.980
this is a good and
straightforward point.

00:32:43.980 --> 00:32:46.800
So these are the good matrices.

00:32:46.800 --> 00:32:48.990
Symmetric matrices.

00:32:48.990 --> 00:32:53.160
A transpose equals
A. Their eigenvalues

00:32:53.160 --> 00:32:56.840
lie on the real line.

00:32:56.840 --> 00:33:04.630
And these are all
perpendicular eigenvectors.

00:33:04.630 --> 00:33:08.220
What about antisymmetric?

00:33:08.220 --> 00:33:12.030
OK, that means A
transpose is minus A.

00:33:12.030 --> 00:33:16.320
They also fall in this
noble family of matrices,

00:33:16.320 --> 00:33:19.130
and where are their eigenvalues?

00:33:19.130 --> 00:33:22.140
Pure imaginary, right up here.

00:33:22.140 --> 00:33:26.120
Now do you want to know,
who else is in this family?

00:33:26.120 --> 00:33:28.150
What's the other, this
is the complex plane;

00:33:28.150 --> 00:33:30.480
there's one more piece
of the complex plane

00:33:30.480 --> 00:33:32.240
that you know I'm going to put.

00:33:32.240 --> 00:33:35.090
Which is?

00:33:35.090 --> 00:33:39.200
What else to make that
complex plane look familiar,

00:33:39.200 --> 00:33:43.080
it's going to have
the unit circle.

00:33:43.080 --> 00:33:46.260
Every complex plane has got
to have the unit circle.

00:33:46.260 --> 00:33:51.510
OK, so these guys went with
the, and now what do you think

00:33:51.510 --> 00:33:55.640
goes with the, this will
be the matrices-- Can

00:33:55.640 --> 00:33:59.230
I call them Q instead, because
I called them Q this morning.

00:33:59.230 --> 00:34:02.340
Q transpose is Q inverse.

00:34:02.340 --> 00:34:05.990
Q transpose Q, and they're
the orthogonal matrices.

00:34:05.990 --> 00:34:10.380
So those matrices again,
beautiful matrices

00:34:10.380 --> 00:34:11.580
in the best class.

00:34:11.580 --> 00:34:15.940
And their eigenvalues
are on the unit circle.

00:34:15.940 --> 00:34:17.980
And that would be--

00:34:17.980 --> 00:34:20.370
Why don't I just show you why?

00:34:20.370 --> 00:34:22.410
Because orthogonal
matrices, there

00:34:22.410 --> 00:34:25.440
are not so many that are
really worth knowing.

00:34:25.440 --> 00:34:31.440
So, let me take Qx=lambda*x,
and what is it that I want

00:34:31.440 --> 00:34:33.550
to prove?

00:34:33.550 --> 00:34:38.290
I want to prove that the
eigenvalues have absolute value

00:34:38.290 --> 00:34:39.090
one.

00:34:39.090 --> 00:34:40.600
That's the unit circle.

00:34:40.600 --> 00:34:42.760
So how do I show
that the eigenvalues

00:34:42.760 --> 00:34:45.080
have absolute value of one?

00:34:45.080 --> 00:34:49.960
Let me take the dot
product with Qx transpose.

00:34:49.960 --> 00:34:56.820
So both sides, I'll do Qx
transpose Qx and I'll do

00:34:56.820 --> 00:35:01.510
lambda*x transpose
lambda*x, right?

00:35:01.510 --> 00:35:04.080
Only these are complex.

00:35:04.080 --> 00:35:08.610
I've got to take complex stuff.

00:35:08.610 --> 00:35:10.700
OK.

00:35:10.700 --> 00:35:17.240
I just took the length squared
of both sides, and kept in mind

00:35:17.240 --> 00:35:19.430
the possibility that
this x and lambda

00:35:19.430 --> 00:35:22.500
could be, and probably
will be, complex numbers.

00:35:22.500 --> 00:35:25.760
Now, what do I have on the left?

00:35:25.760 --> 00:35:28.060
Do you see it happening?

00:35:28.060 --> 00:35:31.150
I get an x bar
transpose, what do I get?

