1
00:00:00,000 --> 00:00:00,257


2
00:00:00,257 --> 00:00:02,340
The following content is
provided under a Creative

3
00:00:02,340 --> 00:00:03,620
Commons license.

4
00:00:03,620 --> 00:00:05,770
Your support will help
MIT OpenCourseWare

5
00:00:05,770 --> 00:00:10,050
continue to offer high quality
educational resources for free.

6
00:00:10,050 --> 00:00:13,440
To make a donation, or to
view additional materials

7
00:00:13,440 --> 00:00:16,150
from hundreds of MIT courses,
visit MIT OpenCourseWare

8
00:00:16,150 --> 00:00:20,410
at ocw.mit.edu.

9
00:00:20,410 --> 00:00:22,200
PROFESSOR STRANG:
Actually, two things

10
00:00:22,200 --> 00:00:23,760
to say about eigenvalues.

11
00:00:23,760 --> 00:00:28,460
One is about matrices
in general and then

12
00:00:28,460 --> 00:00:30,940
the second is to focus
on our favorites,

13
00:00:30,940 --> 00:00:37,610
those second derivatives
and second differences.

14
00:00:37,610 --> 00:00:39,870
There's a lot to say
about eigenvalues

15
00:00:39,870 --> 00:00:42,810
but then we'll have
the main ideas.

16
00:00:42,810 --> 00:00:46,910
So the central idea
of course is to find

17
00:00:46,910 --> 00:00:51,970
these special directions and
we expect to find n directions,

18
00:00:51,970 --> 00:00:56,880
n eigenvectors y where
this n by n matrix

19
00:00:56,880 --> 00:01:03,320
is acting like a number in
each of those directions.

20
00:01:03,320 --> 00:01:06,640
So we have this for n
different y's and each one

21
00:01:06,640 --> 00:01:09,370
has its own eigenvalue lambda.

22
00:01:09,370 --> 00:01:12,490
And of course the
eig command in MATLAB

23
00:01:12,490 --> 00:01:15,870
will find the y's
and the lambdas.

24
00:01:15,870 --> 00:01:21,320
So it finds the y's and
the lambdas in a matrix.

25
00:01:21,320 --> 00:01:24,350
So that's what I'm going
to do now, straightforward.

26
00:01:24,350 --> 00:01:28,160
Any time I have n vectors, so
I have n of these y's, I've

27
00:01:28,160 --> 00:01:32,000
n y's and n lambdas.

28
00:01:32,000 --> 00:01:35,150
Well, if you give
me n vectors, I

29
00:01:35,150 --> 00:01:37,180
put them into the
columns of a matrix,

30
00:01:37,180 --> 00:01:38,810
almost without thinking.

31
00:01:38,810 --> 00:01:41,340
So can I just do that?

32
00:01:41,340 --> 00:01:44,090
So there is y_1, the
first eigenvector.

33
00:01:44,090 --> 00:01:47,240
That's y_2 to y_n.

34
00:01:47,240 --> 00:01:50,330
Okay, that's my
eigenvector matrix.

35
00:01:50,330 --> 00:01:54,460
Often I call it S. So
I'll stick with that.

36
00:01:54,460 --> 00:01:59,780
S will be the
eigenvector matrix.

37
00:01:59,780 --> 00:02:01,475
Since these are
eigenvectors I'm going

38
00:02:01,475 --> 00:02:05,570
to multiply that
matrix by A. That

39
00:02:05,570 --> 00:02:07,570
should bring out the key point.

40
00:02:07,570 --> 00:02:15,710
I'm just going to repeat
this, which is one at a time,

41
00:02:15,710 --> 00:02:18,800
by doing them all that once.

42
00:02:18,800 --> 00:02:23,350
So what happens if I multiply
a matrix by a bunch of columns?

43
00:02:23,350 --> 00:02:25,220
Matrix multiplication
is wonderful.

44
00:02:25,220 --> 00:02:26,880
It does the right thing.

45
00:02:26,880 --> 00:02:30,540
It multiplies A times
the first column.

46
00:02:30,540 --> 00:02:31,820
So let's put that there.

47
00:02:31,820 --> 00:02:37,230
A times the first column along
to A times the last column.

48
00:02:37,230 --> 00:02:39,380
Just column by column.

49
00:02:39,380 --> 00:02:41,390
But now we recognize these.

50
00:02:41,390 --> 00:02:43,590
They're special y's.

51
00:02:43,590 --> 00:02:45,490
They're special because
they're eigenvectors.

52
00:02:45,490 --> 00:02:53,440
So this is lambda_1*y_1 along
to that column is lambda_n*y_n.

53
00:02:53,440 --> 00:02:56,560
Right?

54
00:02:56,560 --> 00:02:59,280
Now I've used the fact that
they were eigenvectors.

55
00:02:59,280 --> 00:03:03,670
And now, one final neat step
of matrix multiplication

56
00:03:03,670 --> 00:03:10,010
is to factor out this same
eigenvector matrix again

57
00:03:10,010 --> 00:03:12,970
and realize, and
I'll look at it,

58
00:03:12,970 --> 00:03:17,460
that it's being multiplied
by this diagonal, that's

59
00:03:17,460 --> 00:03:23,230
now a diagonal matrix
of eigenvalues.

60
00:03:23,230 --> 00:03:26,610
So let's just look at
that very last step here.

61
00:03:26,610 --> 00:03:29,120
Here I had the first
column was lambda_1*y_1.

62
00:03:29,120 --> 00:03:31,880


63
00:03:31,880 --> 00:03:35,880
I just want to see,
did I get that right?

64
00:03:35,880 --> 00:03:38,260
If I'm looking at
the first column

65
00:03:38,260 --> 00:03:42,170
where that lambda_1 is sitting,
it's going to multiply y_1

66
00:03:42,170 --> 00:03:45,610
and it'll be all zeroes
below so I'll have

67
00:03:45,610 --> 00:03:47,860
none of the other eigenvectors.

68
00:03:47,860 --> 00:03:51,760
So I'll have lambda_1*y_1,
just what I want.

69
00:03:51,760 --> 00:03:56,010
Got a little squeezed near
the end there, but so let

70
00:03:56,010 --> 00:03:57,810
me write above.

71
00:03:57,810 --> 00:04:03,540
The result is just A times
this eigenvector matrix

72
00:04:03,540 --> 00:04:10,040
that I'm going to
call S equals what?

73
00:04:10,040 --> 00:04:15,220
This is Ay=lambda*y
for all n at once.

74
00:04:15,220 --> 00:04:18,290
A times S is, what
have I got here?

75
00:04:18,290 --> 00:04:19,690
What's this?

76
00:04:19,690 --> 00:04:22,980
That's S. And what's
the other guy?

77
00:04:22,980 --> 00:04:25,650
That's the eigenvalue matrix.

78
00:04:25,650 --> 00:04:27,390
So it's just got n numbers.

79
00:04:27,390 --> 00:04:29,880
They automatically
go on the diagonal

80
00:04:29,880 --> 00:04:32,860
and it gets called
capital Lambda.

81
00:04:32,860 --> 00:04:37,340
Capital Lambda for the matrix,
little lambda for the numbers.

82
00:04:37,340 --> 00:04:42,630
So this is n, this
is all n at once.

83
00:04:42,630 --> 00:04:46,630
Straightforward.

84
00:04:46,630 --> 00:04:48,450
Now I'm going to
assume that I've

85
00:04:48,450 --> 00:04:52,390
got these n eigenvectors,
that I've been able to find

86
00:04:52,390 --> 00:04:55,130
n independent directions.

87
00:04:55,130 --> 00:04:57,520
And almost always, you can.

88
00:04:57,520 --> 00:05:01,380
For every symmetric matrix
you automatically can.

89
00:05:01,380 --> 00:05:06,050
So these y's are
independent directions.

90
00:05:06,050 --> 00:05:09,530
If those are the
columns of a matrix,

91
00:05:09,530 --> 00:05:12,900
yeah, here's a key
question about matrices.

92
00:05:12,900 --> 00:05:15,840
What can I say
about this matrix S

93
00:05:15,840 --> 00:05:19,210
if its n columns
are independent?

94
00:05:19,210 --> 00:05:20,790
Whatever that, you
know, we haven't

95
00:05:20,790 --> 00:05:23,830
focused in careful detail,
but we have an idea.

96
00:05:23,830 --> 00:05:28,450
That means sort of none of them
are combinations of the others.

97
00:05:28,450 --> 00:05:30,890
We really have n
separate directions.

98
00:05:30,890 --> 00:05:33,400
Then that matrix is?

99
00:05:33,400 --> 00:05:35,940
Invertible.

100
00:05:35,940 --> 00:05:39,280
A matrix that's got n
columns, independent,

101
00:05:39,280 --> 00:05:41,010
that's what we're hoping for.

102
00:05:41,010 --> 00:05:42,910
That matrix has an inverse.

