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PROFESSOR STRANG: OK, so
this is a review session

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open to questions on homework.

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Open to questions on
topics in the exam

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that's coming tomorrow.

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This morning I wrote down
what the four questions would

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be about, and I'm glad I did.

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I never-- should have done
this many times before.

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So you would know
exactly and get down

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to seeing what
those problems are.

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And of course the
matrices called K,

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and A transpose C A are going to
appear probably more than once.

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So, open for any questions.

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About any topic whatsoever.

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Please.

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Yes, thank you.

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AUDIENCE: [INAUDIBLE]

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PROFESSOR STRANG: The
fourth question on the exam?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR STRANG: I'm glad
you used that word, fun.

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Yes.

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That's exactly what I mean.

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Section 2.4, and
they are fun, yeah.

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So I drew by hand a little
graph with nodes and edges.

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And you want to be able to
take that first basic step.

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So the first step,
which is as far as we

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got by last Wednesday, the
first lecture on Section 2.4,

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was just creating the matrix
A, understanding A transpose A,

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and of course there's more to
understand about A transpose A.

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Actually, why don't
we take one second.

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Suppose I have a graph
with six nodes, let's say.

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Can you imagine a
graph with six nodes?

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And every node connected
to every other node.

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So however many
edges that would be.

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Actually, my grandson just
got that question on his exam.

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He was told there were l islands
with a flight from every island

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to every other island,
and he was asked

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how many flights that makes.

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So I sent him the answer.

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But I was very happy
with his reply.

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He said "that's
exactly what I got."

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So, what do you know.

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It seems to work.

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So anyway.

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Suppose we had, how
many nodes did I say?

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Six?

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OK.

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So we have like a
six-node-- So n is six,

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and it's a complete
graph, this is really just

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to start us off talking
about some of these problems.

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So the matrix A, so I
think it would be 15,

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where did I come up
with that number 15?

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And is it right, actually?

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Yes.

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This is one way, would be the
first node has five edges going

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out and then the
second node would

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have four additional edges,
and three and two and one.

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And five, four, three,
two, one add to 15.

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What would be the shape
of A in that case?

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So it has a row for every edge.

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So 15 by 6, I think.

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OK, and I could create
A transpose A just

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to have a look at it.

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So it would be, what shape
would A transpose A be?

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Six by six.

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Symmetric, of course.

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Will it be singular
or non-singular?

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Singular.

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Singular, because we
haven't grounded any nodes.

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We've got all these nodes,
all these edges, nothing.

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We haven't taken
out that column;

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when I reduce it to five by
five, then it'll be invertible.

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But six by six, so what will
be the diagonal of this?

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This'll be now six by six,
the size will be six by six.

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And what will go on the diagonal
is the degrees of every node.

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That means how many
edges are coming in,

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and what number is that?

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Five.

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So I'll have five down the
diagonal, and what else,

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what will be off the diagonal?

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Minus, a whole lot of
minus ones, a minus one

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above and below for every edge.

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And since we have a
complete graph, how many

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minus ones have we got?

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All of them.

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All minus ones.

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So all minus ones
and all minus ones.

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That's fine to cross over
if you need to, sure.

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I'm not sure what else
to say about that matrix.

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Well, it's not invertible.

98
00:05:38,460 --> 00:05:43,530
Now let's take the next step
which, I'm now going probably

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beyond the exam part.

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Just really to get us started,
I ground the sixth node.

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Suppose I ground
node number six,

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that wipes out a row and
a column, is that right?

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So I'm now left with
a five by five matrix.

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It still has all minus
ones there and there,

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00:06:04,070 --> 00:06:06,010
but now it's five
by five, now it

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is what kind of a matrix,
what are its properties?

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00:06:09,280 --> 00:06:14,780
Square, obviously, symmetric
obviously, and now invertible.

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00:06:14,780 --> 00:06:17,080
Positive definite, OK.

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So it's fives there and now
I would have four minus ones.

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Let's just write them in here.

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Typical row, now, in
this five by five matrix

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would have four minus ones and
of course more here, and more

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here, and one there.

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And symmetric.

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All I want to say
is that that matrix,

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we don't often write down
the inverses of matrices,

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but that one I think we could.

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I think we could actually, and
it's a little bit interesting

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to know, for that special
matrix, everything about it.

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We could find its eigenvalues,
its determinant, its pivots,

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the whole works for that matrix.

122
00:07:01,780 --> 00:07:04,780
And that's one page
of the book, maybe

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00:07:04,780 --> 00:07:07,450
at the end of
Section 2.4, I think,

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00:07:07,450 --> 00:07:11,290
comes more detail
about that matrix.

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So in a way that special guy
is like our special K matrix,

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-1, 2, -1, for
second differences.

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Somehow this is taking,
all nodes are connected.

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Instead of in a line, springs
in a line, points in a line,

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we now have everybody
connected to everybody.

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00:07:35,040 --> 00:07:38,330
So this is sort of the
special matrix when

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everybody is
connected to everybody

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00:07:40,200 --> 00:07:43,750
and we could learn all
about that particular one.

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But then, of course, if
some edges are not in then

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some zeroes will
appear off the diagonal

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in the adjacency matrix part.

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The degrees will drop a little
if we're missing some edges

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00:07:58,010 --> 00:08:03,960
and the inverse will be
not some simple expression.

138
00:08:03,960 --> 00:08:05,470
Anyway, that's to
get us started.

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00:08:05,470 --> 00:08:14,740
So that's really where the last
lecture, Friday, brought us

140
00:08:14,740 --> 00:08:16,780
to this point.

141
00:08:16,780 --> 00:08:21,290
And I'll take this chance
to add in the block matrix

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just because I think of
it as quite important.

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So for this case,
C is the identity.

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00:08:26,950 --> 00:08:31,240
So I would have the identity up
in that block, A in that block,

145
00:08:31,240 --> 00:08:33,850
A transpose in this block.

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00:08:33,850 --> 00:08:39,590
That would be my mixed
method matrix, you could say.

147
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My saddle point matrix.

148
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It starts out very
positive definite.

149
00:08:45,810 --> 00:08:48,010
But it ends up
negative definite.

150
00:08:48,010 --> 00:08:50,450
And that's typical of
mixed methods, when

151
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both unknowns, the currents
as well as the potentials,

152
00:08:57,070 --> 00:08:59,920
are included in the system.

153
00:08:59,920 --> 00:09:04,330
So A transpose w, that was
f, I think, and this is b.

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00:09:04,330 --> 00:09:07,810
I just mentioned that again,
it was in Friday's lecture

155
00:09:07,810 --> 00:09:10,750
and it's in the book
but I would just

156
00:09:10,750 --> 00:09:14,010
want to say I
often refer to this

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as the fundamental problem
of numerical analysis,

158
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is how do you solve that system.

159
00:09:20,190 --> 00:09:23,120
And of course elimination
is one way to do it.

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When I eliminate w,
that will lead me

161
00:09:26,990 --> 00:09:32,080
to the equation A
transpose Au equals,

162
00:09:32,080 --> 00:09:39,020
I think it'll be, there'll be an
A transpose b minus f, I think.

163
00:09:39,020 --> 00:09:41,220
C being the identity there.

164
00:09:41,220 --> 00:09:44,770
So that's the mixed method,
this is the displacement method,

165
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and this is the popular one.

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Because it's all at once.

167
00:09:49,960 --> 00:09:55,240
But people think
about this one, too.

168
00:09:55,240 --> 00:09:59,860
So that's like saying again
what was in Friday's lecture

169
00:09:59,860 --> 00:10:04,100
and will be used going forward.

170
00:10:04,100 --> 00:10:06,130
OK, that was just
to get us started.

171
00:10:06,130 --> 00:10:12,080
Now, please let's
have some questions.

172
00:10:12,080 --> 00:10:14,180
We need another question.

173
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Who else?

174
00:10:15,250 --> 00:10:16,010
Yes, thank you.

175
00:10:16,010 --> 00:10:17,527
AUDIENCE: [INAUDIBLE]

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PROFESSOR STRANG: The
first on the homework.

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What number was that?

178
00:10:23,590 --> 00:10:27,540
Section 2.2, number six?

179
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About the trapezoidal rule?

180
00:10:30,250 --> 00:10:31,470
Yes.

181
00:10:31,470 --> 00:10:38,040
OK, now I did speak about that
a little in the last review

182
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session, but can I just say
a couple words more about it

183
00:10:42,501 --> 00:10:43,000
here?

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00:10:43,000 --> 00:10:44,432
AUDIENCE: [INAUDIBLE]

185
00:10:44,432 --> 00:10:45,890
PROFESSOR STRANG:
What's it asking?

186
00:10:45,890 --> 00:10:48,560
Yes.

187
00:10:48,560 --> 00:10:50,920
People often ask me
that about my problems.