00:35:31.150 --> 00:35:37.630
Q transpose Qx on the left side.

00:35:37.630 --> 00:35:40.540
That's the combination
I'm looking for.

00:35:40.540 --> 00:35:42.500
For an orthogonal matrix.

00:35:42.500 --> 00:35:45.290
Let's imagine the
matrix itself is real,

00:35:45.290 --> 00:35:48.800
otherwise I would
just conjugate it.

00:35:48.800 --> 00:35:53.960
What's nice about
that left side?

00:35:53.960 --> 00:35:57.840
What fact am I going
to use about Q?

00:35:57.840 --> 00:36:00.340
Q transpose Q is the identity.

00:36:00.340 --> 00:36:04.100
So this thing is nothing
but x bar transpose x.

00:36:04.100 --> 00:36:05.700
That's the length of x squared.

00:36:05.700 --> 00:36:07.920
What have I got
on the right side?

00:36:07.920 --> 00:36:11.830
I've got the length of x
squared times a number,

00:36:11.830 --> 00:36:16.550
lambda bar times lambda squared.

00:36:16.550 --> 00:36:17.980
It's there, now.

00:36:17.980 --> 00:36:20.862
On the left side I have
the length of x squared.

00:36:20.862 --> 00:36:22.320
On the right side
I have the length

00:36:22.320 --> 00:36:25.340
of x squared times that
number, mod lambda squared.

00:36:25.340 --> 00:36:27.300
Therefore, that
number has to be one

00:36:27.300 --> 00:36:30.750
and the eigenvalues
are on the unit circle.

00:36:30.750 --> 00:36:37.300
So, I've given you the three big
important classes of matrices

00:36:37.300 --> 00:36:41.790
with perpendicular eigenvectors.

00:36:41.790 --> 00:36:44.820
I think anybody
would wonder, OK,

00:36:44.820 --> 00:36:48.840
what about other eigenvalues.

00:36:48.840 --> 00:36:52.210
What's the condition for
perpendicular eigenvectors

00:36:52.210 --> 00:36:53.890
that includes this.

00:36:53.890 --> 00:36:55.080
And includes this.

00:36:55.080 --> 00:36:57.770
And includes this,
and also allows

00:36:57.770 --> 00:37:01.460
eigenvalues all over the place.

00:37:01.460 --> 00:37:04.310
Would you like to
know that condition?

00:37:04.310 --> 00:37:06.650
What the heck.

00:37:06.650 --> 00:37:11.660
That condition, that
includes all these

00:37:11.660 --> 00:37:17.620
is this, that A transpose times
A equals A times A transpose.

00:37:17.620 --> 00:37:22.580
That's the test for
perpendicular eigenvectors.

00:37:22.580 --> 00:37:26.800
A transpose commutes with A.
So this passes, of course.

00:37:26.800 --> 00:37:28.420
This passes, of course.

00:37:28.420 --> 00:37:31.450
This passes because both
sides are the identity,

00:37:31.450 --> 00:37:34.880
and then some more
matrices pass also.

00:37:34.880 --> 00:37:36.764
OK.

00:37:36.764 --> 00:37:37.430
Is that alright?

00:37:37.430 --> 00:37:41.080
You asked for some linear
algebra and you got it.

00:37:41.080 --> 00:37:42.880
Now I'm ready, yes, thanks.

00:37:42.880 --> 00:37:45.820
AUDIENCE: [INAUDIBLE]

00:37:45.820 --> 00:37:50.560
PROFESSOR STRANG: 2.4.19.

00:37:50.560 --> 00:37:52.200
Oh, let me look.

00:37:52.200 --> 00:37:54.680
2.4.19.

00:37:54.680 --> 00:37:59.600
Ah.

00:37:59.600 --> 00:38:06.680
OK, yes, sorry and I should
have done better with that.

00:38:06.680 --> 00:38:23.640
So one graphs that are
important are grids like this.

00:38:23.640 --> 00:38:28.950
And we'll see them-- Two, three,
four, one, two, three, four.

00:38:28.950 --> 00:38:32.970
That would be where--
These are the nodes.