103
00:05:42,910 --> 00:05:47,080
We can produce, well all the
good facts about matrices.

104
00:05:47,080 --> 00:05:48,610
This is a square matrix.

105
00:05:48,610 --> 00:05:50,950
So I can invert it if you like.

106
00:05:50,950 --> 00:05:53,920
And I can write A as S lambda.

107
00:05:53,920 --> 00:05:56,950
I'm multiplying on the
right by this S inverse.

108
00:05:56,950 --> 00:06:03,620
And there I have the
diagonalization of a matrix.

109
00:06:03,620 --> 00:06:06,070
The matrix has
been diagonalized.

110
00:06:06,070 --> 00:06:07,410
And what does that mean?

111
00:06:07,410 --> 00:06:13,100
Well this is, of course the
diagonal that we're headed for.

112
00:06:13,100 --> 00:06:16,440
And what it means is that
if I look at my matrix

113
00:06:16,440 --> 00:06:23,150
and I separate out the different
eigendirections, I could say,

114
00:06:23,150 --> 00:06:26,100
that the matrix in
those directions

115
00:06:26,100 --> 00:06:28,750
is just this diagonal matrix.

116
00:06:28,750 --> 00:06:36,400
So that's a short
way of saying it.

117
00:06:36,400 --> 00:06:40,810
Let me just carry
one step further.

118
00:06:40,810 --> 00:06:44,650
What would A squared be?

119
00:06:44,650 --> 00:06:50,620
Well now that I have it in this
cool form, S*lambda*S inverse,

120
00:06:50,620 --> 00:06:53,870
I would multiply two of those
together and what would I

121
00:06:53,870 --> 00:06:55,650
learn?

122
00:06:55,650 --> 00:06:59,440
If I do that multiplication
what comes out?

123
00:06:59,440 --> 00:07:02,330
First an S from here.

124
00:07:02,330 --> 00:07:04,420
And then what?

125
00:07:04,420 --> 00:07:05,490
Lambda squared.

126
00:07:05,490 --> 00:07:07,780
Why lambda squared?

127
00:07:07,780 --> 00:07:10,710
Because in the middle is
the S S inverse that's

128
00:07:10,710 --> 00:07:14,250
giving the identity matrix.

129
00:07:14,250 --> 00:07:16,960
So then the lambda
multiplies the lambda

130
00:07:16,960 --> 00:07:18,990
and now here is S inverse.

131
00:07:18,990 --> 00:07:22,860
Well A squared is S*lambda
squared*S inverse.

132
00:07:22,860 --> 00:07:25,370
What does that tell me in words?

133
00:07:25,370 --> 00:07:31,920
That tells me that the
eigenvectors of A squared are?

134
00:07:31,920 --> 00:07:33,300
The same.

135
00:07:33,300 --> 00:07:37,800
As for A. And it tells me that
the eigenvalues of A squared

136
00:07:37,800 --> 00:07:39,120
are?

137
00:07:39,120 --> 00:07:41,400
The squares.

138
00:07:41,400 --> 00:07:43,650
So I could do this.

139
00:07:43,650 --> 00:07:45,600
Maybe I did it
before, one at a time.

140
00:07:45,600 --> 00:07:53,760
Ay=lambda*y, multiply again by
A. A squared*y is lambda*Ay,

141
00:07:53,760 --> 00:07:59,720
but Ay is lambda*y so I'm
up to lambda squared*y.

142
00:07:59,720 --> 00:08:02,040
You should just see
that when you've

143
00:08:02,040 --> 00:08:06,460
got these directions then
your matrix is really simple.

144
00:08:06,460 --> 00:08:08,880
Effectively it's
a diagonal matrix

145
00:08:08,880 --> 00:08:11,530
in these good directions.

146
00:08:11,530 --> 00:08:17,280
So that just shows one way
of seeing-- And of course

147
00:08:17,280 --> 00:08:18,400
what about A inverse?

148
00:08:18,400 --> 00:08:20,960
We might as well
mention A inverse.

149
00:08:20,960 --> 00:08:24,540
Suppose A is invertible.

150
00:08:24,540 --> 00:08:28,710
Then what do I learn
about A inverse?

151
00:08:28,710 --> 00:08:32,380
Can I just invert that?

152
00:08:32,380 --> 00:08:34,000
I'm just playing
with that formula,

153
00:08:34,000 --> 00:08:39,800
so you'll kind of,
like, get handy with it.

154
00:08:39,800 --> 00:08:42,770
What would the inverse
be if I have three things

155
00:08:42,770 --> 00:08:45,330
in a row multiplied together?

156
00:08:45,330 --> 00:08:48,480
What's the inverse?

157
00:08:48,480 --> 00:08:51,480
So I'm going to take the
inverses in the opposite order,

158
00:08:51,480 --> 00:08:52,740
right?

159
00:08:52,740 --> 00:08:56,150
So the inverse of
that will come first.

160
00:08:56,150 --> 00:08:58,640
So what's that?

161
00:08:58,640 --> 00:09:02,390
Just S. The lambda
in the middle gets

162
00:09:02,390 --> 00:09:06,120
inverted and then
the S at the left,

163
00:09:06,120 --> 00:09:09,610
its inverse comes at the right.

164
00:09:09,610 --> 00:09:13,190
Well what do I learn from that?

165
00:09:13,190 --> 00:09:19,820
I learn that the eigenvector
matrix for A inverse is?

166
00:09:19,820 --> 00:09:21,360
Same thing, again.

167
00:09:21,360 --> 00:09:22,180
Same.

168
00:09:22,180 --> 00:09:24,380
Let me put just "Same".

169
00:09:24,380 --> 00:09:31,860
What's the eigenvalue
matrix for A inverse?

170
00:09:31,860 --> 00:09:37,070
It's the inverse of this guy,
so what does it look like?

171
00:09:37,070 --> 00:09:39,900
It's got one over lambdas.

172
00:09:39,900 --> 00:09:42,720
That's all it says.

173
00:09:42,720 --> 00:09:45,980
The eigenvalues for
A inverse are just

174
00:09:45,980 --> 00:09:49,460
one over the eigenvalues for A.

175
00:09:49,460 --> 00:09:54,210
If that is so, and it can't
be difficult, we could again,

176
00:09:54,210 --> 00:10:01,380
we could prove it sort
of like, one at a time.

177
00:10:01,380 --> 00:10:03,280
This is my starting
point, always.

178
00:10:03,280 --> 00:10:07,610
How would I get to A inverse
now and recover this fact

179
00:10:07,610 --> 00:10:13,510
that the eigenvalues for the
inverse, just turn them up.

180
00:10:13,510 --> 00:10:16,950
If A has an eigenvalue
seven, A inverse

181
00:10:16,950 --> 00:10:20,700
will have an eigenvalue 1/7.

182
00:10:20,700 --> 00:10:23,570
What do I do?

183
00:10:23,570 --> 00:10:26,700
Usually multiply both sides
by something sensible.

184
00:10:26,700 --> 00:10:28,630
Right?

185
00:10:28,630 --> 00:10:31,030
What shall I multiply
both sides by?

186
00:10:31,030 --> 00:10:34,990
A inverse sounds like
a good idea, right.

187
00:10:34,990 --> 00:10:36,930
So I'm multiplying both
sides by A inverse,

188
00:10:36,930 --> 00:10:40,370
so that just leaves y and
here is that number, here

189
00:10:40,370 --> 00:10:44,300
is A inverse times y.

190
00:10:44,300 --> 00:10:46,530
Well, maybe I should
do one more thing.

191
00:10:46,530 --> 00:10:48,910
What else shall I do?

192
00:10:48,910 --> 00:10:52,130
Divide by lambda.

193
00:10:52,130 --> 00:10:55,940
Take that number lambda and
put it over here as one lambda.

194
00:10:55,940 --> 00:11:00,030
Well, just exactly
what we're looking for.

195
00:11:00,030 --> 00:11:04,813
The same y has A
inverse, the same y

196
00:11:04,813 --> 00:11:08,210
as an eigenvector of A
inverse and the eigenvalue

197
00:11:08,210 --> 00:11:10,230
is one over lambda.

198
00:11:10,230 --> 00:11:13,660
Oh, and of course,
I should have said

199
00:11:13,660 --> 00:11:16,510
before I inverted
anything, what should I

200
00:11:16,510 --> 00:11:19,920
have said about the lambdas?

201
00:11:19,920 --> 00:11:21,600
Not zero.

202
00:11:21,600 --> 00:11:22,640
Right?

203
00:11:22,640 --> 00:11:28,870
A zero eigenvalue is a signal
the matrix isn't invertible.

204
00:11:28,870 --> 00:11:32,340
So that's perfect test.

205
00:11:32,340 --> 00:11:37,460
If the matrix is invertible, all
its eigenvalues are not zero.

206
00:11:37,460 --> 00:11:40,610
If it's singular, it's
got a zero eigenvalue.