188
00:10:50,920 --> 00:10:54,030
I don't know.

189
00:10:54,030 --> 00:10:55,320
You can't read my mind?

190
00:10:55,320 --> 00:10:56,550
You should.

191
00:10:56,550 --> 00:11:03,930
OK, so the point is that
for special differential

192
00:11:03,930 --> 00:11:07,090
equations-- So let me just
summarize what we did there.

193
00:11:07,090 --> 00:11:10,600
So this we did before, but
I didn't do everything.

194
00:11:10,600 --> 00:11:16,030
So what we did before was point
out that the system du/dt =

195
00:11:16,030 --> 00:11:24,100
Au conserves
energy. u squared, u

196
00:11:24,100 --> 00:11:27,910
of time-- for all
time, u of time

197
00:11:27,910 --> 00:11:40,220
squared stays constant if A
transpose equals minus A. OK,

198
00:11:40,220 --> 00:11:42,780
essentially you
take the derivative,

199
00:11:42,780 --> 00:11:45,900
it's got two terms because
we've got a product there,

200
00:11:45,900 --> 00:11:47,520
a product rule.

201
00:11:47,520 --> 00:11:52,170
The derivative will be,
one term will involve A,

202
00:11:52,170 --> 00:11:54,730
the other term will
involve A transpose,

203
00:11:54,730 --> 00:11:57,730
and if our matrix has this
antisymmetric property,

204
00:11:57,730 --> 00:12:01,010
those terms will cancel;
the derivative will be zero,

205
00:12:01,010 --> 00:12:03,590
and that'll mean that
this is a constant.

206
00:12:03,590 --> 00:12:06,010
OK, so that's the
differential equation.

207
00:12:06,010 --> 00:12:09,090
Now, the question was about
the difference equation.

208
00:12:09,090 --> 00:12:13,540
So we're taking the
trapezoidal rule

209
00:12:13,540 --> 00:12:20,410
and we want to show that
u_n squared stays constant

210
00:12:20,410 --> 00:12:22,390
for the trapezoidal rule.

211
00:12:22,390 --> 00:12:30,640
And so what that means, in
other words, is step by step,

212
00:12:30,640 --> 00:12:38,330
u_(n+1), and I could
write it u_(n+1) squared,

213
00:12:38,330 --> 00:12:43,500
but the other way to write that
and the way we have to work

214
00:12:43,500 --> 00:12:47,410
with it is that, is the same.

215
00:12:47,410 --> 00:12:51,010
Now, that was just an identity,
that's just the meaning.

216
00:12:51,010 --> 00:12:56,570
Now, I want to show
that that's the key.

217
00:12:56,570 --> 00:12:59,280
That's what we
would want to prove.

218
00:12:59,280 --> 00:13:05,950
That the trapezoidal rule copies
the property of constant energy

219
00:13:05,950 --> 00:13:08,030
of the differential equations.

220
00:13:08,030 --> 00:13:11,880
And of course, you know
that in oscillating springs

221
00:13:11,880 --> 00:13:16,600
when there's no source, no
forces coming from outside,

222
00:13:16,600 --> 00:13:19,070
the total energy
will stay constant.

223
00:13:19,070 --> 00:13:22,300
And you could think of
many other situations.

224
00:13:22,300 --> 00:13:26,320
You have a spacecraft, where
you've turned off the engines.

225
00:13:26,320 --> 00:13:29,220
It's just going there,
it's possibly rotating.

226
00:13:29,220 --> 00:13:35,390
So there you've got
angular velocity

227
00:13:35,390 --> 00:13:37,540
included in the total energy.

228
00:13:37,540 --> 00:13:41,180
Important fact, if energy stays
constant you want to know it.

229
00:13:41,180 --> 00:13:44,960
And you're very happy
if the finite difference

230
00:13:44,960 --> 00:13:46,610
method copies it.

231
00:13:46,610 --> 00:13:53,330
OK, so then it was just
a question of-- Here

232
00:13:53,330 --> 00:13:56,360
we took derivatives
to do that one.

233
00:13:56,360 --> 00:13:58,540
Here we're going to be
playing with differences,

234
00:13:58,540 --> 00:14:04,080
and my suggestion
was that the good way

235
00:14:04,080 --> 00:14:08,600
to get it was to
take that vector

236
00:14:08,600 --> 00:14:17,040
times the trapezoidal equation
and show that this turned out

237
00:14:17,040 --> 00:14:21,410
to-- The trapezoidal equation
is something equals zero,

238
00:14:21,410 --> 00:14:26,860
and you hope, and it takes a
few lines of jiggling around,

239
00:14:26,860 --> 00:14:31,480
that when you do that
you'll get the difference,

240
00:14:31,480 --> 00:14:32,920
you get exactly this.

241
00:14:32,920 --> 00:14:40,220
You get u_(n+1) transpose
u_(n+1) minus u_n transpose

242
00:14:40,220 --> 00:14:41,160
u_n.

243
00:14:41,160 --> 00:14:44,100
That's the goal.

244
00:14:44,100 --> 00:14:47,000
We know that the
trapezoidal equation--

245
00:14:47,000 --> 00:14:49,730
maybe I move everything onto
one side so I have something

246
00:14:49,730 --> 00:14:51,420
equals zero.

247
00:14:51,420 --> 00:14:57,340
Then my trick is take
that vector equation,

248
00:14:57,340 --> 00:15:02,030
multiply by that, play
around with those terms

249
00:15:02,030 --> 00:15:03,370
and you'll get this.

250
00:15:03,370 --> 00:15:07,380
So, since that is
zero, this is zero.

251
00:15:07,380 --> 00:15:11,060
And that's exactly what
our goal was to prove.

252
00:15:11,060 --> 00:15:13,570
So it's just in
the jiggling around

253
00:15:13,570 --> 00:15:17,840
and maybe we don't want to
take the full time because I'll

254
00:15:17,840 --> 00:15:18,700
post that.

255
00:15:18,700 --> 00:15:22,680
Actually, I may post some
of these solutions even

256
00:15:22,680 --> 00:15:24,040
before the quiz.

257
00:15:24,040 --> 00:15:26,890
And therefore before
the homework is due,

258
00:15:26,890 --> 00:15:29,600
just because this particular
homework, as I say,

259
00:15:29,600 --> 00:15:32,500
is not-- The graders
are just going

260
00:15:32,500 --> 00:15:39,020
to be so busy with
all the quizzes.

261
00:15:39,020 --> 00:15:40,680
This is for learning.

262
00:15:40,680 --> 00:15:43,620
Now, here's the one thing
you want to learn out

263
00:15:43,620 --> 00:15:46,940
of this messy computation.

264
00:15:46,940 --> 00:15:55,070
You also have a term,
you'll also find a term U_n,

265
00:15:55,070 --> 00:15:57,310
when you just do
this mechanically,

266
00:15:57,310 --> 00:16:00,830
you'll find a u_(n+1)
transpose u_n,

267
00:16:00,830 --> 00:16:06,410
and you'll find a u_n
transpose u_(n+1),

268
00:16:06,410 --> 00:16:10,270
and they'll come in
with opposite signs.

269
00:16:10,270 --> 00:16:13,130
That'll be when you've
plugged in the fact

270
00:16:13,130 --> 00:16:17,580
that A transpose equals minus
A, and all I wanted to do

271
00:16:17,580 --> 00:16:20,900
is ask you, what does
that term amount to?

272
00:16:20,900 --> 00:16:23,510
Because that term will show up.

273
00:16:23,510 --> 00:16:24,620
One way or another.

274
00:16:24,620 --> 00:16:26,620
And what does it equal?

275
00:16:26,620 --> 00:16:27,480
Zero.

276
00:16:27,480 --> 00:16:28,800
Everybody should know that.

277
00:16:28,800 --> 00:16:31,350
That's the one
thing, that you have

278
00:16:31,350 --> 00:16:34,230
to add to just
mechanically computing,

279
00:16:34,230 --> 00:16:39,950
is the fact that the dot product
of that vector with that,

280
00:16:39,950 --> 00:16:46,870
v transpose w is the same
as w transpose v. So,

281
00:16:46,870 --> 00:16:50,550
it's good to just call attention
to the easy things that

282
00:16:50,550 --> 00:16:51,510
are like that.

283
00:16:51,510 --> 00:16:53,780
Why is that?

284
00:16:53,780 --> 00:16:59,280
That's because both sides,
this is equal to what? v_1*w_1,

285
00:16:59,280 --> 00:17:03,440
v_2*w_2, v_3*w_3,
component by component.

286
00:17:03,440 --> 00:17:05,360
And this is w_1*v_1.

287
00:17:05,360 --> 00:17:08,250
But we're just talking
numbers at that point.

288
00:17:08,250 --> 00:17:11,070
So the numbers of v_1 times
w_1 are certainly the same

289
00:17:11,070 --> 00:17:12,850
as w_1 times v_1.