00:38:32.970 --> 00:38:38.510
So this is a grid.

00:38:38.510 --> 00:38:41.400
I meant to draw them all
in, but I won't. n squared.

00:38:41.400 --> 00:38:47.690
So n is six, and
I'd have 36 nodes.

00:38:47.690 --> 00:38:50.840
And you can see
the edges in there.

00:38:50.840 --> 00:38:53.480
So that's the graph
I have in mind.

00:38:53.480 --> 00:38:56.510
And the reason that
problem is there

00:38:56.510 --> 00:39:02.050
is that last year, I
think it was last year

00:39:02.050 --> 00:39:06.600
or the year before, we spent
some time with figuring out

00:39:06.600 --> 00:39:11.320
resistances and currents and
so on for these problems.

00:39:11.320 --> 00:39:15.170
And we needed some
fast way to generate A,

00:39:15.170 --> 00:39:19.870
because this matrix A is now,
what size is the matrix A?

00:39:19.870 --> 00:39:23.310
It's got, I don't know how
many edges does it have?

00:39:23.310 --> 00:39:28.610
One, two, three, four, five,
maybe 30 edges going across

00:39:28.610 --> 00:39:29.840
and 30 coming down.

00:39:29.840 --> 00:39:38.570
It'll be 60 by how many
columns in this matrix?

00:39:38.570 --> 00:39:40.330
You know the answer
now and it's worth

00:39:40.330 --> 00:39:42.550
knowing, for the quiz of course.

00:39:42.550 --> 00:39:44.170
36.

00:39:44.170 --> 00:39:48.500
OK.

00:39:48.500 --> 00:39:55.290
Anyway, the class rebelled
about creating these matrices

00:39:55.290 --> 00:40:03.180
and working with the matrices,
with 2,160 non-zeroes.

00:40:03.180 --> 00:40:04.970
People were dropping the course.

00:40:04.970 --> 00:40:10.100
So we needed a command that
would create A pretty quickly.

00:40:10.100 --> 00:40:14.500
And so that's what
the book, and so this

00:40:14.500 --> 00:40:17.870
was like the 18.085 command.

00:40:17.870 --> 00:40:21.030
After we stumbled
around for a while,

00:40:21.030 --> 00:40:24.730
we discovered that
a MATLAB command

00:40:24.730 --> 00:40:31.060
called kron was a quick
way to create the matrix.

00:40:31.060 --> 00:40:38.160
We'll see that when
we get to that point.

00:40:38.160 --> 00:40:40.620
This is an important graph.

00:40:40.620 --> 00:40:44.840
And it's closely connected
to Laplace's-- You remember

00:40:44.840 --> 00:40:45.820
Laplace's--?

00:40:45.820 --> 00:40:47.450
I'll just tell you.

00:40:47.450 --> 00:40:52.590
Laplace's equation is this,
you have a second x derivative,

00:40:52.590 --> 00:40:54.380
we know how to deal with those.

00:40:54.380 --> 00:40:59.750
But it also has a
second y derivative.

00:40:59.750 --> 00:41:02.650
So I'm really looking ahead
at the most important equation

00:41:02.650 --> 00:41:05.970
of Chapter 3,
Laplace's equation.

00:41:05.970 --> 00:41:11.260
And suppose I use
finite differences.

00:41:11.260 --> 00:41:16.700
I want a matrix K_(2D) that
deals with this 2D problem.

00:41:16.700 --> 00:41:19.140
And let me just say
what it would be.

00:41:19.140 --> 00:41:21.430
At a typical point
this x derivative

00:41:21.430 --> 00:41:25.100
is giving me a minus one,
a two and a minus one.

00:41:25.100 --> 00:41:27.790
And the y derivative is
giving me a minus one

00:41:27.790 --> 00:41:34.530
that moves this guy up
to four and minus one.

00:41:34.530 --> 00:41:39.220
So instead of -1, 2,
-1 along a typical row,

00:41:39.220 --> 00:41:43.390
we'll now have a four on the
diagonal and four minus one

00:41:43.390 --> 00:41:45.600
in a certain pattern.

