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PROFESSOR STRANG:
Shall we start?

10
00:00:26,300 --> 00:00:31,390
The main job of today is
eigenvalues and eigenvectors.

11
00:00:31,390 --> 00:00:34,330
The next section in the
book and a very big topic

12
00:00:34,330 --> 00:00:37,170
and things to say about it.

13
00:00:37,170 --> 00:00:41,170
I do want to begin with a
recap of what I didn't quite

14
00:00:41,170 --> 00:00:45,130
finish last time.

15
00:00:45,130 --> 00:00:50,360
So what we did was solve this
very straightforward equation.

16
00:00:50,360 --> 00:00:53,330
Straightforward except
that it has a point source,

17
00:00:53,330 --> 00:00:54,910
a delta function.

18
00:00:54,910 --> 00:00:58,350
And we solved it, both
the fixed-fixed case

19
00:00:58,350 --> 00:01:02,720
when a straight line
went up and back down

20
00:01:02,720 --> 00:01:05,520
and in the free-fixed
case when it

21
00:01:05,520 --> 00:01:11,790
was a horizontal line and then
down with slope minus one.

22
00:01:11,790 --> 00:01:15,060
And there are different
ways to get to this answer.

23
00:01:15,060 --> 00:01:18,570
But once you have it,
you can look at it

24
00:01:18,570 --> 00:01:20,230
and say, well is it right?

25
00:01:20,230 --> 00:01:22,830
Certainly the boundary
conditions are correct.

26
00:01:22,830 --> 00:01:26,060
Zero slope, went through
zero, that's good.

27
00:01:26,060 --> 00:01:28,470
And then the only
thing you really

28
00:01:28,470 --> 00:01:33,230
have to check is does
the slope drop by one

29
00:01:33,230 --> 00:01:36,780
at the point of the impulse?

30
00:01:36,780 --> 00:01:40,580
Because that's what this
is forcing us to do.

31
00:01:40,580 --> 00:01:43,180
It's saying the slope
should drop by one.

32
00:01:43,180 --> 00:01:46,550
And here the slope
is 1-a going up.

33
00:01:46,550 --> 00:01:51,330
And if I take the derivative,
it's -a going down.

34
00:01:51,330 --> 00:01:54,740
1-a dropped to -a, good.

35
00:01:54,740 --> 00:01:56,640
Here the slope was zero.

36
00:01:56,640 --> 00:02:00,100
Here the slope was
minus one, good.

37
00:02:00,100 --> 00:02:01,850
So those are the right answers.

38
00:02:01,850 --> 00:02:11,120
And this is simple, but
really a great example.

39
00:02:11,120 --> 00:02:14,330
And then, what I
wanted to do was

40
00:02:14,330 --> 00:02:17,500
catch the same thing
for the matrices.

41
00:02:17,500 --> 00:02:24,640
So those matrices, we all
know what K is and what T is.

42
00:02:24,640 --> 00:02:27,380
So I'm solving,
I'm really solving

43
00:02:27,380 --> 00:02:31,890
K K inverse equal identity.

44
00:02:31,890 --> 00:02:33,520
That's the equation I'm solving.

45
00:02:33,520 --> 00:02:37,340
So I'm looking for
K inverse and trying

46
00:02:37,340 --> 00:02:40,090
to get the columns
of the identity.

47
00:02:40,090 --> 00:02:42,930
And you realize the
columns of the identity

48
00:02:42,930 --> 00:02:45,660
are just like delta vectors.

49
00:02:45,660 --> 00:02:48,000
They've got a one
in one spot, they're

50
00:02:48,000 --> 00:02:52,620
a point load just
like this thing.

51
00:02:52,620 --> 00:02:56,610
So can I just say how
I remember K inverse?

52
00:02:56,610 --> 00:02:58,640
I finally, you
know-- again there

53
00:02:58,640 --> 00:03:01,790
are different ways to get to it.

54
00:03:01,790 --> 00:03:03,900
One way is MATLAB, just do it.

55
00:03:03,900 --> 00:03:08,620
But I guess maybe
the whole point

56
00:03:08,620 --> 00:03:13,100
is, the whole point of these
and the eigenvalues that

57
00:03:13,100 --> 00:03:16,370
are coming too, is this.