290
00:17:12,850 --> 00:17:15,880
Every component by component,
they're exactly the same

291
00:17:15,880 --> 00:17:19,760
and of course then the
dot products are the same.

292
00:17:19,760 --> 00:17:26,770
So that's the fact which
for this v and that w,

293
00:17:26,770 --> 00:17:31,130
make the term go away,
that's still sitting there.

294
00:17:31,130 --> 00:17:35,870
Other terms go away
because of this property.

295
00:17:35,870 --> 00:17:38,740
Having written that
and recognizing

296
00:17:38,740 --> 00:17:42,100
that we have Fourier
stuff coming up

297
00:17:42,100 --> 00:17:48,860
in the last third of the course,
where we have complex numbers.

298
00:17:48,860 --> 00:17:57,220
I have to say that if when
I have complex vectors,

299
00:17:57,220 --> 00:18:01,080
do you know about those?

300
00:18:01,080 --> 00:18:04,610
The dot product, or
the length squared,

301
00:18:04,610 --> 00:18:09,450
if this was a vector of complex,
with possibly complex numbers,

302
00:18:09,450 --> 00:18:12,060
I wouldn't take the
length squared just

303
00:18:12,060 --> 00:18:14,390
by adding up these squares.

304
00:18:14,390 --> 00:18:19,410
Suppose my-- Yes, I'm really
anticipating weeks ahead,

305
00:18:19,410 --> 00:18:23,230
but suppose my
vector was [1,  i].

306
00:18:23,230 --> 00:18:27,790
What's the length of
that particular vector v?

307
00:18:27,790 --> 00:18:31,620
Well, if I do v transpose
v, what do I get?

308
00:18:31,620 --> 00:18:36,770
For v equals [1, i], what does
v transpose v turn out to be?

309
00:18:36,770 --> 00:18:37,880
Zero.

310
00:18:37,880 --> 00:18:40,300
One squared plus
i squared is zero.

311
00:18:40,300 --> 00:18:41,570
No good.

312
00:18:41,570 --> 00:18:45,060
So obviously some rule
has to change a little bit

313
00:18:45,060 --> 00:18:46,780
to get the correct number.

314
00:18:46,780 --> 00:18:51,660
The correct length squared,
I would rather expect two.

315
00:18:51,660 --> 00:18:54,610
The size of that squared plus
the size of that squared.

316
00:18:54,610 --> 00:18:56,315
So I don't want to
square i, I want

317
00:18:56,315 --> 00:18:58,410
to square its absolute value.

318
00:18:58,410 --> 00:19:01,930
And the way to do
that is conjugate one

319
00:19:01,930 --> 00:19:03,920
of the two things.

320
00:19:03,920 --> 00:19:08,710
Now I'm taking [1, i], and on
the other side I have [1, -i]

321
00:19:08,710 --> 00:19:11,660
and that gives me
the two that I want.

322
00:19:11,660 --> 00:19:17,740
So what I'm doing, when
I've got complex vectors

323
00:19:17,740 --> 00:19:21,000
then I would really
do that, and now that

324
00:19:21,000 --> 00:19:23,190
is not the same as that.

325
00:19:23,190 --> 00:19:24,241
Right.

326
00:19:24,241 --> 00:19:24,740
Yeah.

327
00:19:24,740 --> 00:19:30,720
If in one case if I'm
doing the conjugate of v

328
00:19:30,720 --> 00:19:32,730
and the other case it's
the conjugate of w,

329
00:19:32,730 --> 00:19:35,880
then I've got a
complex conjugate.

330
00:19:35,880 --> 00:19:40,830
OK, that's a throwaway
comment that just is relevant

331
00:19:40,830 --> 00:19:47,260
because it keeps us focused
for a moment on the real case,

332
00:19:47,260 --> 00:19:52,640
where we do have equals.

333
00:19:52,640 --> 00:19:55,960
Now, I don't know if that
was sufficient answer?

334
00:19:55,960 --> 00:19:58,520
It wasn't a complete
answer because I

335
00:19:58,520 --> 00:20:03,740
didn't do the manipulations,
but the solutions

336
00:20:03,740 --> 00:20:06,150
posted should show you those.

337
00:20:06,150 --> 00:20:09,360
And, of course, you can organize
them a little different.

338
00:20:09,360 --> 00:20:11,760
OK, good for that one.

339
00:20:11,760 --> 00:20:12,410
Yes, please.

340
00:20:12,410 --> 00:20:16,450
AUDIENCE: [INAUDIBLE]

341
00:20:16,450 --> 00:20:18,410
PROFESSOR STRANG: The
other two terms here?

342
00:20:18,410 --> 00:20:22,870
AUDIENCE: [INAUDIBLE]

343
00:20:22,870 --> 00:20:25,970
PROFESSOR STRANG: Yes.

344
00:20:25,970 --> 00:20:28,810
You couldn't cancel them.

345
00:20:28,810 --> 00:20:34,710
Well, I recommend just, that's
how the best mathematics is

346
00:20:34,710 --> 00:20:35,320
done, right?

347
00:20:35,320 --> 00:20:39,200
You want zero,
you just X it out.

348
00:20:39,200 --> 00:20:40,700
Anyway.

349
00:20:40,700 --> 00:20:44,160
Let me leave the posted
solutions to be a hint

350
00:20:44,160 --> 00:20:46,140
and come back to it.

351
00:20:46,140 --> 00:20:47,390
Yeah.

352
00:20:47,390 --> 00:20:52,150
AUDIENCE: [INAUDIBLE]

353
00:20:52,150 --> 00:20:54,000
PROFESSOR STRANG:
The next problem was?

354
00:20:54,000 --> 00:20:56,850
AUDIENCE: [INAUDIBLE]

355
00:20:56,850 --> 00:20:58,200
PROFESSOR STRANG: Oh, yes.

356
00:20:58,200 --> 00:21:01,010
OK, that one I spoke a
little bit about, but now

357
00:21:01,010 --> 00:21:08,850
let me read from the
problem set that I got.

358
00:21:08,850 --> 00:21:12,870
I noticed that was quite brief.

359
00:21:12,870 --> 00:21:16,510
Oh, to find that actual angle?

360
00:21:16,510 --> 00:21:20,170
Somehow that's a little
interesting, isn't it?

361
00:21:20,170 --> 00:21:24,660
AUDIENCE: [INAUDIBLE]

362
00:21:24,660 --> 00:21:28,160
PROFESSOR STRANG: To tell the
truth, I meant numerically.

363
00:21:28,160 --> 00:21:33,500
I meant, what's the
point of that question.

364
00:21:33,500 --> 00:21:38,750
The point is we're trying to
solve, this isn't a big deal.

365
00:21:38,750 --> 00:21:44,530
But it was just if we're
using this trapezoidal method,

366
00:21:44,530 --> 00:21:48,620
the beauty of that, exactly
what our thing proves,

367
00:21:48,620 --> 00:21:54,170
is-- Here the constant
energy surface is the circle.

368
00:21:54,170 --> 00:21:58,310
The point of the trapezoidal
method for this simple equation

369
00:21:58,310 --> 00:22:07,560
u''+u=0, which amounted to the
equation uv' equals something

370
00:22:07,560 --> 00:22:17,280
like, our a matrix
was antisymmetric.

371
00:22:17,280 --> 00:22:19,750
So it fit perfectly
in that problem,

372
00:22:19,750 --> 00:22:23,010
and if we started on the
circle we stay on the circle.

373
00:22:23,010 --> 00:22:28,400
And if I take 32 steps I come
back pretty closely to here,

374
00:22:28,400 --> 00:22:33,600
and I just thought it might be
fun to figure out numerically,

375
00:22:33,600 --> 00:22:37,570
with MATLAB or a
calculator or something,

376
00:22:37,570 --> 00:22:42,260
we take an angle, theta, is that
what the problem asks, and then

377
00:22:42,260 --> 00:22:46,800
come around here to
32 theta, and 32 theta

378
00:22:46,800 --> 00:22:49,160
will not be exactly 2pi.

379
00:22:49,160 --> 00:22:51,000
But darned close.

380
00:22:51,000 --> 00:22:53,750
Because you could see in
the figure in the book

381
00:22:53,750 --> 00:22:56,230
it wasn't too far off.

382
00:22:56,230 --> 00:22:59,030
So the question was,
what is that theta?

383
00:22:59,030 --> 00:23:02,870
So I think that the
formula turned out

384
00:23:02,870 --> 00:23:09,070
to be that each step multiplied
by this one plus i delta t,

385
00:23:09,070 --> 00:23:15,330
or h, on two divided by one
minus i delta t over two.

386
00:23:15,330 --> 00:23:21,590
And when you plug in
delta t to be 2pi over 32,

387
00:23:21,590 --> 00:23:26,100
so that's the, what
did I say, that's

388
00:23:26,100 --> 00:23:28,260
the tangent of
theta or something?