207
00:11:40,610 --> 00:11:47,060
If a matrix is singular,
then Ay would be 0y for some,

208
00:11:47,060 --> 00:11:49,670
there'd be a vector
that that matrix kills.

209
00:11:49,670 --> 00:11:52,570
If A is not invertible,
there's a reason for it.

210
00:11:52,570 --> 00:11:55,730
It's because it takes
some vector to zero,

211
00:11:55,730 --> 00:12:01,580
and of course, you can't
bring it back to life.

212
00:12:01,580 --> 00:12:03,320
So shall I just
put that up here?

213
00:12:03,320 --> 00:12:08,290
Lambda=0 would tell me I
have a singular matrix.

214
00:12:08,290 --> 00:12:12,120
All lambda not equal
zero would tell me

215
00:12:12,120 --> 00:12:17,510
I have an invertible matrix.

216
00:12:17,510 --> 00:12:19,990
These are straightforward facts.

217
00:12:19,990 --> 00:12:25,310
It's taken down in this row
and it's just really handy

218
00:12:25,310 --> 00:12:30,690
to have up here.

219
00:12:30,690 --> 00:12:34,110
Well now I'm ready to move
toward the specific matrices,

220
00:12:34,110 --> 00:12:35,600
our favorites.

221
00:12:35,600 --> 00:12:38,190
Now, those are symmetric.

222
00:12:38,190 --> 00:12:41,780
So maybe before I
leave this picture,

223
00:12:41,780 --> 00:12:46,500
we better recall what is special
when the matrix is symmetric.

224
00:12:46,500 --> 00:12:49,480
So that's going to
be the next thing.

225
00:12:49,480 --> 00:12:53,200
So if A is symmetric I get
some extra good things.

226
00:12:53,200 --> 00:12:58,710
So let me take instead
of A, I'll use K.

227
00:12:58,710 --> 00:13:02,750
So that'll be my letter
for the best matrices.

228
00:13:02,750 --> 00:13:07,460
So symmetric.

229
00:13:07,460 --> 00:13:10,660
So now what's the deal
with symmetric matrices?

230
00:13:10,660 --> 00:13:12,930
The eigenvalues, the lambdas.

231
00:13:12,930 --> 00:13:16,030
I'll just call them the
lambdas and the y's.

232
00:13:16,030 --> 00:13:20,940
The lambdas are, do you
remember from last time?

233
00:13:20,940 --> 00:13:25,960
If I have a symmetric matrix,
the eigenvalues are all?

234
00:13:25,960 --> 00:13:26,900
Anybody remember?

235
00:13:26,900 --> 00:13:28,900
They're all real numbers.

236
00:13:28,900 --> 00:13:32,210
You can never run into
complex eigenvalues

237
00:13:32,210 --> 00:13:34,570
if you start with
a symmetric matrix.

238
00:13:34,570 --> 00:13:39,250
We didn't prove that but it's
just a few steps like those.

239
00:13:39,250 --> 00:13:42,870
And what about, most
important, what about the y's?

240
00:13:42,870 --> 00:13:44,110
The eigenvectors.

241
00:13:44,110 --> 00:13:48,940
They are, or can be chosen
to be, or whatever, anybody

242
00:13:48,940 --> 00:13:50,200
remember that fact?

243
00:13:50,200 --> 00:13:53,030
These are, like,
the golden facts.

244
00:13:53,030 --> 00:14:00,980
Every sort of bunch of
matrices reveals itself

245
00:14:00,980 --> 00:14:03,560
through what its
eigenvalues are like

246
00:14:03,560 --> 00:14:05,420
and what its
eigenvectors are like.

247
00:14:05,420 --> 00:14:08,590
And the most important class
is symmetric and that reveals

248
00:14:08,590 --> 00:14:12,730
itself through real
eigenvalues and...?

249
00:14:12,730 --> 00:14:13,990
Orthogonal, good.

250
00:14:13,990 --> 00:14:18,760
Orthogonal eigenvectors,
orthogonal.

251
00:14:18,760 --> 00:14:23,580
And in fact, since
I'm an eigenvector,

252
00:14:23,580 --> 00:14:27,030
I can adjust its
length as I like.

253
00:14:27,030 --> 00:14:29,380
Right?

254
00:14:29,380 --> 00:14:32,400
If y is an eigenvector,
11y is an eigenvector

255
00:14:32,400 --> 00:14:35,660
because I would just
multiply both sides by 11.

256
00:14:35,660 --> 00:14:40,860
That whole line of eigenvectors
is getting stretched by lambda.

257
00:14:40,860 --> 00:14:45,970
So what I want to do is
make them unit vectors.

258
00:14:45,970 --> 00:14:49,750
MATLAB will
automatically produce,

259
00:14:49,750 --> 00:14:53,100
eig would automatically
give you vectors that

260
00:14:53,100 --> 00:15:00,740
have been normalized to unit.

261
00:15:00,740 --> 00:15:03,630
Here's something good.

262
00:15:03,630 --> 00:15:05,820
So what does orthogonal mean?

263
00:15:05,820 --> 00:15:10,240
That means that one of them,
the dot product of one of them

264
00:15:10,240 --> 00:15:14,220
with another one is?

265
00:15:14,220 --> 00:15:17,180
Now that's not, I didn't
do the dot product yet.

266
00:15:17,180 --> 00:15:21,700
What symbol do I have to
write on left-hand side?

267
00:15:21,700 --> 00:15:23,440
Well you could say,
just put a dot.

268
00:15:23,440 --> 00:15:25,150
Of course.

269
00:15:25,150 --> 00:15:34,320
But dots are not cool, right?

270
00:15:34,320 --> 00:15:36,940
So maybe I should say
inner product, that's

271
00:15:36,940 --> 00:15:41,990
the more upper-class word.

272
00:15:41,990 --> 00:15:44,200
But I want to use transpose.

273
00:15:44,200 --> 00:15:45,720
So it's the transpose.

274
00:15:45,720 --> 00:15:47,130
That's the dot product.

275
00:15:47,130 --> 00:15:50,010
And that's the test
for perpendicular.

276
00:15:50,010 --> 00:15:51,450
So what's the answer then?

277
00:15:51,450 --> 00:15:55,300
I get a zero if i
is different from j.

278
00:15:55,300 --> 00:15:58,220
If I'm taking two
different eigenvectors

279
00:15:58,220 --> 00:16:01,520
and I take their dot product,
that's what you told me,

280
00:16:01,520 --> 00:16:02,960
they're orthogonal.

281
00:16:02,960 --> 00:16:05,780
And now what if i equals j?

282
00:16:05,780 --> 00:16:09,400
If I'm taking the dot
product with itself,

283
00:16:09,400 --> 00:16:11,960
each eigenvector with itself.

284
00:16:11,960 --> 00:16:16,920
So what does the dot product
of a vector with itself give?

285
00:16:16,920 --> 00:16:19,030
It'll be one because I'm normal.

286
00:16:19,030 --> 00:16:21,420
Exactly.

287
00:16:21,420 --> 00:16:23,520
What it always gives,
the dot product

288
00:16:23,520 --> 00:16:25,700
of a vector with
itself, you just

289
00:16:25,700 --> 00:16:28,880
realize that that'll be
y_1 squared, y_2 squared,

290
00:16:28,880 --> 00:16:31,960
it'll be the length squared.

291
00:16:31,960 --> 00:16:36,820
And here we're making
the length to be one.

292
00:16:36,820 --> 00:16:39,720
Well once again, if
I write something

293
00:16:39,720 --> 00:16:43,110
down like this which
is straightforward

294
00:16:43,110 --> 00:16:46,800
I want to express it
as a matrix statement.

295
00:16:46,800 --> 00:16:54,090
So I want to multiply, it'll
involve my good eigenvector

296
00:16:54,090 --> 00:16:55,670
matrix.

297
00:16:55,670 --> 00:17:00,250
And this will be what?

298
00:17:00,250 --> 00:17:03,740
I want to take all these
dots products at once.

299
00:17:03,740 --> 00:17:07,440
I want to take the dot product
of every y with every other y.

300
00:17:07,440 --> 00:17:08,790
Well here you go.

301
00:17:08,790 --> 00:17:14,480
Just put these guys in the rows,
now that we see that it really

302
00:17:14,480 --> 00:17:19,500
was the transpose
multiplying y, do you

303
00:17:19,500 --> 00:17:21,670
see that that's just done it?

304
00:17:21,670 --> 00:17:24,380
In fact, you'll tell me
what the answer is here.

305
00:17:24,380 --> 00:17:28,800
Don't shout it out, but let's
take it two or three entries

306
00:17:28,800 --> 00:17:31,680
and then you can shout it out.

307
00:17:31,680 --> 00:17:37,020
So what's the (1,
1) entry here of I

308
00:17:37,020 --> 00:17:39,690
guess that's what
we called S. And now

309
00:17:39,690 --> 00:17:41,680
this would be its transpose.

