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PROFESSOR STRANG: OK, so review
session on the first part of
10
00:00:26 --> 00:00:31
the Fourier chapter, these two
topics that we've done and that
11
00:00:31 --> 00:00:34
homework is now coming on.
12
00:00:34 --> 00:00:37
Fourier series, the
classical facts.
13
00:00:37 --> 00:00:44
Plus, paying attention to the
rate of decay of the Fourier
14
00:00:44 --> 00:00:48
coefficients, it's an aspect
not always mentioned.
15
00:00:48 --> 00:00:52
And the energy equality
is important.
16
00:00:52 --> 00:00:56
So it's not just here's the
function, find the coefficient.
17
00:00:56 --> 00:01:02
That's part of it
but not all of it.
18
00:01:02 --> 00:01:07
And then the discrete
series, we're doing today.
19
00:01:07 --> 00:01:11
OK, so those are the two topics
for today, and then the next
20
00:01:11 --> 00:01:15
review session, which would be
two weeks from now, would focus
21
00:01:15 --> 00:01:20
especially on convolution
and Fourier integrals.
22
00:01:20 --> 00:01:25
OK, so I'm open to questions
on the homework, or
23
00:01:25 --> 00:01:27
off the homework.
24
00:01:27 --> 00:01:32
Always fine.
25
00:01:32 --> 00:01:34
I didn't know how many
questions to ask you
26
00:01:34 --> 00:01:35
on the homework.
27
00:01:35 --> 00:01:41
I wanted you to have enough
practice doing this stuff
28
00:01:41 --> 00:01:46
because the time for this
Fourier part of the course
29
00:01:46 --> 00:01:47
is a little shorter.
30
00:01:47 --> 00:01:54
Thanksgiving comes into it, so
needed to do some exercise.
31
00:01:54 --> 00:01:56
And you've got a good question.
32
00:01:56 --> 00:01:56
Thanks.
33
00:01:56 --> 00:02:00
AUDIENCE: [INAUDIBLE]
34
00:02:00 --> 00:02:05
PROFESSOR STRANG: Number
18 on the homework, OK.
35
00:02:05 --> 00:02:08
Ah yes, OK.
36
00:02:08 --> 00:02:10
Right, alright.
37
00:02:10 --> 00:02:17
So the idea of that problem,
I'm really asking you to read
38
00:02:17 --> 00:02:22
the two pages, the last
two pages of the section.
39
00:02:22 --> 00:02:26
That use Fourier series to
solve the heat equation.
40
00:02:26 --> 00:02:30
So we've used, briefly,
Fourier series to solve
41
00:02:30 --> 00:02:32
Laplace's equation.
42
00:02:32 --> 00:02:34
So that was just to recall.
43
00:02:34 --> 00:02:37
So Fourier series to solve
Laplace's equation was when
44
00:02:37 --> 00:02:40
the region was a circle.
45
00:02:40 --> 00:02:44
The function was given, the
boundary values were given.
46
00:02:44 --> 00:02:47
It's 2pi periodic
because it is a circle.
47
00:02:47 --> 00:02:53
And we solved Laplace inside.
48
00:02:53 --> 00:02:58
Because on the boundary, the
perfect thing we needed was
49
00:02:58 --> 00:03:00
the Fourier series
to match the path.
50
00:03:00 --> 00:03:04
Now, I'm taking another,
classical, classical
51
00:03:04 --> 00:03:06
application too.
52
00:03:06 --> 00:03:07
The heat equation.
53
00:03:07 --> 00:03:13
So I made it heat equation,
so this direction is time.
54
00:03:13 --> 00:03:15
This direction is space.
55
00:03:15 --> 00:03:23
The heat equation is u_t=u_xx,
the coefficient, everybody here
56
00:03:23 --> 00:03:26
knows there would
be a c in there.
57
00:03:26 --> 00:03:29
But let's take it to be one.
58
00:03:29 --> 00:03:34
Then what are the solutions,
and how does a Fourier series
59
00:03:34 --> 00:03:37
help help you to match
the initial functions?
60
00:03:37 --> 00:03:43
So I'm matching, I'm
given u(x,0) here.
61
00:03:43 --> 00:03:44
OK.
62
00:03:44 --> 00:03:47
Along this is at time zero.
63
00:03:47 --> 00:03:50
So that says at a time
zero, so I have a bar.
64
00:03:50 --> 00:03:53
I have a conducting bar.
65
00:03:53 --> 00:03:59
And this is such a classical
example that I didn't feel you
66
00:03:59 --> 00:04:02
could miss it completely.
67
00:04:02 --> 00:04:06
Even though we look
beyond formulas.
68
00:04:06 --> 00:04:09
But here's one where
the formula shows us
69
00:04:09 --> 00:04:10
something important.
70
00:04:10 --> 00:04:15
OK, so what are solutions
to this equation?
71
00:04:15 --> 00:04:18
You look for solutions,
so the classical idea
72
00:04:18 --> 00:04:22
of separate variables.
73
00:04:22 --> 00:04:29
Look for solutions that are of
the form, some function of t,
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00:04:29 --> 00:04:32
maybe I'll try to use the
same letters as the text.
75
00:04:32 --> 00:04:38
Some function of t, times
some function of x, OK?
76
00:04:38 --> 00:04:43
And the text uses, look for
solutions. u(x,t) that
77
00:04:43 --> 00:04:44
are of this form.
78
00:04:44 --> 00:04:50
Some function of t, and
I didn't remember.
79
00:04:50 --> 00:04:51
Ah yes, it's A(x)B(t).
80
00:04:51 --> 00:04:56
81
00:04:56 --> 00:04:59
OK, that's what I
mean by separating.
82
00:04:59 --> 00:05:02
So those will be especially
simple solutions.
83
00:05:02 --> 00:05:06
And when we go to match the
initial condition, I'll just
84
00:05:06 --> 00:05:08
plug in t=0 and I'll
see the A(x)'s.
85
00:05:09 --> 00:05:12
Well, what are they?
86
00:05:12 --> 00:05:15
In this case, so their
eigenfunction - oh, I have to
87
00:05:15 --> 00:05:18
tell you about the remaining
boundary conditions, don't I?
88
00:05:18 --> 00:05:25
Because that will decide what
the A(x) has to satisfy, and
89
00:05:25 --> 00:05:28
will decide what those
eigenfunctions are.
90
00:05:28 --> 00:05:29
So, let's see.
91
00:05:29 --> 00:05:35
In this problem I think I
picked free conditions.
92
00:05:35 --> 00:05:40
So I made the interval minus
pi to pi, that's a change.
93
00:05:40 --> 00:05:43
Minus pi to pi just so we
have nice Fourier series.
94
00:05:43 --> 00:05:49
And here I have this boundary
is free, du/dx, u'.
95
00:05:51 --> 00:05:52
At x=-pi.
96
00:05:54 --> 00:05:59
For all time, so up
this line is zero.
97
00:05:59 --> 00:06:05
And also u' at plus pi,
and all time is zero.
98
00:06:05 --> 00:06:09
So up those lines,
heat's going out.
99
00:06:09 --> 00:06:11
That's what that means.
100
00:06:11 --> 00:06:13
Is that what that means,
or does that mean
101
00:06:13 --> 00:06:14
heat can't go out?
102
00:06:14 --> 00:06:18
No, so what's happening?
103
00:06:18 --> 00:06:19
Heat's not going out.
104
00:06:19 --> 00:06:21
Is that right?
105
00:06:21 --> 00:06:23
The slope is zero, right?
106
00:06:23 --> 00:06:26
The slope is the temperature
gradient, we're requiring
107
00:06:26 --> 00:06:27
the temperature gradient.
108
00:06:27 --> 00:06:30
So the ends of the
bar are insulated.
109
00:06:30 --> 00:06:32
So this is insulated.
110
00:06:32 --> 00:06:35
No passage through, right?
111
00:06:35 --> 00:06:36
Is that what that means?
112
00:06:36 --> 00:06:37
Yeah.
113
00:06:37 --> 00:06:43
It's like there's nobody,
it's cut off there.