389
00:23:28,260 --> 00:23:32,330
Sorry, I've forgotten the
way the problem was put.

390
00:23:32,330 --> 00:23:38,220
Oh, it's e to the i theta, yeah.

391
00:23:38,220 --> 00:23:41,350
What's the main point
about that complex number?

392
00:23:41,350 --> 00:23:42,970
When you look at
that complex number

393
00:23:42,970 --> 00:23:47,950
what's the most
essential thing you see?

394
00:23:47,950 --> 00:23:49,296
That it, yeah, tell me again.

395
00:23:49,296 --> 00:23:50,170
AUDIENCE: [INAUDIBLE]

396
00:23:50,170 --> 00:23:52,320
PROFESSOR STRANG:
Magnitude one, great.

397
00:23:52,320 --> 00:23:55,230
It's a number divided by
its complex conjugate,

398
00:23:55,230 --> 00:23:57,270
so it's a number
of magnitude one.

399
00:23:57,270 --> 00:24:01,180
And now tell me, if you see
a complex number of magnitude

400
00:24:01,180 --> 00:24:03,380
one, what jumps to mind?

401
00:24:03,380 --> 00:24:06,950
What form do you
naturally put it in?

402
00:24:06,950 --> 00:24:07,450
e^(i*theta).

403
00:24:07,450 --> 00:24:10,540


404
00:24:10,540 --> 00:24:15,117
Every complex number of absolute
value one is just beautifully

405
00:24:15,117 --> 00:24:16,450
written in the form e^(i*theta).

406
00:24:16,450 --> 00:24:19,780


407
00:24:19,780 --> 00:24:24,070
That complex number is some
point on the unit circle,

408
00:24:24,070 --> 00:24:24,790
so there it is.

409
00:24:24,790 --> 00:24:28,360
Right there, there
it is, e^(i*theta).

410
00:24:28,360 --> 00:24:31,870
With that-- theta
is negative there,

411
00:24:31,870 --> 00:24:33,670
because we're going
the wrong way.

412
00:24:33,670 --> 00:24:35,940
No big deal.

413
00:24:35,940 --> 00:24:38,510
Maybe here theta's positive.

414
00:24:38,510 --> 00:24:43,660
I've forgotten, so
I won't try to go

415
00:24:43,660 --> 00:24:47,450
either the clockwise or the
counterclockwise way around.

416
00:24:47,450 --> 00:24:51,070
So, if I wanted to figure out
what theta was and plugged

417
00:24:51,070 --> 00:24:54,260
in these things, let's see.

418
00:24:54,260 --> 00:25:03,590
So that's pi over 32, delta t
over two would be pi over 32,

419
00:25:03,590 --> 00:25:09,640
and this guy would be its
conjugate. pi over 32,

420
00:25:09,640 --> 00:25:12,670
and then in this
solution that'll

421
00:25:12,670 --> 00:25:23,580
be plugged on the homework
this will be, I think maybe,

422
00:25:23,580 --> 00:25:30,040
maybe the theta comes out to
be, it's kind of cool actually,

423
00:25:30,040 --> 00:25:37,000
twice the arc tangent of
pi over 32 or something.

424
00:25:37,000 --> 00:25:38,130
I didn't know that.

425
00:25:38,130 --> 00:25:44,200
But that'll be in the
solutions for you to check.

426
00:25:44,200 --> 00:25:50,440
So now, why do I like
e^(i*theta) so much?

427
00:25:50,440 --> 00:25:53,470
Because now I could tell
you what this point is,

428
00:25:53,470 --> 00:25:56,870
after you've done it 32 times.

429
00:25:56,870 --> 00:25:59,330
What angle have you reached?

430
00:25:59,330 --> 00:26:05,520
This is the crunch line of using
complex numbers, e^(i*theta),

431
00:26:05,520 --> 00:26:09,950
is that they're absolutely
great for taking powers.

432
00:26:09,950 --> 00:26:15,450
If I take the 32nd power of
x plus iy, I'm lost, right.

433
00:26:15,450 --> 00:26:19,890
x plus iy to the 32nd
power starts out x^32,

434
00:26:19,890 --> 00:26:24,600
ends up i^32 y^32, with
horrible stuff in between.

435
00:26:24,600 --> 00:26:30,187
But what is the 32nd
power of e^(i*theta)?

436
00:26:30,187 --> 00:26:30,770
e^(i*32theta).

437
00:26:30,770 --> 00:26:33,740


438
00:26:33,740 --> 00:26:37,160
Just that angle 32 times
exactly as we've drawn it.

439
00:26:37,160 --> 00:26:39,300
So that's the point
e^(i*32theta).

440
00:26:39,300 --> 00:26:42,250


441
00:26:42,250 --> 00:26:49,790
OK, and therefore if we now
know what theta is, so yeah.

442
00:26:49,790 --> 00:26:55,900
So it must be pretty near
2pi, but not exactly.

443
00:26:55,900 --> 00:26:59,750
I guess that's about right.

444
00:26:59,750 --> 00:27:03,200
In fact, having got
this far, the tangent

445
00:27:03,200 --> 00:27:09,210
of a very small angle
is approximately what?

446
00:27:09,210 --> 00:27:11,470
It's approximately
the angle, right?

447
00:27:11,470 --> 00:27:14,760
The sine of a very small angle
is approximately the angle.

448
00:27:14,760 --> 00:27:16,910
The cosine is approximately one.

449
00:27:16,910 --> 00:27:19,090
The tangent is
approximately the angle.

450
00:27:19,090 --> 00:27:25,380
So this, 32 theta,
is 32 times two times

451
00:27:25,380 --> 00:27:28,180
approximately the angle.

452
00:27:28,180 --> 00:27:33,760
And what answer do you get?

453
00:27:33,760 --> 00:27:35,250
2pi.

454
00:27:35,250 --> 00:27:38,820
Which makes sense.

455
00:27:38,820 --> 00:27:47,430
So you could say what the
trapezoidal method has done

456
00:27:47,430 --> 00:27:52,390
is to replace the exact
angle by the inverse tangent

457
00:27:52,390 --> 00:27:53,810
approximately.

458
00:27:53,810 --> 00:27:56,090
That's sort of nice.

459
00:27:56,090 --> 00:27:59,090
In this example you
can get as far as that

460
00:27:59,090 --> 00:28:03,220
and you could actually
find out how near that is.

461
00:28:03,220 --> 00:28:06,540
And, by the way, how near
would I expect it to be?

462
00:28:06,540 --> 00:28:11,450
I would expect it,
so what do we know

463
00:28:11,450 --> 00:28:16,190
about the trapezoidal method
without having proved it?

464
00:28:16,190 --> 00:28:19,530
It's second order
accurate, right?

465
00:28:19,530 --> 00:28:21,340
If it was first
order accurate, I

466
00:28:21,340 --> 00:28:26,120
would expect it to miss by
something of the size of theta.

467
00:28:26,120 --> 00:28:28,380
Maybe a fraction of theta.

468
00:28:28,380 --> 00:28:30,150
But being second
order accurate, I'm

469
00:28:30,150 --> 00:28:31,890
expecting it to
miss by something

470
00:28:31,890 --> 00:28:34,820
of size theta squared.

471
00:28:34,820 --> 00:28:40,190
So it would be pretty
near zero, right.

472
00:28:40,190 --> 00:28:47,080
And actually, another way I
know it's around-- So the error

473
00:28:47,080 --> 00:28:49,870
would be something
like, it would have a 32

474
00:28:49,870 --> 00:28:51,740
squared in the denominator.

475
00:28:51,740 --> 00:28:59,660
And I've just thought of
another way to see that.

476
00:28:59,660 --> 00:29:02,220
We just said that the
first term in the arc

477
00:29:02,220 --> 00:29:05,750
tangent of a small angle,
theta, of a small angle,

478
00:29:05,750 --> 00:29:10,150
alpha, whatever that is, pi over
32, the first term in the arc

479
00:29:10,150 --> 00:29:11,490
tangent is?

480
00:29:11,490 --> 00:29:12,560
The angle.

481
00:29:12,560 --> 00:29:14,310
That's what we just said.

482
00:29:14,310 --> 00:29:16,520
Then, do you know
what would come next?

483
00:29:16,520 --> 00:29:20,890
Now we're looking at the error.

484
00:29:20,890 --> 00:29:27,300
So that of a very small
angle will start theta,

485
00:29:27,300 --> 00:29:30,460
and I want to ask you about
how many theta squareds

486
00:29:30,460 --> 00:29:32,810
and theta cubes.

487
00:29:32,810 --> 00:29:39,300
You're seeing what you can do
with paper and pencils type

488
00:29:39,300 --> 00:29:41,520
stuff.

489
00:29:41,520 --> 00:29:46,999
Here's my main question, how
many theta squareds in there?