310
00:17:41,680 --> 00:17:44,950
And what I'm saying is if I
take-- Yeah, this is important

311
00:17:44,950 --> 00:17:47,530
because throughout
this course we're

312
00:17:47,530 --> 00:17:53,250
going to be taking A
transpose A, S transpose S,

313
00:17:53,250 --> 00:17:56,100
Q transpose Q,
often, often, often.

314
00:17:56,100 --> 00:17:58,340
So here we got the
first time at it.

315
00:17:58,340 --> 00:18:02,370
So why did I put a zero
there, because it's not it.

316
00:18:02,370 --> 00:18:03,880
What is it?

317
00:18:03,880 --> 00:18:06,670
What is that first entry?

318
00:18:06,670 --> 00:18:07,740
One.

319
00:18:07,740 --> 00:18:11,010
Because that's the row times
the column, that's a one.

320
00:18:11,010 --> 00:18:13,780
And what's the entry next to it?

321
00:18:13,780 --> 00:18:14,320
Zero.

322
00:18:14,320 --> 00:18:19,880
Right? y_1 dot product with
y_2 is, we're saying, zero.

323
00:18:19,880 --> 00:18:23,440
So what matrix have I got here?

324
00:18:23,440 --> 00:18:24,370
I've got the identity.

325
00:18:24,370 --> 00:18:28,690
Because y_2 with y_2 will
put a one there and all

326
00:18:28,690 --> 00:18:30,340
zeroes elsewhere.

327
00:18:30,340 --> 00:18:31,510
Zero, zero.

328
00:18:31,510 --> 00:18:34,760
And y_3 times y_3
will be the one.

329
00:18:34,760 --> 00:18:40,180
I get the identity.

330
00:18:40,180 --> 00:18:48,370
So this is for
symmetric matrices.

331
00:18:48,370 --> 00:18:53,520
In general, we can't expect the
eigenvectors to be orthogonal.

332
00:18:53,520 --> 00:18:57,160
It's these special
ones that are.

333
00:18:57,160 --> 00:19:02,640
But they're so important
that we notice.

334
00:19:02,640 --> 00:19:05,420
Now so this is the
eigenvector matrix

335
00:19:05,420 --> 00:19:09,680
S and this is its transpose.

336
00:19:09,680 --> 00:19:12,550
So I'm saying that for
a symmetric matrix,

337
00:19:12,550 --> 00:19:19,500
S transpose times S is I.
Well that's pretty important.

338
00:19:19,500 --> 00:19:22,920
In fact, that's
important enough that I'm

339
00:19:22,920 --> 00:19:27,530
going to give an
extra name to S,

340
00:19:27,530 --> 00:19:34,440
the eigenvector matrix when it
comes from a symmetric matrix,

341
00:19:34,440 --> 00:19:38,460
when it has a matrix with S
transpose times S equaling

342
00:19:38,460 --> 00:19:44,620
the identity is really
a good matrix to know.

343
00:19:44,620 --> 00:19:50,060
So let's just focus
on those guys.

344
00:19:50,060 --> 00:19:52,630
I can put that up here.

345
00:19:52,630 --> 00:19:57,340
So here's a matrix.

346
00:19:57,340 --> 00:20:00,410
Can I introduce a
different letter than S?

347
00:20:00,410 --> 00:20:05,540
It just helps you to remember
that this remarkable property

348
00:20:05,540 --> 00:20:06,710
is in force.

349
00:20:06,710 --> 00:20:10,090
That we've got it.

350
00:20:10,090 --> 00:20:15,310
So I'm going to call it--
When K is a symmetric matrix,

351
00:20:15,310 --> 00:20:24,370
I'll just repeat that,
then its eigenvector matrix

352
00:20:24,370 --> 00:20:29,520
has this S transpose
times S-- I'm

353
00:20:29,520 --> 00:20:35,590
going to call it Q. I'm going
to call the eigenvectors,

354
00:20:35,590 --> 00:20:44,920
so for this special
situation, A times--

355
00:20:44,920 --> 00:20:48,370
So I'm going to call the
eigenvector matrix Q.

356
00:20:48,370 --> 00:20:54,110
It's the S but it's worth
giving it this special notation

357
00:20:54,110 --> 00:21:02,710
to remind us that this is, so
Q is, an orthogonal matrix.

358
00:21:02,710 --> 00:21:06,950
There's a name for matrices
with this important property.

359
00:21:06,950 --> 00:21:09,840
And there's a letter
Q that everybody uses.

360
00:21:09,840 --> 00:21:12,690
An orthogonal matrix.

361
00:21:12,690 --> 00:21:14,170
And what does that mean?

362
00:21:14,170 --> 00:21:18,530
Means just what we said,
Q transpose times Q

363
00:21:18,530 --> 00:21:27,970
is I. What I've done here
is just giving a special,

364
00:21:27,970 --> 00:21:31,860
introducing a special
letter Q, a special name,

365
00:21:31,860 --> 00:21:36,690
orthogonal matrix for
what we found in the good,

366
00:21:36,690 --> 00:21:40,710
in this-- for eigenvectors
of a symmetric matrix.

367
00:21:40,710 --> 00:21:44,290
And this tells me
one thing more.

368
00:21:44,290 --> 00:21:46,210
Look what's happening here.

369
00:21:46,210 --> 00:21:51,160
Q transpose times Q is
giving the identity.

370
00:21:51,160 --> 00:21:54,600
What does that tell me
about the inverse of Q?

371
00:21:54,600 --> 00:22:00,710
That tells me here some matrix
is multiplying Q and giving I.

372
00:22:00,710 --> 00:22:02,050
So what is this matrix?

373
00:22:02,050 --> 00:22:06,600
What's another name
for this Q transpose?

374
00:22:06,600 --> 00:22:09,740
Is also Q inverse.

375
00:22:09,740 --> 00:22:12,190
Because that's what
defines the inverse matrix,

376
00:22:12,190 --> 00:22:23,580
that times Q should give I.
So Q transpose is Q inverse.

377
00:22:23,580 --> 00:22:26,140
I'm moving along here.

378
00:22:26,140 --> 00:22:33,170
Yes, please.

379
00:22:33,170 --> 00:22:37,990
The question was, shouldn't I
call it an orthonormal matrix?

380
00:22:37,990 --> 00:22:40,090
The answer is yes, I should.

381
00:22:40,090 --> 00:22:42,020
But nobody does.

382
00:22:42,020 --> 00:22:42,540
Dammit!

383
00:22:42,540 --> 00:22:45,340
So I'm stuck with that name.

384
00:22:45,340 --> 00:22:47,800
But orthonormal is
the proper name.

385
00:22:47,800 --> 00:22:50,170
If you call it an
orthonormal matrix,

386
00:22:50,170 --> 00:22:53,150
I'm happy because that's
really the right name

387
00:22:53,150 --> 00:22:54,870
for that matrix, orthonormal.

388
00:22:54,870 --> 00:22:58,780
Because orthogonal would just
mean orthogonal columns but

389
00:22:58,780 --> 00:23:01,380
we've taken this
extra little step

390
00:23:01,380 --> 00:23:03,500
to make all the lengths one.

391
00:23:03,500 --> 00:23:07,220
And then that gives us
this great property.

392
00:23:07,220 --> 00:23:09,260
Q transpose is Q inverse.

393
00:23:09,260 --> 00:23:14,780
Orthogonal matrices
are like rotations.

394
00:23:14,780 --> 00:23:19,270
I better give an example
of an orthogonal matrix.

395
00:23:19,270 --> 00:23:20,630
I'll do it right under here.

396
00:23:20,630 --> 00:23:22,970
Here is an orthogonal matrix.

397
00:23:22,970 --> 00:23:25,470
So what's the point?

398
00:23:25,470 --> 00:23:28,170
It's supposed to be a unit
vector in the first column

399
00:23:28,170 --> 00:23:31,940
so I'll put
cos(theta), sin(theta).

400
00:23:31,940 --> 00:23:34,390
And now what can go
in the second column

401
00:23:34,390 --> 00:23:36,990
of this orthogonal matrix?

402
00:23:36,990 --> 00:23:40,940
It's gotta be a unit vector
again because we've normalized

403
00:23:40,940 --> 00:23:46,500
and it's gotta be, what's the
connection to the first column?

404
00:23:46,500 --> 00:23:48,970
Orthogonal, gotta be orthogonal.

405
00:23:48,970 --> 00:23:51,400
So I just wanted to
put something here

406
00:23:51,400 --> 00:23:53,980
that sum of squares
is one, so I'll

407
00:23:53,980 --> 00:23:56,700
think cos(theta) and
sin(theta) again.

408
00:23:56,700 --> 00:23:59,430
But then I've got to
flip them a little

409
00:23:59,430 --> 00:24:01,060
to make it orthogonal to this.

410
00:24:01,060 --> 00:24:08,040
So if I put minus sin(theta)
there and plus cos(theta) there

411
00:24:08,040 --> 00:24:10,600
that certainly has
length one, good.