114
00:06:43 --> 00:06:46
The rod isn't extended
for heat to go further.
115
00:06:46 --> 00:06:55
OK, so that tells us what
the x and the t, what
116
00:06:55 --> 00:06:57
the A(x) and B(t) are.
117
00:06:57 --> 00:06:58
So here's the point.
118
00:06:58 --> 00:07:01
You plug that hoped-for
solution into the
119
00:07:01 --> 00:07:04
equation, right?
120
00:07:04 --> 00:07:07
So I plug it in here.
121
00:07:07 --> 00:07:10
What do I have on the left
side, just the time derivative?
122
00:07:10 --> 00:07:16
So that's A times A(x), B'(t),
you see taking the time
123
00:07:16 --> 00:07:17
derivative doesn't touch A(x).
124
00:07:18 --> 00:07:22
On the right-hand side, I
don't touch B(t), but I
125
00:07:22 --> 00:07:25
have a second derivative.
126
00:07:25 --> 00:07:27
Of the x part.
127
00:07:27 --> 00:07:30
So far, so good.
128
00:07:30 --> 00:07:35
Now, a little trick.
129
00:07:35 --> 00:07:39
If I divide both sides
by A and by B, I get
130
00:07:39 --> 00:07:43
B'/B equaling A''/A.
131
00:07:43 --> 00:07:47
132
00:07:47 --> 00:07:48
Right?
133
00:07:48 --> 00:07:52
Just, put the A under here,
put the B under here.
134
00:07:52 --> 00:07:53
Now what?
135
00:07:53 --> 00:07:58
This is neat, because
that function is only
136
00:07:58 --> 00:08:00
depending on time.
137
00:08:00 --> 00:08:05
This function depends only on
x, so that the both sides
138
00:08:05 --> 00:08:06
have to be constant.
139
00:08:06 --> 00:08:11
One can't actually change with
time, because this side is
140
00:08:11 --> 00:08:13
not changing with time.
141
00:08:13 --> 00:08:15
And this couldn't actually
change with x, because
142
00:08:15 --> 00:08:17
that's not changing with x.
143
00:08:17 --> 00:08:20
So those are both constants.
144
00:08:20 --> 00:08:23
So both constants.
145
00:08:23 --> 00:08:27
Let's see, I'll maybe
just put a constant.
146
00:08:27 --> 00:08:30
And various constants
are possible.
147
00:08:30 --> 00:08:32
OK, so now you see
the point here.
148
00:08:32 --> 00:08:36
Now I have two separate
equations, I have an equation
149
00:08:36 --> 00:08:39
for the B part, dB/dt, B'.
150
00:08:40 --> 00:08:46
If I bring the B up there,
I have equals, the
151
00:08:46 --> 00:08:48
constant times B.
152
00:08:48 --> 00:08:50
And I know the
solution to that.
153
00:08:50 --> 00:08:56
B(t) is, as everybody knows,
what's the solution to
154
00:08:56 --> 00:08:59
a first order constant
coefficient equation?
155
00:08:59 --> 00:08:59
Just e^(ct)*B(0).
156
00:08:59 --> 00:09:03
157
00:09:03 --> 00:09:06
Good, we've got B.
158
00:09:06 --> 00:09:09
We've got a B(t) that
works, and now what's
159
00:09:09 --> 00:09:11
the A that also works?
160
00:09:11 --> 00:09:12
That has A''.
161
00:09:14 --> 00:09:20
And I bring the A up, so now
I have A''=cA, so the A
162
00:09:20 --> 00:09:25
that goes with it is?
163
00:09:25 --> 00:09:32
Oh, OK, I've used a c
there, so what's the good?
164
00:09:32 --> 00:09:35
I want two derivatives
should bring down a c.
165
00:09:35 --> 00:09:42
Let me change c to
minus lambda squared.
166
00:09:42 --> 00:09:46
How about if I look ahead,
change this constant to minus
167
00:09:46 --> 00:09:50
lambda squared, because I want
something where two derivatives
168
00:09:50 --> 00:09:55
bring down minus lambda
squared, and what will do that?
169
00:09:55 --> 00:10:00
Any amount of
cos(lambda*x), right?
170
00:10:00 --> 00:10:02
Because two derivatives
will bring down a
171
00:10:02 --> 00:10:04
minus lambda squared.
172
00:10:04 --> 00:10:06
And any amount of
sin(lambda*x).
173
00:10:07 --> 00:10:18
And this is now e to the
minus lambda squared t.
174
00:10:18 --> 00:10:21
I'm doing this fast, but
actually it's totally simple.
175
00:10:21 --> 00:10:25
The conclusion is that I now
have a bunch of solutions of
176
00:10:25 --> 00:10:28
this special separated form.
177
00:10:28 --> 00:10:33
Where B(t) could be that and
A(x) could be either of those
178
00:10:33 --> 00:10:35
or any combination of those.
179
00:10:35 --> 00:10:40
And I have to use the same
lambda for each, so that the
180
00:10:40 --> 00:10:46
two equations will work
in the original problem.
181
00:10:46 --> 00:10:47
Good.
182
00:10:47 --> 00:10:51
Now, so far, no
boundary conditions.
183
00:10:51 --> 00:10:54
What I've got so far is
just a lot of solutions.
184
00:10:54 --> 00:10:56
These times that.
185
00:10:56 --> 00:10:58
With any lambda.
186
00:10:58 --> 00:11:01
But of course the boundary
conditions will tell me the
187
00:11:01 --> 00:11:03
lambdas, first of all.
188
00:11:03 --> 00:11:04
And how do they tell me?
189
00:11:04 --> 00:11:08
The only boundary condition
I have is this free stuff.
190
00:11:08 --> 00:11:14
So it's, free the slope should
be zero at pi, and zero at?
191
00:11:14 --> 00:11:16
So that's the x direction.
192
00:11:16 --> 00:11:19
So that's going to tell
me - I've forgotten.
193
00:11:19 --> 00:11:23
Do I want cosines or sines?
194
00:11:23 --> 00:11:24
Cosines.
195
00:11:24 --> 00:11:29
I want the derivative
to be zero at pi.
196
00:11:29 --> 00:11:32
Yeah, so I think I
want cosines, good.
197
00:11:32 --> 00:11:35
And then lambdas can't
be anything at all.
198
00:11:35 --> 00:11:41
Because, should lambda be an
integer or something like that?
199
00:11:41 --> 00:11:44
I think maybe lambda should
be an integer, because
200
00:11:44 --> 00:11:47
I want to plug in pi.
201
00:11:47 --> 00:11:50
So let me take the
second derivative.
202
00:11:50 --> 00:11:53
Is minus lambda squared
cos(lambda*x).
203
00:11:54 --> 00:11:56
And then I want to
plug in lambda=x=pi.
204
00:11:58 --> 00:12:04
And I want this to be zero.
205
00:12:04 --> 00:12:10
So lambda should be an
integer, is that right?
206
00:12:10 --> 00:12:13
At multiples of pi, the
cosine is zero, yes.
207
00:12:13 --> 00:12:16
Is it? no.
208
00:12:16 --> 00:12:19
Did I want sine?
209
00:12:19 --> 00:12:23
Maybe I wanted sine.
210
00:12:23 --> 00:12:26
Oh, it's the first derivative
I'm looking at, thank you.
211
00:12:26 --> 00:12:27
Thank you.
212
00:12:27 --> 00:12:31
OK, good.
213
00:12:31 --> 00:12:33
OK, now I've got it.
214
00:12:33 --> 00:12:34
Thank you.
215
00:12:34 --> 00:12:37
And now I see lambda
should be an integer.
216
00:12:37 --> 00:12:41
Lambda should be, zero
is, yeah zero's alright.
217
00:12:41 --> 00:12:43
That's the constant term,
yeah, we need that. zero,
218
00:12:43 --> 00:12:46
one, two, and so on.
219
00:12:46 --> 00:12:52
So do you see that I've now
got, now I can take, I've
220
00:12:52 --> 00:12:53
got a lot of solutions.
221
00:12:53 --> 00:12:55
And I have a linear problem.