490
00:29:46,999 --> 00:29:48,040
You want to make a guess?

491
00:29:48,040 --> 00:29:51,960
A mathematician's
favorite number, zero.

492
00:29:51,960 --> 00:29:55,820
Right, there will be no
theta squared terms in.

493
00:29:55,820 --> 00:29:59,860
That's an odd function, so
I'm expecting only odd powers

494
00:29:59,860 --> 00:30:02,190
and therefore I won't
be surprised to see

495
00:30:02,190 --> 00:30:04,230
theta cubed come up first.

496
00:30:04,230 --> 00:30:07,170
And then when I
multiply by the 32,

497
00:30:07,170 --> 00:30:14,890
I get the theta squared that
I guessed we would have.

498
00:30:14,890 --> 00:30:22,040
OK, once again I'll stop there
because that's very narrow path

499
00:30:22,040 --> 00:30:26,070
to be following but
it shows you how.

500
00:30:26,070 --> 00:30:28,760
You know, there's
a lot of room still

501
00:30:28,760 --> 00:30:31,440
for what you can do
with paper and pencil

502
00:30:31,440 --> 00:30:35,320
to understand a model problem.

503
00:30:35,320 --> 00:30:38,030
And then the computer
would tell us

504
00:30:38,030 --> 00:30:42,510
about a serious problem of
following the solar system

505
00:30:42,510 --> 00:30:44,560
for a million years.

506
00:30:44,560 --> 00:30:48,740
OK, another totally
different question, if I can.

507
00:30:48,740 --> 00:30:49,510
Yes, thank you.

508
00:30:49,510 --> 00:30:50,760
AUDIENCE: [INAUDIBLE]

509
00:30:50,760 --> 00:30:51,968
PROFESSOR STRANG: Yeah, sure.

510
00:30:51,968 --> 00:30:53,020
AUDIENCE: [INAUDIBLE]

511
00:30:53,020 --> 00:30:56,000
PROFESSOR STRANG: 2.4.1, right.

512
00:30:56,000 --> 00:30:57,110
A mistake in the book.

513
00:30:57,110 --> 00:30:59,010
AUDIENCE: [INAUDIBLE]

514
00:30:59,010 --> 00:31:01,300
PROFESSOR STRANG:
Or in the, yeah.

515
00:31:01,300 --> 00:31:04,230
It's quite possible.

516
00:31:04,230 --> 00:31:12,680
OK, there's a printed
error in the graph.

517
00:31:12,680 --> 00:31:13,200
Yeah.

518
00:31:13,200 --> 00:31:18,380
So in numbering the
edges, well let's

519
00:31:18,380 --> 00:31:19,800
blame it on the printer, right?

520
00:31:19,800 --> 00:31:20,840
Not the author.

521
00:31:20,840 --> 00:31:26,190
OK, so the diagonal edge,
that five probably was

522
00:31:26,190 --> 00:31:28,950
intended to be a three, yeah.

523
00:31:28,950 --> 00:31:29,940
Thank you.

524
00:31:29,940 --> 00:31:35,840
So we'll catch that
in the next printing.

525
00:31:35,840 --> 00:31:42,000
And you recognize that always,
numbering the edges and nodes

526
00:31:42,000 --> 00:31:43,810
is a pretty
arbitrary thing, it's

527
00:31:43,810 --> 00:31:48,020
just if we number differently
that just reorders

528
00:31:48,020 --> 00:31:50,040
the rows of the matrix.

529
00:31:50,040 --> 00:31:51,580
If we number the
edge differently,

530
00:31:51,580 --> 00:31:55,950
it'll reorder the rows and
it'll reorder rows and columns

531
00:31:55,950 --> 00:31:59,470
of A transpose A. So it won't
make a serious difference

532
00:31:59,470 --> 00:32:00,250
in the matrix.

533
00:32:00,250 --> 00:32:00,750
Yeah.

534
00:32:00,750 --> 00:32:02,932
AUDIENCE: [INAUDIBLE]

535
00:32:02,932 --> 00:32:05,140
PROFESSOR STRANG: Do you
want to go back to this guy?

536
00:32:05,140 --> 00:32:06,400
OK.

537
00:32:06,400 --> 00:32:10,291
AUDIENCE: So if you have
an anti-symmetric matrix,

538
00:32:10,291 --> 00:32:12,166
does it follow that the
eigenvectors used are

539
00:32:12,166 --> 00:32:12,749
perpendicular?

540
00:32:12,749 --> 00:32:16,470
PROFESSOR STRANG: This
is a good question.

541
00:32:16,470 --> 00:32:20,910
This guy, way up here,
with this property,

542
00:32:20,910 --> 00:32:22,860
AUDIENCE: The eigenvectors
are perpendicular?

543
00:32:22,860 --> 00:32:25,068
PROFESSOR STRANG: The
eigenvectors are perpendicular.

544
00:32:25,068 --> 00:32:26,610
Yes, yeah.

545
00:32:26,610 --> 00:32:32,820
So we have, there's this, like,
the nobility among matrices

546
00:32:32,820 --> 00:32:35,610
are the ones with
perpendicular eigenvectors.

547
00:32:35,610 --> 00:32:39,100
So that includes
symmetric matrices,

548
00:32:39,100 --> 00:32:43,980
this is a good and
straightforward point.

549
00:32:43,980 --> 00:32:46,800
So these are the good matrices.

550
00:32:46,800 --> 00:32:48,990
Symmetric matrices.

551
00:32:48,990 --> 00:32:53,160
A transpose equals
A. Their eigenvalues

552
00:32:53,160 --> 00:32:56,840
lie on the real line.

553
00:32:56,840 --> 00:33:04,630
And these are all
perpendicular eigenvectors.

554
00:33:04,630 --> 00:33:08,220
What about antisymmetric?

555
00:33:08,220 --> 00:33:12,030
OK, that means A
transpose is minus A.

556
00:33:12,030 --> 00:33:16,320
They also fall in this
noble family of matrices,

557
00:33:16,320 --> 00:33:19,130
and where are their eigenvalues?

558
00:33:19,130 --> 00:33:22,140
Pure imaginary, right up here.

559
00:33:22,140 --> 00:33:26,120
Now do you want to know,
who else is in this family?

560
00:33:26,120 --> 00:33:28,150
What's the other, this
is the complex plane;

561
00:33:28,150 --> 00:33:30,480
there's one more piece
of the complex plane

562
00:33:30,480 --> 00:33:32,240
that you know I'm going to put.

563
00:33:32,240 --> 00:33:35,090
Which is?

564
00:33:35,090 --> 00:33:39,200
What else to make that
complex plane look familiar,

565
00:33:39,200 --> 00:33:43,080
it's going to have
the unit circle.

566
00:33:43,080 --> 00:33:46,260
Every complex plane has got
to have the unit circle.

567
00:33:46,260 --> 00:33:51,510
OK, so these guys went with
the, and now what do you think

568
00:33:51,510 --> 00:33:55,640
goes with the, this will
be the matrices-- Can

569
00:33:55,640 --> 00:33:59,230
I call them Q instead, because
I called them Q this morning.

570
00:33:59,230 --> 00:34:02,340
Q transpose is Q inverse.

571
00:34:02,340 --> 00:34:05,990
Q transpose Q, and they're
the orthogonal matrices.

572
00:34:05,990 --> 00:34:10,380
So those matrices again,
beautiful matrices

573
00:34:10,380 --> 00:34:11,580
in the best class.

574
00:34:11,580 --> 00:34:15,940
And their eigenvalues
are on the unit circle.

575
00:34:15,940 --> 00:34:17,980
And that would be--

576
00:34:17,980 --> 00:34:20,370
Why don't I just show you why?

577
00:34:20,370 --> 00:34:22,410
Because orthogonal
matrices, there

578
00:34:22,410 --> 00:34:25,440
are not so many that are
really worth knowing.

579
00:34:25,440 --> 00:34:31,440
So, let me take Qx=lambda*x,
and what is it that I want

580
00:34:31,440 --> 00:34:33,550
to prove?

581
00:34:33,550 --> 00:34:38,290
I want to prove that the
eigenvalues have absolute value

582
00:34:38,290 --> 00:34:39,090
one.

583
00:34:39,090 --> 00:34:40,600
That's the unit circle.

584
00:34:40,600 --> 00:34:42,760
So how do I show
that the eigenvalues

585
00:34:42,760 --> 00:34:45,080
have absolute value of one?

586
00:34:45,080 --> 00:34:49,960
Let me take the dot
product with Qx transpose.

587
00:34:49,960 --> 00:34:56,820
So both sides, I'll do Qx
transpose Qx and I'll do

588
00:34:56,820 --> 00:35:01,510
lambda*x transpose
lambda*x, right?

589
00:35:01,510 --> 00:35:04,080
Only these are complex.