58
00:03:16,370 --> 00:03:25,230
That we have here the chance
to see important special cases

59
00:03:25,230 --> 00:03:26,580
that work out.

60
00:03:26,580 --> 00:03:28,710
Normally we don't
find the inverse,

61
00:03:28,710 --> 00:03:30,640
print out the
inverse of a matrix.

62
00:03:30,640 --> 00:03:32,370
It's not nice.

63
00:03:32,370 --> 00:03:36,870
Normally we just let eig
find the eigenvalues.

64
00:03:36,870 --> 00:03:39,590
Because that's an even
worse calculation,

65
00:03:39,590 --> 00:03:42,130
to find eigenvalues, in general.

66
00:03:42,130 --> 00:03:47,240
I'm talking here about our
matrices of all sizes n by n.

67
00:03:47,240 --> 00:03:51,680
Nobody finds the eigenvalues
by hand of n by n matrices.

68
00:03:51,680 --> 00:03:55,880
But these have
terrific eigenvalues

69
00:03:55,880 --> 00:03:58,380
and especially eigenvectors.

70
00:03:58,380 --> 00:04:04,010
So in a way this is a little
bit like, typical of math.

71
00:04:04,010 --> 00:04:07,700
That you ask about
general stuff or you

72
00:04:07,700 --> 00:04:14,690
write the equation
with a matrix A.

73
00:04:14,690 --> 00:04:17,540
So that's the
general information.

74
00:04:17,540 --> 00:04:21,110
And then there's the
specific, special guys

75
00:04:21,110 --> 00:04:22,980
with special functions.

76
00:04:22,980 --> 00:04:27,170
And here there'll be sines
and cosines and exponentials.

77
00:04:27,170 --> 00:04:29,240
Other places in
applied math, there

78
00:04:29,240 --> 00:04:31,710
are Bessel functions
and Legendre functions.

79
00:04:31,710 --> 00:04:33,210
Special guys.

80
00:04:33,210 --> 00:04:37,080
So here, these are special.

81
00:04:37,080 --> 00:04:40,880
And how do I complete K inverse?

82
00:04:40,880 --> 00:04:44,810
So this four, three, two, one.

83
00:04:44,810 --> 00:04:46,180
Let me complete T inverse.

84
00:04:46,180 --> 00:04:48,460
You probably know
T inverse already.

85
00:04:48,460 --> 00:04:52,090
So T, this is, four,
three, two, one,

86
00:04:52,090 --> 00:04:57,090
is when the load is way
over at the far left end

87
00:04:57,090 --> 00:04:59,250
and it's just descending.

88
00:04:59,250 --> 00:05:05,770
And now I'm going to-- Let me
show you how I write it in.

89
00:05:05,770 --> 00:05:07,890
Pay attention here
to the diagonal.

90
00:05:07,890 --> 00:05:15,810
So this will be three,
three, two, one.

91
00:05:15,810 --> 00:05:24,110
Do you see that's the solution
that's sort of like this one?

92
00:05:24,110 --> 00:05:28,310
That's the second column of
the inverse so it's solving,

93
00:05:28,310 --> 00:05:31,820
I'm solving, T T
inverse equals I here.

94
00:05:31,820 --> 00:05:34,700
It's the-- The second
column is the guy

95
00:05:34,700 --> 00:05:39,030
with a one in the second place.

96
00:05:39,030 --> 00:05:42,400
So that's where the load
is, in position number two.

97
00:05:42,400 --> 00:05:46,000
So I'm level, three,
three, up to that load.

98
00:05:46,000 --> 00:05:51,620
And then I'm dropping
after the load.

99
00:05:51,620 --> 00:05:56,720
What's the third
column of T inverse?

100
00:05:56,720 --> 00:05:59,230
I started with that
first column and I

101
00:05:59,230 --> 00:06:01,700
knew that the answer
would be symmetric

102
00:06:01,700 --> 00:06:03,710
because T is symmetric,
so that allowed

103
00:06:03,710 --> 00:06:05,890
me to write the first row.

104
00:06:05,890 --> 00:06:08,760
And now we can fill in the rest.