00:41:45.600 --> 00:41:46.100
Anyway.

00:41:46.100 --> 00:41:50.590
You'll see that, it's
interesting when we get to it.

00:41:50.590 --> 00:41:54.950
That would show up in A
transpose A. So what I've said

00:41:54.950 --> 00:42:00.110
here is what happens with
A transpose A. I guess

00:42:00.110 --> 00:42:05.080
I'm hoping that you begin to
know these matrices, first

00:42:05.080 --> 00:42:07.140
seeing them occasionally
in homeworks

00:42:07.140 --> 00:42:10.950
and then in the lecture.

00:42:10.950 --> 00:42:11.490
Good.

00:42:11.490 --> 00:42:15.040
But that's looking ahead.

00:42:15.040 --> 00:42:19.140
I needed some questions
that are like,

00:42:19.140 --> 00:42:21.760
close to, really
on what we're doing

00:42:21.760 --> 00:42:24.210
or what the quiz would do.

00:42:24.210 --> 00:42:25.620
Any - thank you.

00:42:25.620 --> 00:42:30.680
AUDIENCE: [INAUDIBLE]

00:42:30.680 --> 00:42:33.030
PROFESSOR STRANG: Oh yes.

00:42:33.030 --> 00:42:35.110
A little bit.

00:42:35.110 --> 00:42:39.180
OK, yeah.

00:42:39.180 --> 00:42:44.410
So I wrote down this
equation and what I'm writing

00:42:44.410 --> 00:42:47.300
right there is the new part.

00:42:47.300 --> 00:42:51.780
Sort of new, and I guess--
Equals some right hand side

00:42:51.780 --> 00:42:52.950
f(x).

00:42:52.950 --> 00:42:59.570
And the discrete version
will be an A transpose

00:42:59.570 --> 00:43:02.520
C A equal a vector
f, maybe there

00:43:02.520 --> 00:43:05.910
will be a delta x squared here.

00:43:05.910 --> 00:43:07.000
OK.

00:43:07.000 --> 00:43:11.010
I guess maybe, I
don't want to go far

00:43:11.010 --> 00:43:16.500
but I want you to see that if
we have a coefficient c in here

00:43:16.500 --> 00:43:20.350
it should show up there.

00:43:20.350 --> 00:43:26.260
You could actually, it might be
reasonable to look at 3.1 just

00:43:26.260 --> 00:43:30.550
to look slightly ahead
to see the parallels,

00:43:30.550 --> 00:43:37.950
but you would get them right
without a lecture on it.

00:43:37.950 --> 00:43:41.830
Your coefficient shows up in
the differential equation,

00:43:41.830 --> 00:43:45.900
and it shows up on the
diagonal of C in the difference

00:43:45.900 --> 00:43:49.500
equation.

00:43:49.500 --> 00:43:51.130
I won't give a whole
lecture on that,

00:43:51.130 --> 00:44:01.330
just to say that correspondence
is exactly the one we know.

00:44:01.330 --> 00:44:05.730
A is a difference matrix.

00:44:05.730 --> 00:44:07.340
Like the derivative.

00:44:07.340 --> 00:44:10.080
C will be a diagonal
matrix, A transpose

00:44:10.080 --> 00:44:14.170
will be whatever
that comes out to be.

00:44:14.170 --> 00:44:17.010
And so you've seen
A transpose A,

00:44:17.010 --> 00:44:19.120
but think again about
that difference--

00:44:19.120 --> 00:44:22.540
And ask yourselves this.

00:44:22.540 --> 00:44:27.430
I suggest, take c to be
one, get c out of there.

00:44:27.430 --> 00:44:33.880
And just think, again, what is
the difference matrix A with

00:44:33.880 --> 00:44:38.610
a boundary either
fixed-fixed or fixed-free,

00:44:38.610 --> 00:44:40.420
those will be two different A's.

00:44:40.420 --> 00:44:53.960
What are the A's for
fixed-fixed and for fixed-free?

00:44:53.960 --> 00:44:57.080
This is what we were doing
at the very beginning

00:44:57.080 --> 00:44:59.920
of the course.