590
00:35:04,080 --> 00:35:08,610
I've got to take complex stuff.

591
00:35:08,610 --> 00:35:10,700
OK.

592
00:35:10,700 --> 00:35:17,240
I just took the length squared
of both sides, and kept in mind

593
00:35:17,240 --> 00:35:19,430
the possibility that
this x and lambda

594
00:35:19,430 --> 00:35:22,500
could be, and probably
will be, complex numbers.

595
00:35:22,500 --> 00:35:25,760
Now, what do I have on the left?

596
00:35:25,760 --> 00:35:28,060
Do you see it happening?

597
00:35:28,060 --> 00:35:31,150
I get an x bar
transpose, what do I get?

598
00:35:31,150 --> 00:35:37,630
Q transpose Qx on the left side.

599
00:35:37,630 --> 00:35:40,540
That's the combination
I'm looking for.

600
00:35:40,540 --> 00:35:42,500
For an orthogonal matrix.

601
00:35:42,500 --> 00:35:45,290
Let's imagine the
matrix itself is real,

602
00:35:45,290 --> 00:35:48,800
otherwise I would
just conjugate it.

603
00:35:48,800 --> 00:35:53,960
What's nice about
that left side?

604
00:35:53,960 --> 00:35:57,840
What fact am I going
to use about Q?

605
00:35:57,840 --> 00:36:00,340
Q transpose Q is the identity.

606
00:36:00,340 --> 00:36:04,100
So this thing is nothing
but x bar transpose x.

607
00:36:04,100 --> 00:36:05,700
That's the length of x squared.

608
00:36:05,700 --> 00:36:07,920
What have I got
on the right side?

609
00:36:07,920 --> 00:36:11,830
I've got the length of x
squared times a number,

610
00:36:11,830 --> 00:36:16,550
lambda bar times lambda squared.

611
00:36:16,550 --> 00:36:17,980
It's there, now.

612
00:36:17,980 --> 00:36:20,862
On the left side I have
the length of x squared.

613
00:36:20,862 --> 00:36:22,320
On the right side
I have the length

614
00:36:22,320 --> 00:36:25,340
of x squared times that
number, mod lambda squared.

615
00:36:25,340 --> 00:36:27,300
Therefore, that
number has to be one

616
00:36:27,300 --> 00:36:30,750
and the eigenvalues
are on the unit circle.

617
00:36:30,750 --> 00:36:37,300
So, I've given you the three big
important classes of matrices

618
00:36:37,300 --> 00:36:41,790
with perpendicular eigenvectors.

619
00:36:41,790 --> 00:36:44,820
I think anybody
would wonder, OK,

620
00:36:44,820 --> 00:36:48,840
what about other eigenvalues.

621
00:36:48,840 --> 00:36:52,210
What's the condition for
perpendicular eigenvectors

622
00:36:52,210 --> 00:36:53,890
that includes this.

623
00:36:53,890 --> 00:36:55,080
And includes this.

624
00:36:55,080 --> 00:36:57,770
And includes this,
and also allows

625
00:36:57,770 --> 00:37:01,460
eigenvalues all over the place.

626
00:37:01,460 --> 00:37:04,310
Would you like to
know that condition?

627
00:37:04,310 --> 00:37:06,650
What the heck.

628
00:37:06,650 --> 00:37:11,660
That condition, that
includes all these

629
00:37:11,660 --> 00:37:17,620
is this, that A transpose times
A equals A times A transpose.

630
00:37:17,620 --> 00:37:22,580
That's the test for
perpendicular eigenvectors.

631
00:37:22,580 --> 00:37:26,800
A transpose commutes with A.
So this passes, of course.

632
00:37:26,800 --> 00:37:28,420
This passes, of course.

633
00:37:28,420 --> 00:37:31,450
This passes because both
sides are the identity,

634
00:37:31,450 --> 00:37:34,880
and then some more
matrices pass also.

635
00:37:34,880 --> 00:37:36,764
OK.

636
00:37:36,764 --> 00:37:37,430
Is that alright?

637
00:37:37,430 --> 00:37:41,080
You asked for some linear
algebra and you got it.

638
00:37:41,080 --> 00:37:42,880
Now I'm ready, yes, thanks.

639
00:37:42,880 --> 00:37:45,820
AUDIENCE: [INAUDIBLE]

640
00:37:45,820 --> 00:37:50,560
PROFESSOR STRANG: 2.4.19.

641
00:37:50,560 --> 00:37:52,200
Oh, let me look.

642
00:37:52,200 --> 00:37:54,680
2.4.19.

643
00:37:54,680 --> 00:37:59,600
Ah.

644
00:37:59,600 --> 00:38:06,680
OK, yes, sorry and I should
have done better with that.

645
00:38:06,680 --> 00:38:23,640
So one graphs that are
important are grids like this.

646
00:38:23,640 --> 00:38:28,950
And we'll see them-- Two, three,
four, one, two, three, four.

647
00:38:28,950 --> 00:38:32,970
That would be where--
These are the nodes.

648
00:38:32,970 --> 00:38:38,510
So this is a grid.

649
00:38:38,510 --> 00:38:41,400
I meant to draw them all
in, but I won't. n squared.

650
00:38:41,400 --> 00:38:47,690
So n is six, and
I'd have 36 nodes.

651
00:38:47,690 --> 00:38:50,840
And you can see
the edges in there.

652
00:38:50,840 --> 00:38:53,480
So that's the graph
I have in mind.

653
00:38:53,480 --> 00:38:56,510
And the reason that
problem is there

654
00:38:56,510 --> 00:39:02,050
is that last year, I
think it was last year

655
00:39:02,050 --> 00:39:06,600
or the year before, we spent
some time with figuring out

656
00:39:06,600 --> 00:39:11,320
resistances and currents and
so on for these problems.

657
00:39:11,320 --> 00:39:15,170
And we needed some
fast way to generate A,

658
00:39:15,170 --> 00:39:19,870
because this matrix A is now,
what size is the matrix A?

659
00:39:19,870 --> 00:39:23,310
It's got, I don't know how
many edges does it have?

660
00:39:23,310 --> 00:39:28,610
One, two, three, four, five,
maybe 30 edges going across

661
00:39:28,610 --> 00:39:29,840
and 30 coming down.

662
00:39:29,840 --> 00:39:38,570
It'll be 60 by how many
columns in this matrix?

663
00:39:38,570 --> 00:39:40,330
You know the answer
now and it's worth

664
00:39:40,330 --> 00:39:42,550
knowing, for the quiz of course.

665
00:39:42,550 --> 00:39:44,170
36.

666
00:39:44,170 --> 00:39:48,500
OK.

667
00:39:48,500 --> 00:39:55,290
Anyway, the class rebelled
about creating these matrices

668
00:39:55,290 --> 00:40:03,180
and working with the matrices,
with 2,160 non-zeroes.

669
00:40:03,180 --> 00:40:04,970
People were dropping the course.

670
00:40:04,970 --> 00:40:10,100
So we needed a command that
would create A pretty quickly.

671
00:40:10,100 --> 00:40:14,500
And so that's what
the book, and so this

672
00:40:14,500 --> 00:40:17,870
was like the 18.085 command.

673
00:40:17,870 --> 00:40:21,030
After we stumbled
around for a while,

674
00:40:21,030 --> 00:40:24,730
we discovered that
a MATLAB command

675
00:40:24,730 --> 00:40:31,060
called kron was a quick
way to create the matrix.

676
00:40:31,060 --> 00:40:38,160
We'll see that when
we get to that point.

677
00:40:38,160 --> 00:40:40,620
This is an important graph.

678
00:40:40,620 --> 00:40:44,840
And it's closely connected
to Laplace's-- You remember

679
00:40:44,840 --> 00:40:45,820
Laplace's--?

680
00:40:45,820 --> 00:40:47,450
I'll just tell you.

681
00:40:47,450 --> 00:40:52,590
Laplace's equation is this,
you have a second x derivative,

682
00:40:52,590 --> 00:40:54,380
we know how to deal with those.

683
00:40:54,380 --> 00:40:59,750
But it also has a
second y derivative.

684
00:40:59,750 --> 00:41:02,650
So I'm really looking ahead
at the most important equation

685
00:41:02,650 --> 00:41:05,970
of Chapter 3,
Laplace's equation.

686
00:41:05,970 --> 00:41:11,260
And suppose I use
finite differences.

687
00:41:11,260 --> 00:41:16,700
I want a matrix K_(2D) that
deals with this 2D problem.

688
00:41:16,700 --> 00:41:19,140
And let me just say
what it would be.

689
00:41:19,140 --> 00:41:21,430
At a typical point
this x derivative

690
00:41:21,430 --> 00:41:25,100
is giving me a minus one,
a two and a minus one.

691
00:41:25,100 --> 00:41:27,790
And the y derivative is
giving me a minus one

692
00:41:27,790 --> 00:41:34,530
that moves this guy up
to four and minus one.