105
00:06:08,760 --> 00:06:12,450
So what do you think, if
the point load is-- Now,

106
00:06:12,450 --> 00:06:16,250
I'm looking at the third column,
third column of the identity,

107
00:06:16,250 --> 00:06:19,350
the load has moved down
to position number three.

108
00:06:19,350 --> 00:06:22,080
So what do I have
there and there?

109
00:06:22,080 --> 00:06:23,760
Two and two.

110
00:06:23,760 --> 00:06:26,241
And what do I have last?

111
00:06:26,241 --> 00:06:26,740
One.

112
00:06:26,740 --> 00:06:28,290
It's dropping to zero.

113
00:06:28,290 --> 00:06:31,730
You could put zero in
green here if you wanted.

114
00:06:31,730 --> 00:06:38,810
Zero is the unseen
last boundary,

115
00:06:38,810 --> 00:06:42,370
you know, row at this end.

116
00:06:42,370 --> 00:06:45,990
And finally, what's
happening here?

117
00:06:45,990 --> 00:06:50,080
What do I get from that?

118
00:06:50,080 --> 00:06:54,420
All one, one, one
to the diagonal.

119
00:06:54,420 --> 00:06:57,890
And then sure enough
it drops to zero.

120
00:06:57,890 --> 00:07:01,010
So this would be a case
where the load is there.

121
00:07:01,010 --> 00:07:06,010
It would be one, one,
one, one and then boom.

122
00:07:06,010 --> 00:07:07,112
No, it wouldn't be.

123
00:07:07,112 --> 00:07:08,070
It'd be more like this.

124
00:07:08,070 --> 00:07:14,780
One, one, one, one
and then down to--

125
00:07:14,780 --> 00:07:15,280
Okay.

126
00:07:15,280 --> 00:07:18,380
That's a pretty clean inverse.

127
00:07:18,380 --> 00:07:22,240
That's a very beautiful matrix.

128
00:07:22,240 --> 00:07:24,110
Don't you admire that matrix?

129
00:07:24,110 --> 00:07:27,100
I mean, if they were
all like that, gee,

130
00:07:27,100 --> 00:07:29,630
this would be a great world.

131
00:07:29,630 --> 00:07:40,760
But of course it's not sparse.

132
00:07:40,760 --> 00:07:43,510
That's why we don't
often use the inverse.

133
00:07:43,510 --> 00:07:46,090
Because we had a
sparse matrix T that

134
00:07:46,090 --> 00:07:48,380
was really fast to compute with.

135
00:07:48,380 --> 00:07:50,830
And here, if you
tell me the inverse,

136
00:07:50,830 --> 00:07:52,630
you've actually slowed me down.

137
00:07:52,630 --> 00:07:58,720
Because you've given me now a
dense matrix, no zeroes even

138
00:07:58,720 --> 00:08:04,830
and multiplying T inverse
times the right side

139
00:08:04,830 --> 00:08:08,800
would be slower than
just doing elimination.

140
00:08:08,800 --> 00:08:11,400
Now this is the kind of
more interesting one.

141
00:08:11,400 --> 00:08:16,000
Because this is the one that
has to go up to the diagonal

142
00:08:16,000 --> 00:08:18,490
and then down.

143
00:08:18,490 --> 00:08:22,680
So let me-- can I fill in what
I think-- way this one goes?

144
00:08:22,680 --> 00:08:26,370
I'm going upwards
to the diagonal

145
00:08:26,370 --> 00:08:28,170
and then I'm coming
down to zero.

146
00:08:28,170 --> 00:08:31,600
Remember that I'm coming
down to zero on this K.

147
00:08:31,600 --> 00:08:37,590
So Zero, zero, zero, zero
is kind of the row number.

148
00:08:37,590 --> 00:08:41,540
If that's row number zero,
here's one, two, three, four,

149
00:08:41,540 --> 00:08:42,700
the real thing.

150
00:08:42,700 --> 00:08:48,400
And then row five is
getting back to zero again.

151
00:08:48,400 --> 00:08:52,660
So what do you think, finish
the rest of that column.

152
00:08:52,660 --> 00:08:56,550
So you're telling me now
the response to the load

153
00:08:56,550 --> 00:08:57,850
in position two.

154
00:08:57,850 --> 00:08:59,910
So it's going to look like this.