00:44:59.920 --> 00:45:03.230
So A is a first
difference matrix,

00:45:03.230 --> 00:45:05.890
and A transpose A will
be the second difference.

00:45:05.890 --> 00:45:12.550
So the A transpose A, of course,
I was doing A transpose A,

00:45:12.550 --> 00:45:15.960
then the answer here
would be the matrix K

00:45:15.960 --> 00:45:21.850
and the answer here would
be the matrix T. Or,

00:45:21.850 --> 00:45:26.490
depending which end is free,
but we'd have one change.

00:45:26.490 --> 00:45:30.290
That's A transpose A,
but now think about the A

00:45:30.290 --> 00:45:32.420
that it came from.

00:45:32.420 --> 00:45:38.640
So A will be, A
is the matrix that

00:45:38.640 --> 00:45:40.890
takes differences
of the u's, and then

00:45:40.890 --> 00:45:45.570
A transpose A takes
second differences.

00:45:45.570 --> 00:45:47.170
Of all the questions
asked, this is

00:45:47.170 --> 00:45:53.580
the one most
relevant for the exam

00:45:53.580 --> 00:45:57.050
and for what we've done so far.

00:45:57.050 --> 00:46:01.130
I've gone off onto topics
that we look ahead to,

00:46:01.130 --> 00:46:03.210
but this is where we are.

00:46:03.210 --> 00:46:07.000
So that matrix A is a
first difference matrix,

00:46:07.000 --> 00:46:11.120
and then you put in the
boundary conditions.

00:46:11.120 --> 00:46:12.405
OK.

00:46:12.405 --> 00:46:13.530
There was another question.

00:46:13.530 --> 00:46:14.030
Yeah.

00:46:14.030 --> 00:46:18.080
AUDIENCE: [INAUDIBLE]

00:46:18.080 --> 00:46:19.870
PROFESSOR STRANG:
Of number four?

00:46:19.870 --> 00:46:22.910
Which number four in which?

00:46:22.910 --> 00:46:23.770
Oh, in the quiz.

00:46:23.770 --> 00:46:25.810
Oh yes, right.

00:46:25.810 --> 00:46:27.050
Yes.

00:46:27.050 --> 00:46:28.690
Did I tell you what
problem four was?

00:46:28.690 --> 00:46:29.190
No.

00:46:29.190 --> 00:46:31.740
I hope not.

00:46:31.740 --> 00:46:39.130
OK problem four in the quiz.

00:46:39.130 --> 00:46:41.430
It's about a delta function?

00:46:41.430 --> 00:46:43.620
Yeah.

00:46:43.620 --> 00:46:52.870
What do I know, what do
you want me to tell you?

00:46:52.870 --> 00:46:55.540
So the delta, of
course, comes in

00:46:55.540 --> 00:47:00.060
as, we've seen it, as the right
hand side of a differential

00:47:00.060 --> 00:47:01.080
equation.

00:47:01.080 --> 00:47:05.840
So it might be the right-hand
side even of this equation.

00:47:05.840 --> 00:47:20.330
So if this equation was delta
of x, or x-1/2 or something.

00:47:20.330 --> 00:47:23.680
I mean, the essential
thing is that when

00:47:23.680 --> 00:47:32.050
delta's on the right side, that
gives you a drop in the slope.

00:47:32.050 --> 00:47:35.200
Suppose I just have a first
order equation like d --

00:47:35.200 --> 00:47:41.130
I'll call it z --
dz/dx = delta(x-a).

00:47:41.130 --> 00:47:45.210
Yeah.

00:47:45.210 --> 00:47:53.700
And suppose that I know
that z(0) starts at zero.

00:47:53.700 --> 00:47:55.920
You've got to be able
to solve that equation,

00:47:55.920 --> 00:48:02.420
so this is a useful
prep for that.

00:48:02.420 --> 00:48:08.670
That would be a good equation
to know the solution to.

00:48:08.670 --> 00:48:13.770
And what kind of
function is this?

00:48:13.770 --> 00:48:17.140
What kind of a
function is z(x) there?

00:48:17.140 --> 00:48:20.730
I just use the letter
z to have a new letter.