222
00:12:55 --> 00:12:57
So I can take any combination.
223
00:12:57 --> 00:13:05
So finally I have final
solution is that u(x,t) is any
224
00:13:05 --> 00:13:12
combination with coefficients,
I'm free to choose, of A(x),
225
00:13:12 --> 00:13:18
which is cos(nx), because
lambda had to be an n.
226
00:13:18 --> 00:13:22
And n could be anywhere
from zero on up.
227
00:13:22 --> 00:13:31
Times e to the minus, lambda's
n, so that's n squared t.
228
00:13:31 --> 00:13:32
Did I get that?
229
00:13:32 --> 00:13:34
Let me draw a circle
and step back.
230
00:13:34 --> 00:13:35
What's up?
231
00:13:35 --> 00:13:37
AUDIENCE: [INAUDIBLE]
232
00:13:37 --> 00:13:40
PROFESSOR STRANG: I could have
negative n's, they wouldn't
233
00:13:40 --> 00:13:42
give me anything new, right?
234
00:13:42 --> 00:13:48
I mean cos(nx) and cos(-nx)
are just, one's just the
235
00:13:48 --> 00:13:51
negative of the other.
236
00:13:51 --> 00:13:53
So these are the good guys.
237
00:13:53 --> 00:13:56
I've got a cosine series
because I've got
238
00:13:56 --> 00:13:58
free n's, right.
239
00:13:58 --> 00:14:00
Because of the
boundary conditions.
240
00:14:00 --> 00:14:01
Do you say that?
241
00:14:01 --> 00:14:04
That's pretty nice.
242
00:14:04 --> 00:14:08
There's my A(x), there's
my B(t), I can take
243
00:14:08 --> 00:14:10
any combination.
244
00:14:10 --> 00:14:11
Usual stuff.
245
00:14:11 --> 00:14:14
You get to that solution.
246
00:14:14 --> 00:14:18
OK, and now I have to meet the
initial conditions, right?
247
00:14:18 --> 00:14:21
Boundary conditions are
now built-in because
248
00:14:21 --> 00:14:23
I chose cosine.
249
00:14:23 --> 00:14:24
Or you did.
250
00:14:24 --> 00:14:30
Now, this will tell
me what the c's are.
251
00:14:30 --> 00:14:31
I'm going to set t=0.
252
00:14:33 --> 00:14:38
At t=0, I'm given the initial
condition, u(x,0), and I
253
00:14:38 --> 00:14:40
have the same sum of
c_n*cos(nx), e^0.
254
00:14:40 --> 00:14:45
255
00:14:45 --> 00:14:49
So this will tell me the c's
are the cosine coefficients of
256
00:14:49 --> 00:14:51
the given initial conditions.
257
00:14:51 --> 00:14:56
So I expand, so here's where
Fourier series is paid off.
258
00:14:56 --> 00:15:01
Expand the initial function
in a cosine series.
259
00:15:01 --> 00:15:04
And then go forward in time.
260
00:15:04 --> 00:15:06
This is just the
old e^(lambda*t).
261
00:15:06 --> 00:15:10
262
00:15:10 --> 00:15:14
Only the lambda we're talking
about here is minus n squared.
263
00:15:14 --> 00:15:18
And you see what's
happening here?
264
00:15:18 --> 00:15:21
What's going to happen
for large time?
265
00:15:21 --> 00:15:23
So this is a very
physical problem.
266
00:15:23 --> 00:15:27
That I think you cannot
take 18.085 without
267
00:15:27 --> 00:15:29
seeing this problem.
268
00:15:29 --> 00:15:32
You can't learn about Fourier
series without using it
269
00:15:32 --> 00:15:36
for the initial value.
270
00:15:36 --> 00:15:41
And then propagating in time
with the usual exponentials.
271
00:15:41 --> 00:15:46
For each one, and now as n
increases what do I see?
272
00:15:46 --> 00:15:48
Faster and faster decay.
273
00:15:48 --> 00:15:54
For large n, these are going
to zero extremely fast.
274
00:15:54 --> 00:16:00
So that what you see with a
solid bar, which starts with
275
00:16:00 --> 00:16:08
the temperature u in some
probably not negative unless
276
00:16:08 --> 00:16:10
it's a really cold bar.
277
00:16:10 --> 00:16:16
But, anyway, it starts with
some initial temperature.
278
00:16:16 --> 00:16:18
That flattens out fast.
279
00:16:18 --> 00:16:21
The heat flows, to equilibrate.
280
00:16:21 --> 00:16:24
What I approach is the
constant terms, c_0.
281
00:16:26 --> 00:16:27
This approaches c_0.
282
00:16:28 --> 00:16:33
Because all these other n
positives, they go to zero.
283
00:16:33 --> 00:16:38
So the heat distributes
itself equally.
284
00:16:38 --> 00:16:42
OK, and now I guess the
particular u(0) in the
285
00:16:42 --> 00:16:46
problem was a delta.
286
00:16:46 --> 00:16:53
OK, and so the particular
u(0) was all, was from
287
00:16:53 --> 00:16:58
that really hot point.
288
00:16:58 --> 00:17:00
So we know the coefficients.
289
00:17:00 --> 00:17:05
We know the cosine coefficients
for the delta function, we know
290
00:17:05 --> 00:17:08
these c's, and what were they?
291
00:17:08 --> 00:17:12
1/2pi was c_0.
292
00:17:12 --> 00:17:15
And the other c's
were 1/pi, I think.
293
00:17:15 --> 00:17:18
Is that right?
294
00:17:18 --> 00:17:19
Maybe they're all 1/2pi.
295
00:17:21 --> 00:17:22
Maybe.
296
00:17:22 --> 00:17:22
Yeah.
297
00:17:22 --> 00:17:24
Whatever.
298
00:17:24 --> 00:17:25
They disappear fast.
299
00:17:25 --> 00:17:28
And this is what we approach.
300
00:17:28 --> 00:17:31
So the heat from the
delta function is, yeah.
301
00:17:31 --> 00:17:37
So is that everything
the problem wanted?
302
00:17:37 --> 00:17:37
Yeah.
303
00:17:37 --> 00:17:38
Yeah.
304
00:17:38 --> 00:17:40
I think we've done it.
305
00:17:40 --> 00:17:44
We'll put in the c_n's to
complete that picture.
306
00:17:44 --> 00:17:46
Into here.
307
00:17:46 --> 00:17:51
And then c_0 is the one
that survives over time.
308
00:17:51 --> 00:17:54
Yeah.
309
00:17:54 --> 00:17:56
I guess you've, once I
got rolling I couldn't
310
00:17:56 --> 00:18:01
stop and that's u.
311
00:18:01 --> 00:18:06
For investing time this
afternoon you get a fast look
312
00:18:06 --> 00:18:11
at this classical, classical
problem of separating the
313
00:18:11 --> 00:18:14
variables using the
Fourier series for
314
00:18:14 --> 00:18:16
the initial function.
315
00:18:16 --> 00:18:20
And recognizing that
we're doing this on
316
00:18:20 --> 00:18:24
a finite interval.
317
00:18:24 --> 00:18:32
If the bar was infinitely long,
then we would be talking
318
00:18:32 --> 00:18:34
about Fourier integrals.
319
00:18:34 --> 00:18:37
And that's what's
coming up a bit later.
320
00:18:37 --> 00:18:40
We would integrate
instead of sum.
321
00:18:40 --> 00:18:40
Yeah.
322
00:18:40 --> 00:18:45
But the idea would not be
different, if we had infinite
323
00:18:45 --> 00:18:48
bar then we would not be
restricted to n equals zero,
324
00:18:48 --> 00:18:52
one, two, three, any n, any
cosine, wouldn't have to
325
00:18:52 --> 00:18:54
be an integer at all.
326
00:18:54 --> 00:18:57
Any number, any frequency
would be allowed.
327
00:18:57 --> 00:19:02
And so we would have to
integrate that, yeah.
328
00:19:02 --> 00:19:08
And that's a classical
problem too, again.