155
00:08:59,910 --> 00:09:03,010
In fact, it's going to
look very like this.

156
00:09:03,010 --> 00:09:06,010
There's the three and then
this is in position two.

157
00:09:06,010 --> 00:09:08,750
And then I'm going to have
something here and something

158
00:09:08,750 --> 00:09:12,510
here and it'll drop to zero.

159
00:09:12,510 --> 00:09:14,050
What do I get?

160
00:09:14,050 --> 00:09:15,630
Four, two.

161
00:09:15,630 --> 00:09:17,830
Six, four, two, zero.

162
00:09:17,830 --> 00:09:19,490
It's dropping to zero.

163
00:09:19,490 --> 00:09:21,580
I'm going to finish
this in but then I'm

164
00:09:21,580 --> 00:09:25,660
going to look back and see
have I really got it right.

165
00:09:25,660 --> 00:09:28,880
How does this go now?

166
00:09:28,880 --> 00:09:31,480
Two, let's see.

167
00:09:31,480 --> 00:09:36,800
Now it's going up from
zero to two to four to six.

168
00:09:36,800 --> 00:09:38,330
That's on the diagonal.

169
00:09:38,330 --> 00:09:39,620
Now it starts down.

170
00:09:39,620 --> 00:09:43,300
It's got to get to zero,
so that'll be a three.

171
00:09:43,300 --> 00:09:48,320
Here is a one going up
to two to three to four.

172
00:09:48,320 --> 00:09:49,360
Is that right?

173
00:09:49,360 --> 00:09:52,120
And then dropped fast to zero.

174
00:09:52,120 --> 00:09:55,600
Is that correct?

175
00:09:55,600 --> 00:09:57,410
Think so, yep.

176
00:09:57,410 --> 00:10:01,260
Except, wait a minute now.

177
00:10:01,260 --> 00:10:03,740
We've got the right
overall picture.

178
00:10:03,740 --> 00:10:06,170
Climbing up, dropping down.

179
00:10:06,170 --> 00:10:07,970
Climbing up, dropping down.

180
00:10:07,970 --> 00:10:09,550
Climbing up, dropping down.

181
00:10:09,550 --> 00:10:10,420
All good.

182
00:10:10,420 --> 00:10:17,190
But we haven't yet got,
we haven't checked yet

183
00:10:17,190 --> 00:10:23,130
that the change in the
slope is supposed to be one.

184
00:10:23,130 --> 00:10:24,680
And it's not.

185
00:10:24,680 --> 00:10:29,130
Here the slope is like,
three, It's going up by threes

186
00:10:29,130 --> 00:10:33,840
and then it's
going down by twos.

187
00:10:33,840 --> 00:10:38,250
So we've gone from going
up at a slope of three

188
00:10:38,250 --> 00:10:41,010
to down to a slope of two.

189
00:10:41,010 --> 00:10:44,140
Up three, down just like this.

190
00:10:44,140 --> 00:10:47,520
But that would be a
change in slope of five.

191
00:10:47,520 --> 00:10:51,490
Therefore there's a 1/5.

192
00:10:51,490 --> 00:10:54,330
So this is going up with
a slope of four and down

193
00:10:54,330 --> 00:10:55,760
with a slope of one.

194
00:10:55,760 --> 00:11:00,050
Four dropping to one when I
divide by the five, that's

195
00:11:00,050 --> 00:11:01,070
what I like.

196
00:11:01,070 --> 00:11:04,060
Here is up by twos,
down by threes, again

197
00:11:04,060 --> 00:11:07,090
it's a change of five
so I need the five.

198
00:11:07,090 --> 00:11:09,720
Up by ones, down by four.

199
00:11:09,720 --> 00:11:12,740
Sudden, that's a
fast drop of four.

200
00:11:12,740 --> 00:11:16,190
Again, the slope changed
by five, dividing by five,

201
00:11:16,190 --> 00:11:17,610
that's got it.

202
00:11:17,610 --> 00:11:18,690
So that's my picture.

203
00:11:18,690 --> 00:11:22,420
You could now create K
inverse for any size.

204
00:11:22,420 --> 00:11:30,160
And more than that, sort
of see into K inverse

205
00:11:30,160 --> 00:11:32,250
what those numbers are.