412
00:24:10,600 --> 00:24:14,580
And the dot product, can you do
the dot product of that column

413
00:24:14,580 --> 00:24:15,690
with that column?

414
00:24:15,690 --> 00:24:20,530
It's minus sine cosine,
plus sine cosine, zero.

415
00:24:20,530 --> 00:24:23,500
So there is a two
by two, actually

416
00:24:23,500 --> 00:24:26,740
that's a fantastic
building block out of which

417
00:24:26,740 --> 00:24:32,480
you could build many orthogonal
matrices of all sizes.

418
00:24:32,480 --> 00:24:38,390
That's a rotation by theta.

419
00:24:38,390 --> 00:24:41,400
That's a useful matrix to know.

420
00:24:41,400 --> 00:24:46,500
It takes every vector, swings
it around by an angle theta.

421
00:24:46,500 --> 00:24:47,330
What do I mean?

422
00:24:47,330 --> 00:24:53,980
I mean that Qx, Q times a
vector x, rotates x by theta.

423
00:24:53,980 --> 00:24:55,840
Let me put it.

424
00:24:55,840 --> 00:25:01,650
Qx rotates whatever vector x
you give it, you multiply by Q,

425
00:25:01,650 --> 00:25:07,370
it rotates it around by theta,
it doesn't change the length.

426
00:25:07,370 --> 00:25:16,246
So that would be an eigenvector
matrix of a pretty typical two

427
00:25:16,246 --> 00:25:19,660
by two.

428
00:25:19,660 --> 00:25:22,220
I see as I talk
about eigenvectors,

429
00:25:22,220 --> 00:25:24,650
eigenvalues there's
so much to say.

430
00:25:24,650 --> 00:25:29,150
Because everything you know
about a matrix shows up somehow

431
00:25:29,150 --> 00:25:31,450
in its eigenvectors
and eigenvalues

432
00:25:31,450 --> 00:25:38,030
and we're focusing
on symmetric guys.

433
00:25:38,030 --> 00:25:42,012
What happens to this
A=S*lambda*S inverse?

434
00:25:42,012 --> 00:25:42,970
Let's write that again.

435
00:25:42,970 --> 00:25:50,680
Now we've got K. It's
S*lambda*S inverse like any good

436
00:25:50,680 --> 00:25:57,630
diagonalization but now
I'm giving S a new name,

437
00:25:57,630 --> 00:25:59,220
which is what?

438
00:25:59,220 --> 00:26:04,790
Q. because when I give K,
when I use that letter K

439
00:26:04,790 --> 00:26:09,360
I'm thinking symmetric so
I'm in this special situation

440
00:26:09,360 --> 00:26:10,380
of symmetric.

441
00:26:10,380 --> 00:26:12,860
I have the lambda,
the eigenvalue matrix,

442
00:26:12,860 --> 00:26:16,990
and here I have Q inverse.

443
00:26:16,990 --> 00:26:20,300
But there's another
little way to write it

444
00:26:20,300 --> 00:26:24,970
and it's terrifically important
in mechanics and dynamics,

445
00:26:24,970 --> 00:26:26,600
everywhere.

446
00:26:26,600 --> 00:26:27,980
It's simple now.

447
00:26:27,980 --> 00:26:29,200
We know everything.

448
00:26:29,200 --> 00:26:31,880
Q lambda what?

449
00:26:31,880 --> 00:26:37,660
Q transpose.

450
00:26:37,660 --> 00:26:40,640
Do you see the
beauty of that form?

451
00:26:40,640 --> 00:26:48,190
That's called the principal
axis theorem in mechanics.

452
00:26:48,190 --> 00:26:50,680
It's called the spectral
theorem in mathematics.

453
00:26:50,680 --> 00:26:55,170
It's diagonalization, it's
quantum mechanics, everything.

454
00:26:55,170 --> 00:26:58,300
Any time you have
a symmetric matrix

455
00:26:58,300 --> 00:27:04,000
there's the wonderful
statement of how it breaks up

456
00:27:04,000 --> 00:27:07,440
when you look at its
orthonormal eigenvectors

457
00:27:07,440 --> 00:27:10,160
and its real eigenvalues.

458
00:27:10,160 --> 00:27:17,030
Do you see that once again
the symmetry has reappeared

459
00:27:17,030 --> 00:27:19,560
in the three factors?

460
00:27:19,560 --> 00:27:21,730
The symmetry has
reappeared in the fact

461
00:27:21,730 --> 00:27:25,740
that this vector is the
transpose of this one.

462
00:27:25,740 --> 00:27:34,320
We saw that for elimination
when these were triangular.

463
00:27:34,320 --> 00:27:40,760
That makes me remember what
we had in a different context,

464
00:27:40,760 --> 00:27:46,400
in the elimination when things
were triangular we had K=L*D*L

465
00:27:46,400 --> 00:27:48,540
transpose.

466
00:27:48,540 --> 00:27:55,430
I just squeezed that in to ask
you to sort of think of the two

467
00:27:55,430 --> 00:27:58,730
as two wonderful pieces
of linear algebra

468
00:27:58,730 --> 00:28:04,340
in such a perfect
shorthand, perfect notation.

469
00:28:04,340 --> 00:28:07,450
This was triangular
times the pivot matrix

470
00:28:07,450 --> 00:28:10,020
times the upper triangular.

471
00:28:10,020 --> 00:28:13,550
This is orthogonal times
the eigenvalue matrix

472
00:28:13,550 --> 00:28:17,020
times its transpose.

473
00:28:17,020 --> 00:28:20,130
And the key point
here was triangular

474
00:28:20,130 --> 00:28:28,020
and the key point
here is orthogonal.

475
00:28:28,020 --> 00:28:31,480
That took some time,
but it had to be done.

476
00:28:31,480 --> 00:28:33,440
This is the right
way to understand.

477
00:28:33,440 --> 00:28:37,910
That the central theme, it's a
highlight of a linear algebra

478
00:28:37,910 --> 00:28:42,140
course and we just
went straight to it.

479
00:28:42,140 --> 00:28:50,890
And now what I wanted to do
was look now at the special K.

480
00:28:50,890 --> 00:28:56,980
Oh, that's an awful pun.

481
00:28:56,980 --> 00:29:03,850
The special matrices that we
have, so those are n by n.

482
00:29:03,850 --> 00:29:06,930
And as I said last
time, usually it's

483
00:29:06,930 --> 00:29:10,180
not very likely that we
find all the eigenvalues

484
00:29:10,180 --> 00:29:16,780
and eigenvectors of this family
of bigger and bigger matrices.

485
00:29:16,780 --> 00:29:19,756
So now I'm going to
specialize to my n

486
00:29:19,756 --> 00:29:24,980
by n matrix K equals
twos down the diagonal,

487
00:29:24,980 --> 00:29:30,320
minus ones above and
minus ones below.

488
00:29:30,320 --> 00:29:32,560
What are the eigenvalues
of that matrix

489
00:29:32,560 --> 00:29:36,370
and what are the eigenvectors?

490
00:29:36,370 --> 00:29:39,330
How to tackle that?

491
00:29:39,330 --> 00:29:44,750
The best way is the way
we've done with the inverse

492
00:29:44,750 --> 00:29:47,780
and other ways of
understanding K,

493
00:29:47,780 --> 00:29:52,270
was to compare it with
the continuous problem.

494
00:29:52,270 --> 00:29:56,950
So this is a big matrix
which is a second difference

495
00:29:56,950 --> 00:29:59,820
matrix, fixed-fixed.

496
00:29:59,820 --> 00:30:04,060
Everybody remembers that the
boundary conditions associated

497
00:30:04,060 --> 00:30:05,790
with this are fixed-fixed.

498
00:30:05,790 --> 00:30:12,340
I want to ask you to look at
the corresponding differential

499
00:30:12,340 --> 00:30:13,550
equation.

500
00:30:13,550 --> 00:30:17,530
So you may not have
thought about eigenvectors

501
00:30:17,530 --> 00:30:19,490
of differential equations.

502
00:30:19,490 --> 00:30:21,770
And maybe I have to
call them eigenfunctions

503
00:30:21,770 --> 00:30:24,210
but the idea doesn't
change one bit.

504
00:30:24,210 --> 00:30:29,290
So what shall I look at?

505
00:30:29,290 --> 00:30:32,230
K corresponds to what?

506
00:30:32,230 --> 00:30:36,770
Continuous
differential business,

507
00:30:36,770 --> 00:30:40,200
what derivative, what?

508
00:30:40,200 --> 00:30:42,750
So I would like to
look at Ky=lambda*y.

509
00:30:42,750 --> 00:30:46,550


510
00:30:46,550 --> 00:30:48,390
I'm looking for
the y's and lambdas

511
00:30:48,390 --> 00:30:54,510
and the way I'm going to
get them is to look at,

512
00:30:54,510 --> 00:30:56,850
what did you say it was?