329
00:19:08 --> 00:19:11
It's come up in a modern
way, that the famous
330
00:19:11 --> 00:19:15
Black-Scholes equation.
331
00:19:15 --> 00:19:17
So.
332
00:19:17 --> 00:19:21
The heat equation
is for 18.086.
333
00:19:21 --> 00:19:26
Here, we brought it up because
we could solve it fast.
334
00:19:26 --> 00:19:30
But the actual, yeah.
335
00:19:30 --> 00:19:33
The most important solution I
could give you to the heat
336
00:19:33 --> 00:19:36
equation would be the one
that starts from that
337
00:19:36 --> 00:19:39
point source of heat.
338
00:19:39 --> 00:19:42
But on the whole line.
339
00:19:42 --> 00:19:46
The one that would be
integrals instead of sums.
340
00:19:46 --> 00:19:47
Yeah.
341
00:19:47 --> 00:19:51
So we came pretty close to
solving the most important
342
00:19:51 --> 00:19:53
heat equation problem.
343
00:19:53 --> 00:20:00
But doing the periodic case,
with just cosines, and the
344
00:20:00 --> 00:20:05
infinite line case would be
the most famous of all.
345
00:20:05 --> 00:20:05
Yeah.
346
00:20:05 --> 00:20:09
And it has a beautiful form,
and I was going to say that
347
00:20:09 --> 00:20:12
the heat equation's
pretty classical.
348
00:20:12 --> 00:20:19
But let's see, where
can I write the magic
349
00:20:19 --> 00:20:22
words, Black-Scholes.
350
00:20:22 --> 00:20:27
Next to the heat equation, so
that's the heat equation but
351
00:20:27 --> 00:20:36
it's also - do you know these
names, Black and Scholes?
352
00:20:36 --> 00:20:40
Anybody in Mathematics
of Finance?
353
00:20:40 --> 00:20:47
So these much-despised option,
derivative options, so people
354
00:20:47 --> 00:20:53
on Wall Street, traders, will
carry around a little
355
00:20:53 --> 00:20:58
calculator that solves the
Black-Scholes equation so they
356
00:20:58 --> 00:21:02
can price the options that
they're bidding to buy and
357
00:21:02 --> 00:21:04
sell, so they can
price them fast.
358
00:21:04 --> 00:21:10
And that little calculator does
a finite differences, or a
359
00:21:10 --> 00:21:15
Fourier series solution to this
Black-Scholes equation, which
360
00:21:15 --> 00:21:18
actually, if you change
variables on it, is
361
00:21:18 --> 00:21:19
the heat equation.
362
00:21:19 --> 00:21:25
So what you see here as is
actually important on Wall
363
00:21:25 --> 00:21:28
Street except, it's probably
not the right moment
364
00:21:28 --> 00:21:29
to mention it.
365
00:21:29 --> 00:21:34
Right?
366
00:21:34 --> 00:21:38
So you can blame the whole
meltdown on mathematicians.
367
00:21:38 --> 00:21:41
But that wouldn't
be entirely fair.
368
00:21:41 --> 00:21:43
They didn't mean it, anyway.
369
00:21:43 --> 00:21:50
But that's been the biggest
source of employment, I
370
00:21:50 --> 00:21:55
guess, apart from teaching,
in the last ten years.
371
00:21:55 --> 00:21:59
People who could work out the
partial differential equations,
372
00:21:59 --> 00:22:01
and they get more complicated
than the heat equation,
373
00:22:01 --> 00:22:03
you can be sure.
374
00:22:03 --> 00:22:11
And so the classical one, these
guys are economists at MIT
375
00:22:11 --> 00:22:13
and Harvard, or they were.
376
00:22:13 --> 00:22:18
And I guess maybe the Nobel
Prize in Economics came
377
00:22:18 --> 00:22:20
to part of that group.
378
00:22:20 --> 00:22:23
And also to Merton.
379
00:22:23 --> 00:22:28
Maybe Black, possibly Black
died before the time
380
00:22:28 --> 00:22:31
of the Nobel Prize.
381
00:22:31 --> 00:22:32
Anyway, they were the first.
382
00:22:32 --> 00:22:36
And it's a beautiful paper,
beautiful paper too.
383
00:22:36 --> 00:22:41
Just to figure out how do you
price the, what's the value
384
00:22:41 --> 00:22:47
of an option to buy that
allows you to buy or sell
385
00:22:47 --> 00:22:49
a stock at a later time?
386
00:22:49 --> 00:22:50
Yeah.
387
00:22:50 --> 00:22:54
So it's, well of course you
have to make assumptions on
388
00:22:54 --> 00:22:58
what's going to happen over
that time and that's where Wall
389
00:22:58 --> 00:23:01
Street came to grief, I guess.
390
00:23:01 --> 00:23:10
If you had to put it in a
nutshell, I mean, the options,
391
00:23:10 --> 00:23:13
the standard straightforward
options, those are fine.
392
00:23:13 --> 00:23:18
Using Black-Scholes, and then
what's happened is they now
393
00:23:18 --> 00:23:21
price more and more
complicated things.
394
00:23:21 --> 00:23:27
To the point that the banks
were buying and selling credit
395
00:23:27 --> 00:23:32
default swaps, insurance swaps,
that practically nobody
396
00:23:32 --> 00:23:33
understood what they were.
397
00:23:33 --> 00:23:37
They just assumed that if there
was always a market, for them
398
00:23:37 --> 00:23:39
like insurance, somehow
it wouldn't happen.
399
00:23:39 --> 00:23:41
And you get on.
400
00:23:41 --> 00:23:42
But it happened.
401
00:23:42 --> 00:23:46
So, now we're in trouble.
402
00:23:46 --> 00:23:50
OK, that's not
18.085, fortunately.
403
00:23:50 --> 00:23:55
Or math, but.
404
00:23:55 --> 00:23:58
Anyway, a lot of people got
involved with things they
405
00:23:58 --> 00:24:01
didn't really know about.
406
00:24:01 --> 00:24:06
And then were selling them
as well as of course
407
00:24:06 --> 00:24:07
the mortgage problems.
408
00:24:07 --> 00:24:11
Anyway.
409
00:24:11 --> 00:24:20
Ready for other questions on
our homework, or these topics.
410
00:24:20 --> 00:24:21
Yeah.
411
00:24:21 --> 00:24:22
OK.
412
00:24:22 --> 00:24:24
AUDIENCE: [INAUDIBLE]
413
00:24:24 --> 00:24:24
PROFESSOR STRANG: OK
414
00:24:24 --> 00:24:28
AUDIENCE: [INAUDIBLE]
415
00:24:28 --> 00:24:35
PROFESSOR STRANG:
OK, right, yeah.
416
00:24:35 --> 00:24:37
Have an image of waves, so --
417
00:24:37 --> 00:24:38
AUDIENCE: [INAUDIBLE]
418
00:24:38 --> 00:24:41
PROFESSOR STRANG: Yeah.
419
00:24:41 --> 00:24:45
I suppose if I had to have a
picture of the discrete thing,
420
00:24:45 --> 00:24:52
if my picture of the function
case was a bunch of sines, and
421
00:24:52 --> 00:24:57
cosines, somehow adding
up to my function.
422
00:24:57 --> 00:25:03
And if time, I'm solving the
heat equation then those
423
00:25:03 --> 00:25:09
separate waves are maybe
decaying in time.
424
00:25:09 --> 00:25:09
Here.
425
00:25:09 --> 00:25:12
So that when I add them
up at a later time I get
426
00:25:12 --> 00:25:13
something different.
427
00:25:13 --> 00:25:18
Or if it was the wave equation,
which is probably your image,
428
00:25:18 --> 00:25:20
they're moving in time.
429
00:25:20 --> 00:25:23
So they add up to different
things at different times,
430
00:25:23 --> 00:25:26
because they can move
at different speeds.
431
00:25:26 --> 00:25:31
Yes, so a function is a
sum of waves, right?
432
00:25:31 --> 00:25:34
Then what would the
discrete guy be?
433
00:25:34 --> 00:25:38
I guess I would just have to
imagine the function as only
434
00:25:38 --> 00:25:41
having those n values.