693
00:41:34,530 --> 00:41:39,220
So instead of -1, 2,
-1 along a typical row,

694
00:41:39,220 --> 00:41:43,390
we'll now have a four on the
diagonal and four minus one

695
00:41:43,390 --> 00:41:45,600
in a certain pattern.

696
00:41:45,600 --> 00:41:46,100
Anyway.

697
00:41:46,100 --> 00:41:50,590
You'll see that, it's
interesting when we get to it.

698
00:41:50,590 --> 00:41:54,950
That would show up in A
transpose A. So what I've said

699
00:41:54,950 --> 00:42:00,110
here is what happens with
A transpose A. I guess

700
00:42:00,110 --> 00:42:05,080
I'm hoping that you begin to
know these matrices, first

701
00:42:05,080 --> 00:42:07,140
seeing them occasionally
in homeworks

702
00:42:07,140 --> 00:42:10,950
and then in the lecture.

703
00:42:10,950 --> 00:42:11,490
Good.

704
00:42:11,490 --> 00:42:15,040
But that's looking ahead.

705
00:42:15,040 --> 00:42:19,140
I needed some questions
that are like,

706
00:42:19,140 --> 00:42:21,760
close to, really
on what we're doing

707
00:42:21,760 --> 00:42:24,210
or what the quiz would do.

708
00:42:24,210 --> 00:42:25,620
Any - thank you.

709
00:42:25,620 --> 00:42:30,680
AUDIENCE: [INAUDIBLE]

710
00:42:30,680 --> 00:42:33,030
PROFESSOR STRANG: Oh yes.

711
00:42:33,030 --> 00:42:35,110
A little bit.

712
00:42:35,110 --> 00:42:39,180
OK, yeah.

713
00:42:39,180 --> 00:42:44,410
So I wrote down this
equation and what I'm writing

714
00:42:44,410 --> 00:42:47,300
right there is the new part.

715
00:42:47,300 --> 00:42:51,780
Sort of new, and I guess--
Equals some right hand side

716
00:42:51,780 --> 00:42:52,950
f(x).

717
00:42:52,950 --> 00:42:59,570
And the discrete version
will be an A transpose

718
00:42:59,570 --> 00:43:02,520
C A equal a vector
f, maybe there

719
00:43:02,520 --> 00:43:05,910
will be a delta x squared here.

720
00:43:05,910 --> 00:43:07,000
OK.

721
00:43:07,000 --> 00:43:11,010
I guess maybe, I
don't want to go far

722
00:43:11,010 --> 00:43:16,500
but I want you to see that if
we have a coefficient c in here

723
00:43:16,500 --> 00:43:20,350
it should show up there.

724
00:43:20,350 --> 00:43:26,260
You could actually, it might be
reasonable to look at 3.1 just

725
00:43:26,260 --> 00:43:30,550
to look slightly ahead
to see the parallels,

726
00:43:30,550 --> 00:43:37,950
but you would get them right
without a lecture on it.

727
00:43:37,950 --> 00:43:41,830
Your coefficient shows up in
the differential equation,

728
00:43:41,830 --> 00:43:45,900
and it shows up on the
diagonal of C in the difference

729
00:43:45,900 --> 00:43:49,500
equation.

730
00:43:49,500 --> 00:43:51,130
I won't give a whole
lecture on that,

731
00:43:51,130 --> 00:44:01,330
just to say that correspondence
is exactly the one we know.

732
00:44:01,330 --> 00:44:05,730
A is a difference matrix.

733
00:44:05,730 --> 00:44:07,340
Like the derivative.

734
00:44:07,340 --> 00:44:10,080
C will be a diagonal
matrix, A transpose

735
00:44:10,080 --> 00:44:14,170
will be whatever
that comes out to be.

736
00:44:14,170 --> 00:44:17,010
And so you've seen
A transpose A,

737
00:44:17,010 --> 00:44:19,120
but think again about
that difference--

738
00:44:19,120 --> 00:44:22,540
And ask yourselves this.

739
00:44:22,540 --> 00:44:27,430
I suggest, take c to be
one, get c out of there.

740
00:44:27,430 --> 00:44:33,880
And just think, again, what is
the difference matrix A with

741
00:44:33,880 --> 00:44:38,610
a boundary either
fixed-fixed or fixed-free,

742
00:44:38,610 --> 00:44:40,420
those will be two different A's.

743
00:44:40,420 --> 00:44:53,960
What are the A's for
fixed-fixed and for fixed-free?

744
00:44:53,960 --> 00:44:57,080
This is what we were doing
at the very beginning

745
00:44:57,080 --> 00:44:59,920
of the course.

746
00:44:59,920 --> 00:45:03,230
So A is a first
difference matrix,

747
00:45:03,230 --> 00:45:05,890
and A transpose A will
be the second difference.

748
00:45:05,890 --> 00:45:12,550
So the A transpose A, of course,
I was doing A transpose A,

749
00:45:12,550 --> 00:45:15,960
then the answer here
would be the matrix K

750
00:45:15,960 --> 00:45:21,850
and the answer here would
be the matrix T. Or,

751
00:45:21,850 --> 00:45:26,490
depending which end is free,
but we'd have one change.

752
00:45:26,490 --> 00:45:30,290
That's A transpose A,
but now think about the A

753
00:45:30,290 --> 00:45:32,420
that it came from.

754
00:45:32,420 --> 00:45:38,640
So A will be, A
is the matrix that

755
00:45:38,640 --> 00:45:40,890
takes differences
of the u's, and then

756
00:45:40,890 --> 00:45:45,570
A transpose A takes
second differences.

757
00:45:45,570 --> 00:45:47,170
Of all the questions
asked, this is

758
00:45:47,170 --> 00:45:53,580
the one most
relevant for the exam

759
00:45:53,580 --> 00:45:57,050
and for what we've done so far.

760
00:45:57,050 --> 00:46:01,130
I've gone off onto topics
that we look ahead to,

761
00:46:01,130 --> 00:46:03,210
but this is where we are.

762
00:46:03,210 --> 00:46:07,000
So that matrix A is a
first difference matrix,

763
00:46:07,000 --> 00:46:11,120
and then you put in the
boundary conditions.

764
00:46:11,120 --> 00:46:12,405
OK.

765
00:46:12,405 --> 00:46:13,530
There was another question.

766
00:46:13,530 --> 00:46:14,030
Yeah.

767
00:46:14,030 --> 00:46:18,080
AUDIENCE: [INAUDIBLE]

768
00:46:18,080 --> 00:46:19,870
PROFESSOR STRANG:
Of number four?

769
00:46:19,870 --> 00:46:22,910
Which number four in which?

770
00:46:22,910 --> 00:46:23,770
Oh, in the quiz.

771
00:46:23,770 --> 00:46:25,810
Oh yes, right.

772
00:46:25,810 --> 00:46:27,050
Yes.

773
00:46:27,050 --> 00:46:28,690
Did I tell you what
problem four was?

774
00:46:28,690 --> 00:46:29,190
No.

775
00:46:29,190 --> 00:46:31,740
I hope not.

776
00:46:31,740 --> 00:46:39,130
OK problem four in the quiz.

777
00:46:39,130 --> 00:46:41,430
It's about a delta function?

778
00:46:41,430 --> 00:46:43,620
Yeah.

779
00:46:43,620 --> 00:46:52,870
What do I know, what do
you want me to tell you?

780
00:46:52,870 --> 00:46:55,540
So the delta, of
course, comes in

781
00:46:55,540 --> 00:47:00,060
as, we've seen it, as the right
hand side of a differential

782
00:47:00,060 --> 00:47:01,080
equation.

783
00:47:01,080 --> 00:47:05,840
So it might be the right-hand
side even of this equation.

784
00:47:05,840 --> 00:47:20,330
So if this equation was delta
of x, or x-1/2 or something.

785
00:47:20,330 --> 00:47:23,680
I mean, the essential
thing is that when

786
00:47:23,680 --> 00:47:32,050
delta's on the right side, that
gives you a drop in the slope.

787
00:47:32,050 --> 00:47:35,200
Suppose I just have a first
order equation like d --

788
00:47:35,200 --> 00:47:41,130
I'll call it z --
dz/dx = delta(x-a).

789
00:47:41,130 --> 00:47:45,210
Yeah.

790
00:47:45,210 --> 00:47:53,700
And suppose that I know
that z(0) starts at zero.

791
00:47:53,700 --> 00:47:55,920
You've got to be able
to solve that equation,

792
00:47:55,920 --> 00:48:02,420
so this is a useful
prep for that.

793
00:48:02,420 --> 00:48:08,670
That would be a good equation
to know the solution to.

794
00:48:08,670 --> 00:48:13,770
And what kind of
function is this?