00:48:20.730 --> 00:48:28.960
z(x) will be a step.

00:48:28.960 --> 00:48:33.110
Right. z(x) will be
a step function, yes.

00:48:33.110 --> 00:48:36.860
That's right.

00:48:36.860 --> 00:48:40.210
OK, so the solution
is that at the point

00:48:40.210 --> 00:48:44.230
a, which I'm presuming is beyond
zero, I come along at zero

00:48:44.230 --> 00:48:45.060
and I step up.

00:48:45.060 --> 00:48:45.690
Yep.

00:48:45.690 --> 00:48:46.230
OK.

00:48:46.230 --> 00:48:48.910
That would be a
picture of z(x), yeah.

00:48:48.910 --> 00:48:54.060
So it's basic delta
function material

00:48:54.060 --> 00:49:04.460
that I'm speaking about here.

00:49:04.460 --> 00:49:08.850
So z jumps by one
and if z is a slope,

00:49:08.850 --> 00:49:12.030
then the slope jumps
by one or drops by one,

00:49:12.030 --> 00:49:15.030
depending on a plus
or a minus sign here.

00:49:15.030 --> 00:49:19.370
The things that we've used
to deal with delta functions,

00:49:19.370 --> 00:49:23.170
so that's what
Question 4b is about.

00:49:23.170 --> 00:49:27.440
The drop in slope, or
the jumps, or whatever

00:49:27.440 --> 00:49:31.460
happens when a delta function
shows up on the right side.

00:49:31.460 --> 00:49:33.090
Good question.

00:49:33.090 --> 00:49:33.590
Yep.

00:49:33.590 --> 00:49:36.890
AUDIENCE: [INAUDIBLE]

00:49:36.890 --> 00:49:38.260
PROFESSOR STRANG: 1.1.27.

00:49:38.260 --> 00:49:38.760
Well.

00:49:38.760 --> 00:49:41.460
AUDIENCE: [INAUDIBLE]

00:49:41.460 --> 00:49:44.820
PROFESSOR STRANG: Oh,
and then left a typo.

00:49:44.820 --> 00:49:53.215
AUDIENCE: [INAUDIBLE]

00:49:53.215 --> 00:49:54.090
PROFESSOR STRANG: Oh.

00:49:54.090 --> 00:49:58.500
I'm sorry, OK.

00:49:58.500 --> 00:50:00.620
1.1.27.

00:50:00.620 --> 00:50:05.020
My copy isn't showing it.

00:50:05.020 --> 00:50:07.850
Yeah.

00:50:07.850 --> 00:50:10.330
I may have to punt
on that question.

00:50:10.330 --> 00:50:12.170
Or do you want me to look at it?

00:50:12.170 --> 00:50:18.200
OK, can you maybe just pass
that the book up, alright.

00:50:18.200 --> 00:50:21.020
I'll try to read what
that question was.

00:50:21.020 --> 00:50:23.180
OK.

00:50:23.180 --> 00:50:27.160
Yeah, maybe this is
a question to answer.

00:50:27.160 --> 00:50:33.590
This is probably the one
new question that got added.

00:50:33.590 --> 00:50:39.190
OK, yeah.

00:50:39.190 --> 00:50:41.070
Fair enough.

00:50:41.070 --> 00:50:45.630
So this is continuing
the discussion

00:50:45.630 --> 00:50:50.470
that you asked me to start here
about A, the first difference

00:50:50.470 --> 00:50:50.970
matrix, OK.

00:50:50.970 --> 00:50:57.670
So I'll go a little
more over that.

00:50:57.670 --> 00:51:00.780
So in the book here,
this writes down

00:51:00.780 --> 00:51:10.830
a matrix A_0, which is--
I'll discuss this matrix.

00:51:10.830 --> 00:51:16.210
So there's a difference matrix.

00:51:16.210 --> 00:51:19.110
You see what I mean by
a difference matrix,

00:51:19.110 --> 00:51:25.660
if were to multiply it by
u, [u 0, u 1, u 2, u 3]

00:51:25.660 --> 00:51:28.320
or something, I would
get differences, right?