206
00:11:32,250 --> 00:11:35,350
Because if I wrote the
five by five or six

207
00:11:35,350 --> 00:11:39,470
by six, doing it a
column at a time,

208
00:11:39,470 --> 00:11:42,290
it would look like
a bunch of numbers.

209
00:11:42,290 --> 00:11:44,050
But you see it now.

210
00:11:44,050 --> 00:11:46,880
Do you see the pattern?

211
00:11:46,880 --> 00:11:51,560
Right.

212
00:11:51,560 --> 00:11:54,260
This is one way to
get to those inverses,

213
00:11:54,260 --> 00:11:57,540
and homework problems
are offering other ways.

214
00:11:57,540 --> 00:12:04,240
T, in particular, is
quite easy to invert.

215
00:12:04,240 --> 00:12:07,400
Do I have any other
comment on inverses

216
00:12:07,400 --> 00:12:12,510
before the lecture on
eigenvalues really starts?

217
00:12:12,510 --> 00:12:18,200
Maybe I do have one comment,
one important comment.

218
00:12:18,200 --> 00:12:20,910
It's this, and I won't
develop it in full,

219
00:12:20,910 --> 00:12:24,530
but let's just say it.

220
00:12:24,530 --> 00:12:28,730
What if the load is
not a delta function?

221
00:12:28,730 --> 00:12:31,860
What if I have other loads?

222
00:12:31,860 --> 00:12:35,980
Like the uniform load of
all ones or any other load?

223
00:12:35,980 --> 00:12:45,660
What if the discrete load
here is not a delta vector?

224
00:12:45,660 --> 00:12:50,350
I now know the responses to each
column of the identity, right?

225
00:12:50,350 --> 00:12:54,140
If I put a load in position
one, there's the response.

226
00:12:54,140 --> 00:12:58,480
If I put a load in position
two, there is the response.

227
00:12:58,480 --> 00:13:03,340
Now, what if I have other loads?

228
00:13:03,340 --> 00:13:05,140
Let me take a typical load.

229
00:13:05,140 --> 00:13:10,190
What if the load was, well,
the one we looked at before.

230
00:13:10,190 --> 00:13:13,490
If the load was [1, 1, 1, 1].

231
00:13:13,490 --> 00:13:20,760
So that I had, the bar was
hanging by its own weight,

232
00:13:20,760 --> 00:13:24,930
let's say.

233
00:13:24,930 --> 00:13:29,090
In other words, could
I solve all problems

234
00:13:29,090 --> 00:13:31,480
by knowing these answers?

235
00:13:31,480 --> 00:13:33,510
That's what I'm
trying to get to.

236
00:13:33,510 --> 00:13:36,920
If I know these
special delta loads,

237
00:13:36,920 --> 00:13:41,340
then can I get the
solution for every load?

238
00:13:41,340 --> 00:13:42,250
Yes, no?

239
00:13:42,250 --> 00:13:43,490
What do you think?

240
00:13:43,490 --> 00:13:45,730
Yes, right.

241
00:13:45,730 --> 00:13:47,890
Now with this
matrix it's kind of

242
00:13:47,890 --> 00:13:51,040
easy to see because if you
know the inverse matrix, well

243
00:13:51,040 --> 00:13:53,020
you're obviously in business.

244
00:13:53,020 --> 00:13:59,390
If I had another load, say
another load f for load,

245
00:13:59,390 --> 00:14:03,450
I would just multiply by
K inverse, no problem.

246
00:14:03,450 --> 00:14:05,490
But I want to look
a little deeper.

247
00:14:05,490 --> 00:14:11,800
Because if I had other loads
here than a delta function,

248
00:14:11,800 --> 00:14:14,710
obviously if I had
two delta functions

249
00:14:14,710 --> 00:14:17,700
I could just combine
the two solutions.

250
00:14:17,700 --> 00:14:20,470
That's linearity that
we're using all the time.

251
00:14:20,470 --> 00:14:23,570
If I had ten delta functions
I could combine them.