513
00:30:56,850 --> 00:31:00,950
K, now I'm going to write down
a differential equation that's

514
00:31:00,950 --> 00:31:05,910
like this but we'll
solve it quickly.

515
00:31:05,910 --> 00:31:07,570
So what will it be?

516
00:31:07,570 --> 00:31:11,490
K is like, tell me again.

517
00:31:11,490 --> 00:31:17,490
Second derivative of y
with respect to x squared.

518
00:31:17,490 --> 00:31:20,650
And there's one more thing
you have to remember.

519
00:31:20,650 --> 00:31:22,610
Minus.

520
00:31:22,610 --> 00:31:25,820
And here we have lambda*y(x).

521
00:31:25,820 --> 00:31:32,240


522
00:31:32,240 --> 00:31:36,860
That's an eigenvalue
and an eigenfunction

523
00:31:36,860 --> 00:31:40,660
that we're looking at for
this differential equation.

524
00:31:40,660 --> 00:31:44,180
Now there's another thing
you have to remember.

525
00:31:44,180 --> 00:31:48,200
And you'll know what it
is and you'll tell me.

526
00:31:48,200 --> 00:31:51,130
I could look for
all the solutions.

527
00:31:51,130 --> 00:31:53,770
Well, let me
momentarily do that.

528
00:31:53,770 --> 00:32:00,650
What functions have minus
the second derivative is

529
00:32:00,650 --> 00:32:02,580
a multiple of the function?

530
00:32:02,580 --> 00:32:05,380
Can you just tell me a few?

531
00:32:05,380 --> 00:32:07,250
Sine and cosine.

532
00:32:07,250 --> 00:32:10,350
I mean this is a fantastic
eigenvalue problem

533
00:32:10,350 --> 00:32:18,210
because its solutions
are sines and cosines.

534
00:32:18,210 --> 00:32:22,570
And of course we could combine
them into exponentials.

535
00:32:22,570 --> 00:32:29,810
We could have sine(omega*x)
or cos(omega*x) or we could

536
00:32:29,810 --> 00:32:33,480
combine those into
e^(i*omega*x),

537
00:32:33,480 --> 00:32:36,790
would be a combination of
those, or e^(-i*omega*x).

538
00:32:36,790 --> 00:32:39,310


539
00:32:39,310 --> 00:32:45,400
Those are combinations of
these, so those are not new.

540
00:32:45,400 --> 00:32:47,750
We've gotten lots
of eigenfunctions.

541
00:32:47,750 --> 00:32:51,600
Oh, for every frequency omega
this solves the equation.

542
00:32:51,600 --> 00:32:54,100
What's the eigenvalue?

543
00:32:54,100 --> 00:32:57,710
If you guess the eigenfunction
you've got the eigenvalue just

544
00:32:57,710 --> 00:32:59,850
by seeing what happens.

545
00:32:59,850 --> 00:33:05,040
So what would the eigenvalue be?

546
00:33:05,040 --> 00:33:06,470
Tell me again.

547
00:33:06,470 --> 00:33:07,580
Omega squared.

548
00:33:07,580 --> 00:33:11,040
Because I take the second
derivative of the sine, that'll

549
00:33:11,040 --> 00:33:14,520
give me the cosine back to the
sine, omega squared comes out,

550
00:33:14,520 --> 00:33:17,160
omega comes out twice.

551
00:33:17,160 --> 00:33:19,440
Comes out with a minus
sign from the cosine

552
00:33:19,440 --> 00:33:25,480
and that minus sign is
just right to make it plus.

553
00:33:25,480 --> 00:33:27,700
Lambda is omega squared.

554
00:33:27,700 --> 00:33:29,470
So omega squared.

555
00:33:29,470 --> 00:33:31,680
All the way of course.

556
00:33:31,680 --> 00:33:38,080
Those are the eigenvalues.

557
00:33:38,080 --> 00:33:41,880
All our differential examples
had something more than just

558
00:33:41,880 --> 00:33:43,530
the differential equation.

559
00:33:43,530 --> 00:33:48,240
What's the additional thing that
a differential equation comes

560
00:33:48,240 --> 00:33:49,690
with?

561
00:33:49,690 --> 00:33:51,990
Boundary conditions.

562
00:33:51,990 --> 00:33:53,530
With boundary conditions.

563
00:33:53,530 --> 00:33:55,360
Otherwise we got too many.

564
00:33:55,360 --> 00:33:58,520
I mean we don't want
all of these guys.

565
00:33:58,520 --> 00:34:00,580
What boundary conditions?

566
00:34:00,580 --> 00:34:03,590
If we're thinking about
K, our boundary conditions

567
00:34:03,590 --> 00:34:09,530
should be fixed and fixed.

568
00:34:09,530 --> 00:34:11,930
So that's the full problem.

569
00:34:11,930 --> 00:34:17,060
This is part of the problem
not just an afterthought.

570
00:34:17,060 --> 00:34:21,710
Now these conditions,
that will be perfect.

571
00:34:21,710 --> 00:34:25,420
Instead of having all
these sines and cosines

572
00:34:25,420 --> 00:34:33,740
we're going to narrow down
to a family that satisfies

573
00:34:33,740 --> 00:34:36,330
the boundary conditions.

574
00:34:36,330 --> 00:34:40,480
First boundary condition is
it has to be zero at x=0.

575
00:34:40,480 --> 00:34:43,030
What does that eliminate now?

576
00:34:43,030 --> 00:34:45,770
Cosines are gone,
keeps the sines.

577
00:34:45,770 --> 00:34:49,430
Cosines are gone by that
first boundary condition.

578
00:34:49,430 --> 00:34:52,390
These are guys that are left.

579
00:34:52,390 --> 00:34:59,300
I won't deal with these at this
point because I'm down to sines

580
00:34:59,300 --> 00:35:01,610
already from one
boundary condition.

581
00:35:01,610 --> 00:35:06,820
And now, the other
boundary condition.

582
00:35:06,820 --> 00:35:12,620
The other boundary
condition has to at x=1,

583
00:35:12,620 --> 00:35:16,610
if it's going to work
sin(omega*x) has to be?

584
00:35:16,610 --> 00:35:22,470
Nope, what do I put now?
sin(omega), right? x is one.

585
00:35:22,470 --> 00:35:23,820
I'm plugging in here.

586
00:35:23,820 --> 00:35:26,690
I'm just plugging
in x=1 to satisfy.

587
00:35:26,690 --> 00:35:30,260
And it has to equal zero.

588
00:35:30,260 --> 00:35:37,590
So that means, that
pins down omega.

589
00:35:37,590 --> 00:35:39,430
Doesn't give me just
one omega, well tell me

590
00:35:39,430 --> 00:35:43,280
one omega that's okay then.

591
00:35:43,280 --> 00:35:45,920
The first omega that
occurs to you is?

592
00:35:45,920 --> 00:35:47,070
Pi.

593
00:35:47,070 --> 00:35:48,530
The sine comes back to pi.

594
00:35:48,530 --> 00:35:50,340
So we've got one. y_1.

595
00:35:50,340 --> 00:35:54,490
Our first guy is with
omega=pi is sin(pi*x).

596
00:35:54,490 --> 00:36:00,540


597
00:36:00,540 --> 00:36:03,230
That's our fundamental mode.

598
00:36:03,230 --> 00:36:08,810
That's the number
one eigenfunction.

599
00:36:08,810 --> 00:36:12,050
And it is an eigenfunction,
it satisfies the boundary

600
00:36:12,050 --> 00:36:13,720
condition.

601
00:36:13,720 --> 00:36:15,500
Everybody would know
its picture, just

602
00:36:15,500 --> 00:36:17,710
one arch of the sine function.

603
00:36:17,710 --> 00:36:20,760
And the lambda that
goes with it, lambda_1,

604
00:36:20,760 --> 00:36:23,230
so this is the first
eigenfunction, what's

605
00:36:23,230 --> 00:36:25,960
the first eigenvalue?

606
00:36:25,960 --> 00:36:27,510
Pi squared, right.

607
00:36:27,510 --> 00:36:29,790
Because omega, we took to be pi.

608
00:36:29,790 --> 00:36:33,460
So lambda_1 is pi squared.

609
00:36:33,460 --> 00:36:36,510
We've got one.

610
00:36:36,510 --> 00:36:39,960
We were able to do it
because we could solve

611
00:36:39,960 --> 00:36:45,050
this equation in an easy way.

612
00:36:45,050 --> 00:36:47,450
Ready for a second one?

613
00:36:47,450 --> 00:36:49,580
What will the next one be?

614
00:36:49,580 --> 00:36:54,850
The next eigenfunction it's got
to, whatever its frequency is,

615
00:36:54,850 --> 00:36:57,640
omega, it's got to
have sin(omega)=0.

616
00:36:57,640 --> 00:37:00,260
What's your choice?

617
00:37:00,260 --> 00:37:01,530
2pi.