435
00:25:41 --> 00:25:45
And my wave would just
be, a wave might be
436
00:25:45 --> 00:25:51
just n values there.
437
00:25:51 --> 00:25:58
But still, if I have a time-
dependent problem maybe
438
00:25:58 --> 00:26:01
that thing is pulsing
thing up and down.
439
00:26:01 --> 00:26:05
It's just that it's only got
a fixed number of points.
440
00:26:05 --> 00:26:09
And I'm not looking at the
whole wave on a, yeah.
441
00:26:09 --> 00:26:15
But I don't think it's
essentially different.
442
00:26:15 --> 00:26:19
And of course, the fast Fourier
transform and the discrete case
443
00:26:19 --> 00:26:23
is used to approximate
the continuous one.
444
00:26:23 --> 00:26:24
Yeah.
445
00:26:24 --> 00:26:29
You can look in numerical
recipes for a discussion of
446
00:26:29 --> 00:26:32
approximating the Fourier
series by discrete
447
00:26:32 --> 00:26:33
Fourier series.
448
00:26:33 --> 00:26:34
I mean, that's an
important question.
449
00:26:34 --> 00:26:37
Because of course, Fourier
series has got all
450
00:26:37 --> 00:26:39
these integrals.
451
00:26:39 --> 00:26:46
The coefficients come from
an integral formula.
452
00:26:46 --> 00:26:50
We're not going to do those
integrals exactly, so we
453
00:26:50 --> 00:26:53
have to do them some
approximate way.
454
00:26:53 --> 00:26:57
And one way would be
to use equally spaced
455
00:26:57 --> 00:26:58
points and do the DFT.
456
00:27:00 --> 00:27:04
Can you just remind me what
that integral formula is?
457
00:27:04 --> 00:27:07
I don't want you to, it
was on the board today.
458
00:27:07 --> 00:27:11
What's the formula for
the coefficient c_k in
459
00:27:11 --> 00:27:15
the Fourier series.
460
00:27:15 --> 00:27:20
I'm really just asking you this
because I think you should have
461
00:27:20 --> 00:27:26
it in some memory cache, you
know in fast memory, rather
462
00:27:26 --> 00:27:28
than in the textbook.
463
00:27:28 --> 00:27:32
OK, so what's the
formula for c_k, in the
464
00:27:32 --> 00:27:34
Fourier series case?
465
00:27:34 --> 00:27:36
Everybody think about it.
466
00:27:36 --> 00:27:39
It's going to be an
integral, right?
467
00:27:39 --> 00:27:42
And I'll take it over zero
to 2pi, I don't mind.
468
00:27:42 --> 00:27:44
Or minus pi, pi.
469
00:27:44 --> 00:27:46
And what do I integrate?
470
00:27:46 --> 00:27:48
As the Fourier coefficients
of the function f(x)?
471
00:27:49 --> 00:27:55
So I take f(x), I remember to
divide by 2pi, I'm doing the
472
00:27:55 --> 00:28:00
continuous one, what do I
multiply by to get the
473
00:28:00 --> 00:28:03
coefficients? e^(-ikx).
474
00:28:03 --> 00:28:12
475
00:28:12 --> 00:28:16
So I've forgotten whether
I signed some of these
476
00:28:16 --> 00:28:19
very early questions.
477
00:28:19 --> 00:28:20
It just gave you the
function and said
478
00:28:20 --> 00:28:22
what's the coefficient.
479
00:28:22 --> 00:28:24
If I just look at one or two.
480
00:28:24 --> 00:28:26
Suppose my function is f(x)=x.
481
00:28:27 --> 00:28:31
I guess in that problem,
in Problem 1, I made
482
00:28:31 --> 00:28:33
it minus pi to pi.
483
00:28:33 --> 00:28:34
Suppose f(x)=x.
484
00:28:34 --> 00:28:40
485
00:28:40 --> 00:28:43
So you have to integrate
x times e^(-ikx).
486
00:28:43 --> 00:28:48
487
00:28:48 --> 00:28:50
Well, you got an
integral to do.
488
00:28:50 --> 00:28:55
OK, it's doable but
not instantly doable.
489
00:28:55 --> 00:28:58
Let me ask you some questions
and you tell me about it.
490
00:28:58 --> 00:29:01
So I draw the function.
491
00:29:01 --> 00:29:05
The function is x
from minus pi to pi.
492
00:29:05 --> 00:29:08
And tell me about the
coefficients, how
493
00:29:08 --> 00:29:13
quickly do they decay?
494
00:29:13 --> 00:29:18
This is like some constant
over k to some power
495
00:29:18 --> 00:29:23
p, and what's p?
496
00:29:23 --> 00:29:27
What rate of decay are you
expecting for the coefficients?
497
00:29:27 --> 00:29:29
Well you say to yourself, it
looks like a pretty nice
498
00:29:29 --> 00:29:32
function, smooth as can be.
499
00:29:32 --> 00:29:37
But, what's the answer here?
500
00:29:37 --> 00:29:43
The rate of decay will be,
what will that power be?
501
00:29:43 --> 00:29:44
One.
502
00:29:44 --> 00:29:49
Because the function jumps.
503
00:29:49 --> 00:29:53
The function has a jump there,
and the Fourier coefficients
504
00:29:53 --> 00:29:55
have got to deal with it.
505
00:29:55 --> 00:29:59
So the Fourier series for this
is going to be, if I took 100
506
00:29:59 --> 00:30:01
terms it would be really close.
507
00:30:01 --> 00:30:09
And then it'll go down here to
the, but it's got to get there.
508
00:30:09 --> 00:30:13
And got to start, and
pick up below there.
509
00:30:13 --> 00:30:18
So it's got the same issue
that the Gibbs phenomenon,
510
00:30:18 --> 00:30:21
that the square wave had.
511
00:30:21 --> 00:30:22
It's got that jump.
512
00:30:22 --> 00:30:25
So it'll go like 1/k.
513
00:30:26 --> 00:30:29
OK, at coming back.
514
00:30:29 --> 00:30:31
Could you actually
find those numbers?
515
00:30:31 --> 00:30:34
And do you remember how to
do integral like that?
516
00:30:34 --> 00:30:38
Well, look it up, I guess
is the best answer.
517
00:30:38 --> 00:30:44
But whoever did it the first
time, well, it's integration
518
00:30:44 --> 00:30:46
by parts, somehow.
519
00:30:46 --> 00:30:50
The derivative of this makes it
real simple, and this we can
520
00:30:50 --> 00:30:52
integrate really easily, right?
521
00:30:52 --> 00:30:55
So we integrate that, take
the derivative of that.
522
00:30:55 --> 00:31:02
We get a boundary term,
so I don't exactly
523
00:31:02 --> 00:31:03
remember the formula.
524
00:31:03 --> 00:31:09
But it'll have a couple
of terms, but not bad.
525
00:31:09 --> 00:31:12
And we'll see this 1/k.
526
00:31:13 --> 00:31:16
OK, and that's the formula.
527
00:31:16 --> 00:31:21
If I changed x to something
else, let's see.
528
00:31:21 --> 00:31:24
As long as I'm looking at
number one, what if I took e^x?
529
00:31:24 --> 00:31:25
Oh, easy.
530
00:31:25 --> 00:31:31
Right? e^x, yeah, let's do e^x,
because that's an easy one.
531
00:31:31 --> 00:31:35
Now, e^x looks like, so
what's my graph of e^x?
532
00:31:36 --> 00:31:41
It's quite small here at minus
pi, and pretty large at e^pi.
533
00:31:42 --> 00:31:45
What's the rate of decay of
the Fourier coefficients
534
00:31:45 --> 00:31:48
for that guy?
535
00:31:48 --> 00:31:50
Same.
536
00:31:50 --> 00:31:51
Got to jump again.
537
00:31:51 --> 00:31:52
Drops down here.
538
00:31:52 --> 00:31:55
So let's find it.
539
00:31:55 --> 00:31:58
Let's find those
Fourier, that integral.