00:51:28.320 --> 00:51:33.590
I'd get u_1-u_0,
u_2-u_1, and u_3-u_2.

00:51:36.170 --> 00:51:37.420
Good.

00:51:37.420 --> 00:51:39.681
So that's A_0 times u.

00:51:39.681 --> 00:51:40.180
Alright.

00:51:40.180 --> 00:51:45.640
So that's a difference matrix.

00:51:45.640 --> 00:51:47.650
What graph would that come from?

00:51:47.650 --> 00:51:52.190
That's also the incidence
matrix of a very simple graph.

00:51:52.190 --> 00:51:56.340
This is connecting
Chapter 1 with Chapter 2.

00:51:56.340 --> 00:51:59.230
It would be a line of
springs, it would be a graph.

00:51:59.230 --> 00:52:02.840
It's got edges and nodes.

00:52:02.840 --> 00:52:05.890
It's got three edges,
so I've got three rows.

00:52:05.890 --> 00:52:13.120
It's got four nodes so
I've got four columns.

00:52:13.120 --> 00:52:16.500
Are the columns
independent here?

00:52:16.500 --> 00:52:18.090
No, they never are.

00:52:18.090 --> 00:52:23.500
The vector of all ones would
have differences of all zeroes.

00:52:23.500 --> 00:52:26.110
So what would that, that
would be the difference

00:52:26.110 --> 00:52:31.170
matrix for fixed?

00:52:31.170 --> 00:52:33.100
Free?

00:52:33.100 --> 00:52:38.240
Fixed, free, circular what
would that difference matrix

00:52:38.240 --> 00:52:41.350
correspond to?

00:52:41.350 --> 00:52:42.410
Everybody's saying it.

00:52:42.410 --> 00:52:44.630
Say it a little louder just to.

00:52:44.630 --> 00:52:46.020
Free-free.

00:52:46.020 --> 00:52:48.890
That's a free-free problem,
because they're all in there.

00:52:48.890 --> 00:52:50.523
We haven't knocked any out.

00:52:50.523 --> 00:52:52.360
There are no boundary
conditions yet.

00:52:52.360 --> 00:52:57.740
That's a free-free, so that
A_0 would be free-free.

00:52:57.740 --> 00:53:03.390
OK.

00:53:03.390 --> 00:53:06.716
I'll take one more case and
then I think we're at time.

00:53:06.716 --> 00:53:07.840
Suppose it was fixed-fixed?

00:53:10.740 --> 00:53:14.540
What would be the difference
matrix that would correspond

00:53:14.540 --> 00:53:18.040
to, how would I
change that matrix

00:53:18.040 --> 00:53:20.860
if my problem
became fixed-fixed?

00:53:20.860 --> 00:53:26.220
So now I'm fixing that
u, I'm fixing that u,

00:53:26.220 --> 00:53:31.790
in the mass spring case
I'm adding supports there.

00:53:31.790 --> 00:53:34.680
How would that
change the matrix?

00:53:34.680 --> 00:53:36.680
First and fourth, good.

00:53:36.680 --> 00:53:37.980
Say it again?

00:53:37.980 --> 00:53:43.790
First and fourth
columns would go.

00:53:43.790 --> 00:53:48.290
So fixed-free would
then be three by two.

00:53:48.290 --> 00:53:52.040
Free-free was three by four.

00:53:52.040 --> 00:53:54.280
Yeah, I'm glad this
question came up

00:53:54.280 --> 00:53:56.580
because this is the
right thing for you to be

00:53:56.580 --> 00:54:02.960
thinking about in connection
with the recent question

00:54:02.960 --> 00:54:04.390
you asked.

00:54:04.390 --> 00:54:05.900
OK.

00:54:05.900 --> 00:54:07.710
Maybe that's the
right place to stop,

00:54:07.710 --> 00:54:11.170
because now you've asked
questions that are really

00:54:11.170 --> 00:54:16.510
on target for what we've done,
and I hope useful to you.

00:54:16.510 --> 00:54:23.580
OK, see you guys tomorrow
evening at 7:30 in 54-100, OK.