618
00:37:01,530 --> 00:37:06,950
So the next one is
going to be sin(2pi*x).

619
00:37:06,950 --> 00:37:10,960
And what will be the eigenvalue
that goes with that guy?

620
00:37:10,960 --> 00:37:14,900
lambda_2 will be
omega squared, which

621
00:37:14,900 --> 00:37:20,470
is 2pi squared, 2pi all squared,
so that's four pi squared.

622
00:37:20,470 --> 00:37:24,550
You see the whole list.

623
00:37:24,550 --> 00:37:28,600
The sines with these
correct frequencies

624
00:37:28,600 --> 00:37:33,520
are the eigenfunctions
of the second derivative

625
00:37:33,520 --> 00:37:36,900
with fixed-fixed
boundary conditions.

626
00:37:36,900 --> 00:37:39,890
And this is entirely typical.

627
00:37:39,890 --> 00:37:43,870
We don't have just n of them.

628
00:37:43,870 --> 00:37:45,770
The list goes on forever, right?

629
00:37:45,770 --> 00:37:48,680
The list goes on forever
because we're talking here

630
00:37:48,680 --> 00:37:50,890
about a differential equation.

631
00:37:50,890 --> 00:37:52,420
A differential
equation's somehow

632
00:37:52,420 --> 00:37:55,990
like a matrix of infinite size.

633
00:37:55,990 --> 00:38:04,370
And somehow these sines are the
columns of the infinite size

634
00:38:04,370 --> 00:38:06,140
eigenvector matrix.

635
00:38:06,140 --> 00:38:09,230
And these numbers, pi
squared, four pi squared,

636
00:38:09,230 --> 00:38:13,040
nine pi squared, 16pi
squared are the eigenvalues

637
00:38:13,040 --> 00:38:22,340
of the infinite
eigenvalue matrix.

638
00:38:22,340 --> 00:38:24,140
We got those answers quickly.

639
00:38:24,140 --> 00:38:30,710
And let's just mention that
if I changed to free-fixed

640
00:38:30,710 --> 00:38:36,240
or to free-free I could repeat.

641
00:38:36,240 --> 00:38:38,290
I'd get different y's.

642
00:38:38,290 --> 00:38:41,620
If I have different boundary
conditions I expect to get

643
00:38:41,620 --> 00:38:43,320
different y's.

644
00:38:43,320 --> 00:38:50,890
In fact, what would it look like
if that was y'=0 as the left

645
00:38:50,890 --> 00:38:55,410
end?

646
00:38:55,410 --> 00:38:59,770
What would you expect the
eigenfunctions to look like?

647
00:38:59,770 --> 00:39:01,300
They'd be cosines.

648
00:39:01,300 --> 00:39:02,650
They'd be cosines.

649
00:39:02,650 --> 00:39:06,040
And then we would have to
adjust the omegas to make

650
00:39:06,040 --> 00:39:11,410
them come out right
at the right-hand end.

651
00:39:11,410 --> 00:39:17,570
So this y(0)=0, the
fixed ones gave us sines,

652
00:39:17,570 --> 00:39:24,710
the free ones give us cosines,
the periodic ones if I had

653
00:39:24,710 --> 00:39:29,800
y(0)=y(1) so that I'm
just circling around,

654
00:39:29,800 --> 00:39:36,160
then I would expect these
e^(ikx)'s -- the textbook will,

655
00:39:36,160 --> 00:39:38,890
so I'm in the eigenvalue
section of course,

656
00:39:38,890 --> 00:39:42,630
and the textbook lists
the answers for the other

657
00:39:42,630 --> 00:39:44,040
possibilities.

658
00:39:44,040 --> 00:39:46,330
Let's go with this one.

659
00:39:46,330 --> 00:39:53,310
Because this is the one
that corresponds to K.

660
00:39:53,310 --> 00:40:04,700
We're now ready for
the final moment.

661
00:40:04,700 --> 00:40:14,030
And it is can we guess the
eigenvectors for the matrix?

662
00:40:14,030 --> 00:40:18,800
Now I'm going back to
the matrix question.

663
00:40:18,800 --> 00:40:22,480
And as I say, normally
the answer's no.

664
00:40:22,480 --> 00:40:24,170
Who could guess?

665
00:40:24,170 --> 00:40:25,810
But you can always hope.

666
00:40:25,810 --> 00:40:28,980
You can try.

667
00:40:28,980 --> 00:40:33,200
So what will I try?

668
00:40:33,200 --> 00:40:38,640
Here, let me draw
sin(x), sin(pi*x).

669
00:40:38,640 --> 00:40:42,150
And let me remember
that my matrix K was

670
00:40:42,150 --> 00:40:45,130
a finite difference matrix.

671
00:40:45,130 --> 00:40:47,360
Let's make it four by four.

672
00:40:47,360 --> 00:40:57,450
One, two, three, four let's say.

673
00:40:57,450 --> 00:40:59,180
What would be the
best I could hope for,

674
00:40:59,180 --> 00:41:02,970
for the eigenvector,
the first eigenvector?

675
00:41:02,970 --> 00:41:07,520
I'm hoping that the first
eigenvector of K is very,

676
00:41:07,520 --> 00:41:10,040
very like the
first eigenfunction

677
00:41:10,040 --> 00:41:15,950
in the differential equation,
which was this sin(pi*x),

678
00:41:15,950 --> 00:41:17,430
so that's sin(pi*x).

679
00:41:17,430 --> 00:41:20,350
Well, what do you hope for?

680
00:41:20,350 --> 00:41:23,640
What shall I hope
for as the components

681
00:41:23,640 --> 00:41:29,950
of y_1, the first eigenvector?

682
00:41:29,950 --> 00:41:32,170
It's almost too good.

683
00:41:32,170 --> 00:41:34,710
And as far as I know,
basically it only

684
00:41:34,710 --> 00:41:38,030
happens with these sines
and cosines example.

685
00:41:38,030 --> 00:41:41,830
These heights, I just
picked these, what I might

686
00:41:41,830 --> 00:41:46,710
call samples, of the thing.

687
00:41:46,710 --> 00:41:51,730
Those four values and of course
zero at that end and zero

688
00:41:51,730 --> 00:41:56,070
at that end, so
because K, the matrix K

689
00:41:56,070 --> 00:42:01,750
is building in the fixed-fixed.

690
00:42:01,750 --> 00:42:06,650
These four heights, these four
numbers, those four sines--

691
00:42:06,650 --> 00:42:12,360
In other words, what I hope
is that for Ky=lambda*y,

692
00:42:12,360 --> 00:42:16,730
I hope that y_1, the
first eigenvector,

693
00:42:16,730 --> 00:42:20,160
it'll be sin(pi*x),
but now what is x?

694
00:42:20,160 --> 00:42:24,100
So this is x here
from zero to one.

695
00:42:24,100 --> 00:42:27,650
So what's x there,
there, there and there?

696
00:42:27,650 --> 00:42:31,170
Instead of sin(pi*x),
the whole curve,

697
00:42:31,170 --> 00:42:33,770
I'm picking out
those four samples.

698
00:42:33,770 --> 00:42:39,420
So it'll be the sine
of, what'll it be here?

699
00:42:39,420 --> 00:42:42,080
Pi.

700
00:42:42,080 --> 00:42:46,880
Pi divided by n+1.

701
00:42:46,880 --> 00:42:48,800
Which in my picture
would be, we'll

702
00:42:48,800 --> 00:42:51,790
make it completely explicit.

703
00:42:51,790 --> 00:42:53,560
Five.

704
00:42:53,560 --> 00:42:57,630
It's 1/5 away along.

705
00:42:57,630 --> 00:43:00,650
Maybe I should make these
y's into column vectors

706
00:43:00,650 --> 00:43:03,540
since we're thinking
of them as columns.

707
00:43:03,540 --> 00:43:05,054
So here's y_1.

708
00:43:05,054 --> 00:43:06,970
sin(pi/5), sin(2pi/5),
sin(3pi/5), sin(4pi/5).

709
00:43:06,970 --> 00:43:16,710


710
00:43:16,710 --> 00:43:21,420
That's the first eigenvector.

711
00:43:21,420 --> 00:43:22,990
And it works.

712
00:43:22,990 --> 00:43:27,800
And you could guess
now the general one.

713
00:43:27,800 --> 00:43:36,140
Well when I say it works, I
haven't checked that it works.

714
00:43:36,140 --> 00:43:37,630
I better do that.

715
00:43:37,630 --> 00:43:43,930
But the essential
point is that it works.

716
00:43:43,930 --> 00:43:46,620
I may not even do it today.

717
00:43:46,620 --> 00:43:50,830
So, in fact, tell me
the second eigenvector.

718
00:43:50,830 --> 00:43:55,840
Or tell me the second
eigenfunction over here.

719
00:43:55,840 --> 00:43:57,540
What's the second eigenfunction?

720
00:43:57,540 --> 00:44:02,830
Let me draw it with
this green chalk.