540
00:31:58 --> 00:31:59
That's e^(x-ik).
541
00:32:02 --> 00:32:06
Sorry, let's put down what
I'm integrating here.
542
00:32:06 --> 00:32:08
I'm integrating e^x(1-ik).
543
00:32:08 --> 00:32:13
544
00:32:13 --> 00:32:16
Right?
545
00:32:16 --> 00:32:18
That's what I've
got to integrate.
546
00:32:18 --> 00:32:22
The integral is the same
thing divided by 1-ik.
547
00:32:22 --> 00:32:24
548
00:32:24 --> 00:32:28
I plug in the endpoints,
minus pi and pi, and
549
00:32:28 --> 00:32:29
I've got the answer.
550
00:32:29 --> 00:32:30
And I divide by 2pi.
551
00:32:30 --> 00:32:33
552
00:32:33 --> 00:32:34
Yeah.
553
00:32:34 --> 00:32:36
So that's a totally doable one.
554
00:32:36 --> 00:32:37
Let's plug it in.
555
00:32:37 --> 00:32:37
1/2pi.
556
00:32:37 --> 00:32:40
557
00:32:40 --> 00:32:45
And then the denominator is
this 1-ik, yeah, well there
558
00:32:45 --> 00:32:47
you see it already, right?
559
00:32:47 --> 00:32:51
You see already the 1/k
in the denominator.
560
00:32:51 --> 00:32:54
That's giving us the
slow decay rate.
561
00:32:54 --> 00:32:56
And now I'm plugging in
x=e^pi(1-ik)-e^-pi(1-ik).
562
00:32:56 --> 00:33:10
563
00:33:10 --> 00:33:21
So as k gets big,
this is slow decay.
564
00:33:21 --> 00:33:24
Now, what's happening
here as k gets large?
565
00:33:24 --> 00:33:27
Oh, k is multiplied
by i, there.
566
00:33:27 --> 00:33:33
So this e^(pi*ik) is just
sitting, it may even be
567
00:33:33 --> 00:33:36
just one or something, or
minus one, or whatever.
568
00:33:36 --> 00:33:37
Is it?
569
00:33:37 --> 00:33:41
Yeah. k is just an integer,
this is k*pi*i, that's just
570
00:33:41 --> 00:33:43
minus one to something.
571
00:33:43 --> 00:33:49
So all that numerator is minus
1one to the to the k'th
572
00:33:49 --> 00:33:52
power or something.
573
00:33:52 --> 00:33:53
Times e^pi.
574
00:33:53 --> 00:33:56
575
00:33:56 --> 00:34:05
So e^pi pi there. e^-ik, at
e^(ii*pi*k) is just minus one.
576
00:34:05 --> 00:34:08
And this is maybe e to
the minus pi times
577
00:34:08 --> 00:34:09
the same e^(+i*pi*k).
578
00:34:09 --> 00:34:14
579
00:34:14 --> 00:34:16
I think that's minus
one to the k.
580
00:34:16 --> 00:34:17
Well, wait a minute.
581
00:34:17 --> 00:34:20
Maybe they're not
both, whatever.
582
00:34:20 --> 00:34:22
It's a number.
583
00:34:22 --> 00:34:24
It's a number.
584
00:34:24 --> 00:34:28
That's just of this size.
585
00:34:28 --> 00:34:31
There's the big number,
there's the small one.
586
00:34:31 --> 00:34:37
And divided by that, that's
the thing that shows is
587
00:34:37 --> 00:34:39
yes, there is this jump.
588
00:34:39 --> 00:34:41
Right?
589
00:34:41 --> 00:34:45
OK, that's a couple of sets
of Fourier coefficients.
590
00:34:45 --> 00:34:48
You could ask yourself, because
on the quiz there'll probably
591
00:34:48 --> 00:34:54
be one, and I'll try to pick a
function that's interesting.
592
00:34:54 --> 00:35:00
And I mean, I don't plan
to pick xe^x or anything.
593
00:35:00 --> 00:35:03
Yeah.
594
00:35:03 --> 00:35:05
OK, good.
595
00:35:05 --> 00:35:05
Yes, thanks.
596
00:35:05 --> 00:35:11
AUDIENCE: [INAUDIBLE]
597
00:35:11 --> 00:35:13
Fourier series has twice
the energy as another,
598
00:35:13 --> 00:35:13
what's that mean?
599
00:35:13 --> 00:35:13
PROFESSOR STRANG: I don't know.
600
00:35:13 --> 00:35:14
I guess.
601
00:35:14 --> 00:35:17
Hm.
602
00:35:17 --> 00:35:22
Maybe we're talking about
power, and things like if we
603
00:35:22 --> 00:35:31
were dealing with electronics,
I guess I would interpret that
604
00:35:31 --> 00:35:33
energy in terms of power.
605
00:35:33 --> 00:35:37
So that's what I'd be seeing.
606
00:35:37 --> 00:35:42
I'm not thinking of a really
good answer to say well why
607
00:35:42 --> 00:35:46
is that energy equality,
but it's really useful.
608
00:35:46 --> 00:35:50
You know, so much of signal
processing, and we'll do a bit
609
00:35:50 --> 00:35:57
of signal processing, is simply
based on that energy or
610
00:35:57 --> 00:36:04
equality there, and the
moving into frequency space.
611
00:36:04 --> 00:36:05
And convolution.
612
00:36:05 --> 00:36:09
Actually, they would
call it filtering.
613
00:36:09 --> 00:36:13
So we'll call it filtering
when we use convolution.
614
00:36:13 --> 00:36:15
But it's pure convolution.
615
00:36:15 --> 00:36:22
Pure linear algebra, for
these special bases.
616
00:36:22 --> 00:36:24
So, OK.
617
00:36:24 --> 00:36:29
I could try to come up with a
better answer, or more focused
618
00:36:29 --> 00:36:42
answer than just to say power
or, to use an electric power
619
00:36:42 --> 00:36:45
word is just one step.
620
00:36:45 --> 00:36:48
Yeah, thanks.
621
00:36:48 --> 00:36:48
Yes.
622
00:36:48 --> 00:36:51
AUDIENCE: [INAUDIBLE]
623
00:36:51 --> 00:36:53
PROFESSOR STRANG:
4.3, eight or nine.
624
00:36:53 --> 00:37:00
Let's just have a look and
see what they're about.
625
00:37:00 --> 00:37:02
OK, just some regular guys.
626
00:37:02 --> 00:37:08
Yeah so that, OK, let
me look at, you want
627
00:37:08 --> 00:37:12
me to do 4.3 eight?
628
00:37:12 --> 00:37:19
AUDIENCE: [INAUDIBLE]
629
00:37:19 --> 00:37:20
PROFESSOR STRANG: Ah, yes.
630
00:37:20 --> 00:37:22
Good, good question.
631
00:37:22 --> 00:37:26
OK, so 4.3 eight gave a couple
of vectors, a little bit like
632
00:37:26 --> 00:37:28
the ones I did this morning.
633
00:37:28 --> 00:37:32
I mean here the c and 4.3 eight
has a couple of ones, and this
634
00:37:32 --> 00:37:35
morning I just had
a .
635
00:37:35 --> 00:37:37
But no big deal.
636
00:37:37 --> 00:37:39
Now, yeah.
637
00:37:39 --> 00:37:45
So if two vectors are
orthogonal, are their
638
00:37:45 --> 00:37:49
transforms orthogonal?
639
00:37:49 --> 00:37:51
I think, yes.
640
00:37:51 --> 00:37:52
Yes.
641
00:37:52 --> 00:37:56
The Fourier, so the, yeah.
642
00:37:56 --> 00:38:00
So maybe this is worth
a moment, here.
643
00:38:00 --> 00:38:01
Yeah.
644
00:38:01 --> 00:38:07
Because what do we, let me
write down the letters for
645
00:38:07 --> 00:38:09
my question, and then
answer the question.
646
00:38:09 --> 00:38:18
OK, so suppose I have
two vectors, c and d.
647
00:38:18 --> 00:38:22
And they're orthogonal.