252
00:14:23,570 --> 00:14:29,420
But then suppose I had
instead of a bunch of spikes,

253
00:14:29,420 --> 00:14:31,330
instead of a bunch
of point loads,

254
00:14:31,330 --> 00:14:33,710
I had a distributed load.

255
00:14:33,710 --> 00:14:38,110
Like all ones,
how could I do it?

256
00:14:38,110 --> 00:14:39,690
Main point is I could.

257
00:14:39,690 --> 00:14:40,410
Right?

258
00:14:40,410 --> 00:14:44,220
If I know these answers,
I know all answers.

259
00:14:44,220 --> 00:14:47,780
If I know the response
to a load at each point,

260
00:14:47,780 --> 00:14:50,850
then-- come back to
the discrete one.

261
00:14:50,850 --> 00:14:57,140
What would be the answer if
the load was [1, 1, 1, 1]?

262
00:14:57,140 --> 00:15:06,370
Suppose I now try to solve
the equation Ku=ones(4,1),

263
00:15:06,370 --> 00:15:08,509
so all ones.

264
00:15:08,509 --> 00:15:09,550
What would be the answer?

265
00:15:09,550 --> 00:15:12,960
How would I get it?

266
00:15:12,960 --> 00:15:15,510
I would just add the columns.

267
00:15:15,510 --> 00:15:20,990
Now why would I do that?

268
00:15:20,990 --> 00:15:21,590
Right.

269
00:15:21,590 --> 00:15:24,680
Because this, the
right-hand side,

270
00:15:24,680 --> 00:15:29,630
the input is the sum of
the four columns, the four

271
00:15:29,630 --> 00:15:31,330
special inputs.

272
00:15:31,330 --> 00:15:36,000
So the output is the sum
of the four outputs, right.

273
00:15:36,000 --> 00:15:39,140
In other words, as you saw,
we must know everything.

274
00:15:39,140 --> 00:15:41,950
And that's the way
we really know it.

275
00:15:41,950 --> 00:15:42,520
By linearity.

276
00:15:42,520 --> 00:15:46,580
If the input is a
combination of these,

277
00:15:46,580 --> 00:15:49,950
the output is the same
combination of those.

278
00:15:49,950 --> 00:15:50,770
Right.

279
00:15:50,770 --> 00:16:00,240
So, for example, in this T case,
if input was, if I did Tu=ones,

280
00:16:00,240 --> 00:16:06,760
I would just add those and the
output would be [10 9, 7, 4].

281
00:16:06,760 --> 00:16:10,380
That would be the output
from [1, 1, 1, 1].

282
00:16:10,380 --> 00:16:20,470
And now, oh boy.

283
00:16:20,470 --> 00:16:24,720
Actually, let me just
introduce a guy's name

284
00:16:24,720 --> 00:16:31,550
for these solutions
and not today show you.

285
00:16:31,550 --> 00:16:33,970
You have the idea, of course.

286
00:16:33,970 --> 00:16:37,930
Here we added because
everything was discrete.

287
00:16:37,930 --> 00:16:40,550
So you know what we're
going to do over here.

288
00:16:40,550 --> 00:16:44,290
We'll take integrals, right?

289
00:16:44,290 --> 00:16:51,430
A general load will be an
integral over point loads.

290
00:16:51,430 --> 00:16:53,090
That's the idea.

291
00:16:53,090 --> 00:16:54,220
A fundamental idea.

292
00:16:54,220 --> 00:17:00,960
That some other load, f(x),
is an integral of these guys.

293
00:17:00,960 --> 00:17:05,320
So the solution will be the
same integral of these guys.

294
00:17:05,320 --> 00:17:08,700
Let me not go there except
to tell you the name,

295
00:17:08,700 --> 00:17:11,020
because it's a very famous name.

296
00:17:11,020 --> 00:17:14,830
This solution u with
the delta function

297
00:17:14,830 --> 00:17:17,480
is called the Green's function.

298
00:17:17,480 --> 00:17:21,660
So I've now introduced the idea,
this is the Green's function.

299
00:17:21,660 --> 00:17:30,320
This guy is the Green's function
for the fixed-fixed problem.

300
00:17:30,320 --> 00:17:33,010
And this guy is the
Green's function

301
00:17:33,010 --> 00:17:36,280
for the free-fixed problem.