721
00:44:02,830 --> 00:44:06,440
So I'm going to draw y_2.

722
00:44:06,440 --> 00:44:09,720
Now what does y_2
look like? sin(2pi*x).

723
00:44:09,720 --> 00:44:13,230
What's the new picture here?

724
00:44:13,230 --> 00:44:16,910
It goes up.

725
00:44:16,910 --> 00:44:18,950
What does it do?

726
00:44:18,950 --> 00:44:24,620
By here it's got
back, oh no, damn.

727
00:44:24,620 --> 00:44:27,670
I would've been better with
three points in the middle,

728
00:44:27,670 --> 00:44:30,860
but it's correct.

729
00:44:30,860 --> 00:44:32,860
It comes down here.

730
00:44:32,860 --> 00:44:33,860
Right?

731
00:44:33,860 --> 00:44:37,250
That's sin(2pi*x).

732
00:44:37,250 --> 00:44:46,310
That's halfway along.

733
00:44:46,310 --> 00:44:48,220
I'll finish this guy.

734
00:44:48,220 --> 00:44:56,180
This'll be sin(2pi/5),
sin(4pi/5).

735
00:44:56,180 --> 00:44:58,540
See I'm sampling
this same thing.

736
00:44:58,540 --> 00:45:01,620
I'm sampling 2pi*x
at those same points.

737
00:45:01,620 --> 00:45:05,300
sin(6pi/5) and sin(8pi/5).

738
00:45:05,300 --> 00:45:12,410


739
00:45:12,410 --> 00:45:15,720
Maybe let's accept
this as correct.

740
00:45:15,720 --> 00:45:17,270
It really works.

741
00:45:17,270 --> 00:45:19,250
It's the next eigenvector.

742
00:45:19,250 --> 00:45:23,200
And then there's a third one
and then there's a fourth one.

743
00:45:23,200 --> 00:45:28,120
And how many are
there? n usually.

744
00:45:28,120 --> 00:45:32,870
And in my case, what is n in
the picture I've drawn? n here

745
00:45:32,870 --> 00:45:35,110
is four.

746
00:45:35,110 --> 00:45:39,550
One, two, three, four. n is four
in that picture and that means

747
00:45:39,550 --> 00:45:41,250
that I'm dividing by n+1.

748
00:45:41,250 --> 00:45:45,730


749
00:45:45,730 --> 00:45:49,290
That's really sin(pi*h).

750
00:45:49,290 --> 00:45:53,410
You remember I used
h as the step size.

751
00:45:53,410 --> 00:45:58,630
So h is 1/5, 1/(n+1), 1/5.

752
00:45:58,630 --> 00:46:03,810
So it's sin(pi*h), sin(2pi*h),
4pi*h-- 3pi*h, 4pi*h.

753
00:46:03,810 --> 00:46:07,980
Here's 2, sin(2pi*h),
sin(4pi*h), sin(6pi*h),

754
00:46:07,980 --> 00:46:08,480
sin(8pi*h).

755
00:46:08,480 --> 00:46:15,670


756
00:46:15,670 --> 00:46:18,790
So I have two things to do.

757
00:46:18,790 --> 00:46:22,890
One is to remember what is the
remarkable property of these

758
00:46:22,890 --> 00:46:23,390
y's.

759
00:46:23,390 --> 00:46:26,380
So there's a y
that we've guessed.

760
00:46:26,380 --> 00:46:28,050
Right now you're
taking my word for it

761
00:46:28,050 --> 00:46:32,500
that it is the eigenvector
and this is the next one.

762
00:46:32,500 --> 00:46:34,850
I copied them out
of those functions.

763
00:46:34,850 --> 00:46:38,110
And just remind me, what
is it that I'm claiming

764
00:46:38,110 --> 00:46:42,190
to be true about y_1 and y_2.

765
00:46:42,190 --> 00:46:47,120
They are orthogonal,
there are orthogonal.

766
00:46:47,120 --> 00:46:52,330
Well to check that I'd
have to do some trig stuff.

767
00:46:52,330 --> 00:46:56,270
But what I was going to
do was come over here

768
00:46:56,270 --> 00:47:03,540
and say this was a symmetric
differential equation.

769
00:47:03,540 --> 00:47:07,610
We found its eigenfunctions.

770
00:47:07,610 --> 00:47:11,090
What do you think's
up with those?

771
00:47:11,090 --> 00:47:13,300
Those are orthogonal too.

772
00:47:13,300 --> 00:47:18,490
So this would be a
key fact in any sort

773
00:47:18,490 --> 00:47:24,530
of advanced applied math is
that the sine function is

774
00:47:24,530 --> 00:47:28,080
orthogonal to the sin(2x).

775
00:47:28,080 --> 00:47:30,390
That function as
orthogonal to this one.

776
00:47:30,390 --> 00:47:33,680
And actually that's what
makes the whole world

777
00:47:33,680 --> 00:47:36,320
of Fourier series work.

778
00:47:36,320 --> 00:47:40,610
So that was really
a wonderful fact.

779
00:47:40,610 --> 00:47:42,810
That this is orthogonal to this.

780
00:47:42,810 --> 00:47:47,740
Now you may, quite reasonably,
ask what do I mean by that?

781
00:47:47,740 --> 00:47:52,115
What does it mean for two
functions to be orthogonal?

782
00:47:52,115 --> 00:47:53,990
As long as we're getting
all these parallels,

783
00:47:53,990 --> 00:47:55,800
let's get that one too.

784
00:47:55,800 --> 00:47:57,890
I claim that this
function, which is this,

785
00:47:57,890 --> 00:48:02,350
is orthogonal to this function.

786
00:48:02,350 --> 00:48:05,200
What does that mean?

787
00:48:05,200 --> 00:48:08,200
What should these
functions-- Could I write

788
00:48:08,200 --> 00:48:11,970
dot or transpose or something?

789
00:48:11,970 --> 00:48:15,670
But now I'm doing
it for functions.

790
00:48:15,670 --> 00:48:21,700
I just want you to see
the complete analogy.

791
00:48:21,700 --> 00:48:26,500
So for vectors, what did I do?

792
00:48:26,500 --> 00:48:29,800
If I take a dot product I
multiply the first component

793
00:48:29,800 --> 00:48:32,050
times the first component,
second component times

794
00:48:32,050 --> 00:48:33,630
the second, so on, so on.

795
00:48:33,630 --> 00:48:36,127
Now what'll I do for functions?

796
00:48:36,127 --> 00:48:37,960
I multiply sin(pi*x) *
sin(2pi*x) at each x.

797
00:48:37,960 --> 00:48:42,300


798
00:48:42,300 --> 00:48:44,400
Of course I've got a
whole range of x's.

799
00:48:44,400 --> 00:48:46,620
And then what do I do?

800
00:48:46,620 --> 00:48:48,340
I integrate.

801
00:48:48,340 --> 00:48:49,800
I can't add.

802
00:48:49,800 --> 00:48:52,770
I integrate instead.

803
00:48:52,770 --> 00:48:58,330
So I integrate one function
sin(pi*x) against the other

804
00:48:58,330 --> 00:49:06,060
function, sin(2pi*x), dx, and
I integrate from zero to one

805
00:49:06,060 --> 00:49:08,350
and the answer comes out zero.

806
00:49:08,350 --> 00:49:09,820
The answer comes out zero.

807
00:49:09,820 --> 00:49:13,310
The sine functions
are orthogonal.

808
00:49:13,310 --> 00:49:15,870
The sines are
orthogonal functions.

809
00:49:15,870 --> 00:49:19,610
The sine vectors are
orthogonal vectors.

810
00:49:19,610 --> 00:49:26,420
I normalize to length one and
they go right into my Q. So

811
00:49:26,420 --> 00:49:29,400
if I multiply, if I did that
times that, that dot product

812
00:49:29,400 --> 00:49:31,310
would turn out to be zero.

813
00:49:31,310 --> 00:49:35,200
If I had been a
little less ambitious

814
00:49:35,200 --> 00:49:39,870
and taken n to be two
or three or something

815
00:49:39,870 --> 00:49:41,390
we would have seen
it completely.

816
00:49:41,390 --> 00:49:48,920
But maybe doing
with four is okay.

817
00:49:48,920 --> 00:49:55,650
So great lecture
except for that.

818
00:49:55,650 --> 00:49:58,610
Didn't get there.

819
00:49:58,610 --> 00:50:03,130
So Wednesday's lecture is
sort of the bringing all

820
00:50:03,130 --> 00:50:07,610
these pieces together, positive
eigenvalues, positive pivots,

821
00:50:07,610 --> 00:50:09,120
positive definite.

822
00:50:09,120 --> 00:50:11,700
So come on Wednesday please.

823
00:50:11,700 --> 00:50:13,720
Come Wednesday.

824
00:50:13,720 --> 00:50:15,720
And Wednesday
afternoon I'll have

825
00:50:15,720 --> 00:50:18,160
the review session as usual.