648
00:38:22 --> 00:38:27
And then I want to ask about
their, if I multiply by the
649
00:38:27 --> 00:38:33
Fourier matrices, are
those orthogonal?
650
00:38:33 --> 00:38:36
That's sort of the question.
651
00:38:36 --> 00:38:39
So here the vectors in
frequency space, here they
652
00:38:39 --> 00:38:41
are in physical space.
653
00:38:41 --> 00:38:43
I don't mind if you started
in physical space and
654
00:38:43 --> 00:38:47
went to frequency with F
inverse, same question.
655
00:38:47 --> 00:38:53
Does the Fourier matrix
preserve angles?
656
00:38:53 --> 00:38:55
Does it preserve angles?
657
00:38:55 --> 00:39:02
Do matrices, and F wouldn't
be the only one with this
658
00:39:02 --> 00:39:05
property, do they
preserve length.
659
00:39:05 --> 00:39:06
Right?
660
00:39:06 --> 00:39:09
If you preserve length, do
you preserve angles too?
661
00:39:09 --> 00:39:13
Preserve length, you're just
looking at one vector.
662
00:39:13 --> 00:39:17
We know that we preserve
length, and how
663
00:39:17 --> 00:39:18
do we know that?
664
00:39:18 --> 00:39:20
Let's just remember that.
665
00:39:20 --> 00:39:22
So here's the length question.
666
00:39:22 --> 00:39:24
And this'll be the
angle question.
667
00:39:24 --> 00:39:27
So this is the energy
inequality, coming back
668
00:39:27 --> 00:39:28
to that key thing.
669
00:39:28 --> 00:39:37
This is c bar transpose c, and
over here we have (Fc,Fc) and
670
00:39:37 --> 00:39:39
this is what I did
this morning.
671
00:39:39 --> 00:39:46
So that's c bar transpose F
bar transpose F c, right?
672
00:39:46 --> 00:39:51
That's why the bar as well
as the transpose because
673
00:39:51 --> 00:39:53
we're doing complex.
674
00:39:53 --> 00:39:59
OK, let me make a
little more space.
675
00:39:59 --> 00:40:01
And what did we
do this morning?
676
00:40:01 --> 00:40:07
We replaced that by, well I
wish it were the identity.
677
00:40:07 --> 00:40:11
But it's a multiple of the
identity, that's what matters.
678
00:40:11 --> 00:40:16
So this was N, this is
the identity with an
679
00:40:16 --> 00:40:19
N, c bar transpose c.
680
00:40:19 --> 00:40:24
So the conclusion was
that this thing is just
681
00:40:24 --> 00:40:27
N times this thing.
682
00:40:27 --> 00:40:31
Well, and times this one.
683
00:40:31 --> 00:40:34
I'm expecting oh, but
over here we got a zero.
684
00:40:34 --> 00:40:36
So my N is going to wash out.
685
00:40:36 --> 00:40:39
OK, let's just do the
same thing for angle.
686
00:40:39 --> 00:40:46
This is the key energy
equality for length.
687
00:40:46 --> 00:40:52
And all I want to say is, this
Fourier matrix, like other
688
00:40:52 --> 00:40:57
orthogonal matrices, is
just rotating the space.
689
00:40:57 --> 00:41:01
It's sort of amazing to think
you have one space, physical
690
00:41:01 --> 00:41:05
space, and you rotate it.
691
00:41:05 --> 00:41:07
I'll use that word.
692
00:41:07 --> 00:41:10
Because somehow that's what
an orthogonal matrix does.
693
00:41:10 --> 00:41:13
Well it's complex
N-dimensional space.
694
00:41:13 --> 00:41:17
Sorry, so it's not so easy
to visualize rotations in
695
00:41:17 --> 00:41:21
C^N, but it's a rotation.
696
00:41:21 --> 00:41:24
Angles don't change, and
let's just see why?
697
00:41:24 --> 00:41:28
The inner product of this with
this is c bar transpose, F
698
00:41:28 --> 00:41:31
bar transpose, because I
have to transpose that.
699
00:41:31 --> 00:41:36
Times Fd, what do I do now?
700
00:41:36 --> 00:41:43
This one was c bar
transpose d, with zero.
701
00:41:43 --> 00:41:45
You see you have it?
702
00:41:45 --> 00:41:50
Again, this fact that this
rotation and inverse, and the
703
00:41:50 --> 00:41:55
transpose is the identity,
apart from an N.
704
00:41:55 --> 00:41:59
So again, inner products
are multiplied by N.
705
00:41:59 --> 00:42:04
Not only the length squared,
which is inner product with
706
00:42:04 --> 00:42:08
itself, but all inner products
are just multiplied by N.
707
00:42:08 --> 00:42:11
So if they start zero
they end up zero.
708
00:42:11 --> 00:42:14
Yeah.
709
00:42:14 --> 00:42:15
That was your question, right?
710
00:42:15 --> 00:42:17
Yep, right.
711
00:42:17 --> 00:42:20
So if I have a couple of
vectors, as I guess that
712
00:42:20 --> 00:42:24
problem proposed, it happened
to propose two inputs that
713
00:42:24 --> 00:42:28
happen to be orthogonal, then
you should be able to see that
714
00:42:28 --> 00:42:30
the transforms are, too.
715
00:42:30 --> 00:42:31
Yeah.
716
00:42:31 --> 00:42:33
Yep.
717
00:42:33 --> 00:42:37
Can I ask you about question
two, was that on the homework?
718
00:42:37 --> 00:42:41
Problem two in Section 4.3?
719
00:42:41 --> 00:42:43
Was it?
720
00:42:43 --> 00:42:49
All I want to say is
that's a simple fact.
721
00:42:49 --> 00:42:53
Let me just write that fact
down so we're looking
722
00:42:53 --> 00:42:56
at that fact.
723
00:42:56 --> 00:43:02
If I look at this F matrix,
it's so simple but it leads
724
00:43:02 --> 00:43:05
to lots of good, it's
just a key fact.
725
00:43:05 --> 00:43:12
But if I look at my F matrix,
am I looking at rows here?
726
00:43:12 --> 00:43:15
Yeah, I happened to look
at rows, columns would be
727
00:43:15 --> 00:43:16
the same, it's symmetric.
728
00:43:16 --> 00:43:23
So here's row one, here's row
two, I'm sorry that's row zero.
729
00:43:23 --> 00:43:26
This is row one.
730
00:43:26 --> 00:43:28
And let me look at row N-1.
731
00:43:29 --> 00:43:35
This is w, w^(N-1), this
was w^2, this is w^2(N-1).
732
00:43:35 --> 00:43:38
733
00:43:38 --> 00:43:40
And so on.
734
00:43:40 --> 00:43:42
Everybody's got the idea
of the, and then all
735
00:43:42 --> 00:43:45
these in between rows.
736
00:43:45 --> 00:43:48
Two, three, et cetera.
737
00:43:48 --> 00:43:55
And the question asks, show
that this and this are
738
00:43:55 --> 00:43:57
complex conjugates.
739
00:43:57 --> 00:44:03
That that row and that row are
the same, except conjugates.
740
00:44:03 --> 00:44:09
So if we look at the row number
one in F, we'll be looking
741
00:44:09 --> 00:44:14
at row number N-1 in F bar.
742
00:44:14 --> 00:44:16
Now, why is that row the
conjugate of that row?
743
00:44:16 --> 00:44:19
Why is this the
conjugate of this?
744
00:44:19 --> 00:44:23
Why are those two conjugates?
745
00:44:23 --> 00:44:30
So I'm asking you to explain to
me why w bar is the, why is
746
00:44:30 --> 00:44:40
the conjugate of this, this?
747
00:44:40 --> 00:44:45
So it's just one more neat
fact about these important,
748
00:44:45 --> 00:44:48
crucial, numbers w.
749
00:44:48 --> 00:44:50
And so how do I see this?
750
00:44:50 --> 00:44:54
Well, this is w, this
is one way to do it.
751
00:44:54 --> 00:44:56
This is w^N times w^-1.
752
00:44:57 --> 00:44:58
And what's w^N?