795
00:48:13,770 --> 00:48:17,140
What kind of a
function is z(x) there?

796
00:48:17,140 --> 00:48:20,730
I just use the letter
z to have a new letter.

797
00:48:20,730 --> 00:48:28,960
z(x) will be a step.

798
00:48:28,960 --> 00:48:33,110
Right. z(x) will be
a step function, yes.

799
00:48:33,110 --> 00:48:36,860
That's right.

800
00:48:36,860 --> 00:48:40,210
OK, so the solution
is that at the point

801
00:48:40,210 --> 00:48:44,230
a, which I'm presuming is beyond
zero, I come along at zero

802
00:48:44,230 --> 00:48:45,060
and I step up.

803
00:48:45,060 --> 00:48:45,690
Yep.

804
00:48:45,690 --> 00:48:46,230
OK.

805
00:48:46,230 --> 00:48:48,910
That would be a
picture of z(x), yeah.

806
00:48:48,910 --> 00:48:54,060
So it's basic delta
function material

807
00:48:54,060 --> 00:49:04,460
that I'm speaking about here.

808
00:49:04,460 --> 00:49:08,850
So z jumps by one
and if z is a slope,

809
00:49:08,850 --> 00:49:12,030
then the slope jumps
by one or drops by one,

810
00:49:12,030 --> 00:49:15,030
depending on a plus
or a minus sign here.

811
00:49:15,030 --> 00:49:19,370
The things that we've used
to deal with delta functions,

812
00:49:19,370 --> 00:49:23,170
so that's what
Question 4b is about.

813
00:49:23,170 --> 00:49:27,440
The drop in slope, or
the jumps, or whatever

814
00:49:27,440 --> 00:49:31,460
happens when a delta function
shows up on the right side.

815
00:49:31,460 --> 00:49:33,090
Good question.

816
00:49:33,090 --> 00:49:33,590
Yep.

817
00:49:33,590 --> 00:49:36,890
AUDIENCE: [INAUDIBLE]

818
00:49:36,890 --> 00:49:38,260
PROFESSOR STRANG: 1.1.27.

819
00:49:38,260 --> 00:49:38,760
Well.

820
00:49:38,760 --> 00:49:41,460
AUDIENCE: [INAUDIBLE]

821
00:49:41,460 --> 00:49:44,820
PROFESSOR STRANG: Oh,
and then left a typo.

822
00:49:44,820 --> 00:49:53,215
AUDIENCE: [INAUDIBLE]

823
00:49:53,215 --> 00:49:54,090
PROFESSOR STRANG: Oh.

824
00:49:54,090 --> 00:49:58,500
I'm sorry, OK.

825
00:49:58,500 --> 00:50:00,620
1.1.27.

826
00:50:00,620 --> 00:50:05,020
My copy isn't showing it.

827
00:50:05,020 --> 00:50:07,850
Yeah.

828
00:50:07,850 --> 00:50:10,330
I may have to punt
on that question.

829
00:50:10,330 --> 00:50:12,170
Or do you want me to look at it?

830
00:50:12,170 --> 00:50:18,200
OK, can you maybe just pass
that the book up, alright.

831
00:50:18,200 --> 00:50:21,020
I'll try to read what
that question was.

832
00:50:21,020 --> 00:50:23,180
OK.

833
00:50:23,180 --> 00:50:27,160
Yeah, maybe this is
a question to answer.

834
00:50:27,160 --> 00:50:33,590
This is probably the one
new question that got added.

835
00:50:33,590 --> 00:50:39,190
OK, yeah.

836
00:50:39,190 --> 00:50:41,070
Fair enough.

837
00:50:41,070 --> 00:50:45,630
So this is continuing
the discussion

838
00:50:45,630 --> 00:50:50,470
that you asked me to start here
about A, the first difference

839
00:50:50,470 --> 00:50:50,970
matrix, OK.

840
00:50:50,970 --> 00:50:57,670
So I'll go a little
more over that.

841
00:50:57,670 --> 00:51:00,780
So in the book here,
this writes down

842
00:51:00,780 --> 00:51:10,830
a matrix A_0, which is--
I'll discuss this matrix.

843
00:51:10,830 --> 00:51:16,210
So there's a difference matrix.

844
00:51:16,210 --> 00:51:19,110
You see what I mean by
a difference matrix,

845
00:51:19,110 --> 00:51:25,660
if were to multiply it by
u, [u 0, u 1, u 2, u 3]

846
00:51:25,660 --> 00:51:28,320
or something, I would
get differences, right?

847
00:51:28,320 --> 00:51:33,590
I'd get u_1-u_0,
u_2-u_1, and u_3-u_2.

848
00:51:33,590 --> 00:51:36,170


849
00:51:36,170 --> 00:51:37,420
Good.

850
00:51:37,420 --> 00:51:39,681
So that's A_0 times u.

851
00:51:39,681 --> 00:51:40,180
Alright.

852
00:51:40,180 --> 00:51:45,640
So that's a difference matrix.

853
00:51:45,640 --> 00:51:47,650
What graph would that come from?

854
00:51:47,650 --> 00:51:52,190
That's also the incidence
matrix of a very simple graph.

855
00:51:52,190 --> 00:51:56,340
This is connecting
Chapter 1 with Chapter 2.

856
00:51:56,340 --> 00:51:59,230
It would be a line of
springs, it would be a graph.

857
00:51:59,230 --> 00:52:02,840
It's got edges and nodes.

858
00:52:02,840 --> 00:52:05,890
It's got three edges,
so I've got three rows.

859
00:52:05,890 --> 00:52:13,120
It's got four nodes so
I've got four columns.

860
00:52:13,120 --> 00:52:16,500
Are the columns
independent here?

861
00:52:16,500 --> 00:52:18,090
No, they never are.

862
00:52:18,090 --> 00:52:23,500
The vector of all ones would
have differences of all zeroes.

863
00:52:23,500 --> 00:52:26,110
So what would that, that
would be the difference

864
00:52:26,110 --> 00:52:31,170
matrix for fixed?

865
00:52:31,170 --> 00:52:33,100
Free?

866
00:52:33,100 --> 00:52:38,240
Fixed, free, circular what
would that difference matrix

867
00:52:38,240 --> 00:52:41,350
correspond to?

868
00:52:41,350 --> 00:52:42,410
Everybody's saying it.

869
00:52:42,410 --> 00:52:44,630
Say it a little louder just to.

870
00:52:44,630 --> 00:52:46,020
Free-free.

871
00:52:46,020 --> 00:52:48,890
That's a free-free problem,
because they're all in there.

872
00:52:48,890 --> 00:52:50,523
We haven't knocked any out.

873
00:52:50,523 --> 00:52:52,360
There are no boundary
conditions yet.

874
00:52:52,360 --> 00:52:57,740
That's a free-free, so that
A_0 would be free-free.

875
00:52:57,740 --> 00:53:03,390
OK.

876
00:53:03,390 --> 00:53:06,716
I'll take one more case and
then I think we're at time.

877
00:53:06,716 --> 00:53:07,840
Suppose it was fixed-fixed?

878
00:53:07,840 --> 00:53:10,740


879
00:53:10,740 --> 00:53:14,540
What would be the difference
matrix that would correspond

880
00:53:14,540 --> 00:53:18,040
to, how would I
change that matrix

881
00:53:18,040 --> 00:53:20,860
if my problem
became fixed-fixed?

882
00:53:20,860 --> 00:53:26,220
So now I'm fixing that
u, I'm fixing that u,

883
00:53:26,220 --> 00:53:31,790
in the mass spring case
I'm adding supports there.

884
00:53:31,790 --> 00:53:34,680
How would that
change the matrix?

885
00:53:34,680 --> 00:53:36,680
First and fourth, good.

886
00:53:36,680 --> 00:53:37,980
Say it again?

887
00:53:37,980 --> 00:53:43,790
First and fourth
columns would go.

888
00:53:43,790 --> 00:53:48,290
So fixed-free would
then be three by two.

889
00:53:48,290 --> 00:53:52,040
Free-free was three by four.

890
00:53:52,040 --> 00:53:54,280
Yeah, I'm glad this
question came up

891
00:53:54,280 --> 00:53:56,580
because this is the
right thing for you to be

892
00:53:56,580 --> 00:54:02,960
thinking about in connection
with the recent question

893
00:54:02,960 --> 00:54:04,390
you asked.

894
00:54:04,390 --> 00:54:05,900
OK.

895
00:54:05,900 --> 00:54:07,710
Maybe that's the
right place to stop,

896
00:54:07,710 --> 00:54:11,170
because now you've asked
questions that are really

897
00:54:11,170 --> 00:54:16,510
on target for what we've done,
and I hope useful to you.

898
00:54:16,510 --> 00:54:23,580
OK, see you guys tomorrow
evening at 7:30 in 54-100, OK.

899
00:54:23,580 --> 00:54:25,332