753
00:45:00 --> 00:45:01
One.
754
00:45:01 --> 00:45:04
Everybody knows, w^N is one.
755
00:45:04 --> 00:45:07
So I have w, so I'm
trying to show that.
756
00:45:07 --> 00:45:12
Well, we know that
this is true.
757
00:45:12 --> 00:45:18
So we know that the
conjugate of w, right?
758
00:45:18 --> 00:45:20
There's w.
759
00:45:20 --> 00:45:25
Here's its conjugate, and
it's also the inverse.
760
00:45:25 --> 00:45:30
This w is some e^(i*theta),
this is some e to the
761
00:45:30 --> 00:45:32
the i some angle.
762
00:45:32 --> 00:45:35
Then always, it's sitting
on the unit circle.
763
00:45:35 --> 00:45:40
So its reciprocal is also
on the unit circle.
764
00:45:40 --> 00:45:45
The reciprocal has
e^(-i*theta), so it's
765
00:45:45 --> 00:45:46
just the conjugate.
766
00:45:46 --> 00:45:50
That's a great fact.
767
00:45:50 --> 00:45:54
That's a beautiful fact
about all these w's.
768
00:45:54 --> 00:45:55
And their powers.
769
00:45:55 --> 00:45:59
Conjugate and inverse the same.
770
00:45:59 --> 00:46:01
Right.
771
00:46:01 --> 00:46:04
So that was the key
to problem two.
772
00:46:04 --> 00:46:09
I don't think I asked you to go
through all the steps of
773
00:46:09 --> 00:46:13
problem three, but just in case
you didn't read problem three,
774
00:46:13 --> 00:46:18
let me tell you in one
tiny space what happens.
775
00:46:18 --> 00:46:21
In problem three you discover
that the fourth power
776
00:46:21 --> 00:46:25
of F is the identity.
777
00:46:25 --> 00:46:34
Except there is an N squared,
because from two F's we got an
778
00:46:34 --> 00:46:38
N, so from four F's, so that's
another fantastic fact
779
00:46:38 --> 00:46:41
about the Fourier matrix.
780
00:46:41 --> 00:46:47
That its fourth power, it
must be somehow, the Fourier
781
00:46:47 --> 00:46:50
matrix is rotating, yeah.
782
00:46:50 --> 00:46:54
Somehow, I don't know, how do
you understand that the
783
00:46:54 --> 00:46:58
Fourier matrix is, its fourth
power brings you back
784
00:46:58 --> 00:46:59
to the identity.
785
00:46:59 --> 00:47:03
Apart from, we just
didn't normalize it.
786
00:47:03 --> 00:47:07
So that we wouldn't, to
avoid that N squared,
787
00:47:07 --> 00:47:09
so we had to use it.
788
00:47:09 --> 00:47:10
Yeah.
789
00:47:10 --> 00:47:13
That's pretty amazing.
790
00:47:13 --> 00:47:15
Pretty amazing.
791
00:47:15 --> 00:47:21
So if I had normalized it
right, if I'd took F over the
792
00:47:21 --> 00:47:26
square root of N, that's the
exact normalization to give me
793
00:47:26 --> 00:47:32
a to put the N's where they
belong, you could say.
794
00:47:32 --> 00:47:35
Then the fourth power of that
matrix is the identity.
795
00:47:35 --> 00:47:37
Yeah.
796
00:47:37 --> 00:47:43
New math and new applications
keep coming for these things.
797
00:47:43 --> 00:47:45
I'll tell you, actually.
798
00:47:45 --> 00:47:48
You could tell me
eigenvalues of this matrix.
799
00:47:48 --> 00:47:50
Yeah, this is a good question.
800
00:47:50 --> 00:47:53
What are the eigenvalues
of a matrix whose fourth
801
00:47:53 --> 00:47:57
power is the identity?
802
00:47:57 --> 00:47:58
AUDIENCE: [INAUDIBLE]
803
00:47:58 --> 00:47:59
PROFESSOR STRANG: Yeah, one.
804
00:47:59 --> 00:48:01
What else could the
eigenvalue be?
805
00:48:01 --> 00:48:05
If the fourth power of the
matrix is i, what are the
806
00:48:05 --> 00:48:07
possible eigenvalues?
807
00:48:07 --> 00:48:16
So let me, so this matrix is,
can I call it M for the, or
808
00:48:16 --> 00:48:20
maybe u for the matrix F?
809
00:48:20 --> 00:48:32
So if u^4 is the identity,
what are the eigenvalues?
810
00:48:32 --> 00:48:34
What could, well of
course u could be the
811
00:48:34 --> 00:48:36
identity, but it's not.
812
00:48:36 --> 00:48:41
It's a Fourier matrix.
813
00:48:41 --> 00:48:46
So the eigenvalues could be
one, what else could they be?
814
00:48:46 --> 00:48:48
Minus one is possible.
815
00:48:48 --> 00:48:55
Because minus the identity,
that'd be fine. i, and minus i.
816
00:48:55 --> 00:48:57
Four possible eigenvalues.
817
00:48:57 --> 00:49:01
And when the matrix reaches,
I think if you put the
818
00:49:01 --> 00:49:03
four by four Fourier
819
00:49:03 --> 00:49:06
- I don't know.
820
00:49:06 --> 00:49:10
If you, say, put the four by
four Fourier matrix into
821
00:49:10 --> 00:49:14
MATLAB, and see what you get
for eigenvalues, I've
822
00:49:14 --> 00:49:16
forgotten whether you get
one of each of these.
823
00:49:16 --> 00:49:21
Or whether somebody's repeated
at the level four, but then go
824
00:49:21 --> 00:49:25
up to five or six you'll see
these guys start showing up.
825
00:49:25 --> 00:49:28
Different multiplicity.
826
00:49:28 --> 00:49:28
Right?
827
00:49:28 --> 00:49:34
The 1,024 matrix has
got these guys.
828
00:49:34 --> 00:49:37
Some number of times,
adding up to a 1,024.
829
00:49:37 --> 00:49:38
Right, yeah.
830
00:49:38 --> 00:49:41
So it's quite neat.
831
00:49:41 --> 00:49:45
Now, here's a question, which
I actually just learned
832
00:49:45 --> 00:49:46
a good answer to.
833
00:49:46 --> 00:49:49
What are the eigenvectors?
834
00:49:49 --> 00:49:51
What are the eigenvectors?
835
00:49:51 --> 00:49:55
You could give a sort of
half-baked description.
836
00:49:55 --> 00:49:57
Because once you know
the eigenvalues.
837
00:49:57 --> 00:50:01
But to really get a handle on
the eigenvectors, that's has
838
00:50:01 --> 00:50:08
been a problem that was studied
in IEEE transactions papers.
839
00:50:08 --> 00:50:12
But not really the right, not
a nice specific description
840
00:50:12 --> 00:50:13
of the eigenvectors.
841
00:50:13 --> 00:50:19
And somebody was in my office,
this Fall, a guy, post-doc at
842
00:50:19 --> 00:50:25
Berkeley who's seen the right
way to look at that problem and
843
00:50:25 --> 00:50:28
describe the eigenvectors
of the Fourier matrix.
844
00:50:28 --> 00:50:32
So that's, like, amazing to
me that such a fundamental
845
00:50:32 --> 00:50:34
question was waiting.
846
00:50:34 --> 00:50:40
And turned out to involve
quite important, deep ideas.
847
00:50:40 --> 00:50:42
OK, ready for one
more for today?
848
00:50:42 --> 00:50:45
Anything?
849
00:50:45 --> 00:50:48
Or not.
850
00:50:48 --> 00:50:52
OK, so I hope you're getting
the good stuff now on
851
00:50:52 --> 00:50:54
the Fourier series.
852
00:50:54 --> 00:50:57
Fourier discrete transform.
853
00:50:57 --> 00:51:00
Please come on Friday, because
Friday will be the big
854
00:51:00 --> 00:51:02
day for convolution.
855
00:51:02 --> 00:51:09
And that's the essential thing
that we still have to do.
856
00:51:09 --> 00:51:11
OK, see you Friday.