302
00:17:36,280 --> 00:17:38,820
And the whole point
is, maybe this

303
00:17:38,820 --> 00:17:44,700
is the one point I want you to
sort of see always by analogy.

304
00:17:44,700 --> 00:17:50,450
The Green's function is
just like the inverse.

305
00:17:50,450 --> 00:17:52,080
What is the Green's function?

306
00:17:52,080 --> 00:17:59,070
The Green's function is the
response at x, the u at x,

307
00:17:59,070 --> 00:18:03,010
when the input, when
the impulse is at a.

308
00:18:03,010 --> 00:18:04,800
So it sort of depends
on two things.

309
00:18:04,800 --> 00:18:09,940
It depends on the position a
of the input and it tells you

310
00:18:09,940 --> 00:18:14,990
the response at position x.

311
00:18:14,990 --> 00:18:19,260
And often we would use
the letter G for Green.

312
00:18:19,260 --> 00:18:23,680
So it depends on x and a.

313
00:18:23,680 --> 00:18:29,650
And maybe I'm happy if you
just sort of see in some way

314
00:18:29,650 --> 00:18:33,200
what we did there is just
like what we did here.

315
00:18:33,200 --> 00:18:35,220
And therefore the
Green's function

316
00:18:35,220 --> 00:18:40,820
must be just a differential,
continuous version

317
00:18:40,820 --> 00:18:46,330
of an inverse matrix.

318
00:18:46,330 --> 00:18:52,760
Let's move on to
eigenvalues with that point

319
00:18:52,760 --> 00:18:59,230
sort of made, but not driven
home by many, many examples.

320
00:18:59,230 --> 00:19:15,820
Question, I'll take
a question, shoot.

321
00:19:15,820 --> 00:19:21,220
Why did I increase zero, three,
six and then decrease six?

322
00:19:21,220 --> 00:19:29,350
Well intuitively it's
because this is copying this.

323
00:19:29,350 --> 00:19:32,660
What's wonderful is that
it's a perfect copy.

324
00:19:32,660 --> 00:19:37,460
I mean, intuitively the solution
to our difference equation

325
00:19:37,460 --> 00:19:40,710
should be like the solution
to our differential equation.

326
00:19:40,710 --> 00:19:44,530
That's why if we have
some computational,

327
00:19:44,530 --> 00:19:47,070
some differential equation
that we can't solve,

328
00:19:47,070 --> 00:19:49,930
which would be much more
typical than this one,

329
00:19:49,930 --> 00:19:53,220
that we couldn't solve it
exactly by pencil and paper,

330
00:19:53,220 --> 00:19:58,930
we would replace derivatives
by differences and go over here

331
00:19:58,930 --> 00:20:02,890
and we would hope that they
were like pretty close.

332
00:20:02,890 --> 00:20:07,820
Here they're right,
they're the same.

333
00:20:07,820 --> 00:20:08,880
Oh the other columns?

334
00:20:08,880 --> 00:20:09,630
Absolutely.

335
00:20:09,630 --> 00:20:11,610
These guys?

336
00:20:11,610 --> 00:20:14,880
Zero, two, four, six going up.

337
00:20:14,880 --> 00:20:18,180
Six, three, zero coming back.

338
00:20:18,180 --> 00:20:25,520
So that's a discrete
thing of one like that.

339
00:20:25,520 --> 00:20:27,670
And then the next
guy and the last guy

340
00:20:27,670 --> 00:20:30,310
would be going up
one, two, three, four

341
00:20:30,310 --> 00:20:34,360
and then sudden drop.

342
00:20:34,360 --> 00:20:35,940
Thanks for all questions.

343
00:20:35,940 --> 00:20:39,590
I mean, this sort of,
by adding these guys in,

344
00:20:39,590 --> 00:20:41,750
the first one actually
went up that way.

345
00:20:41,750 --> 00:20:45,500
You see the Green's functions.

346
00:20:45,500 --> 00:20:47,750
But of course this
has a Green's function

347
00:20:47,750 --> 00:20:53,300
for every a. x and a are running
all the way from zero to one.

348
00:20:53,300 --> 00:20:58,550
Here they're just
discrete positions.

349
00:20:58,550 --> 00:21:02,660
Thanks.

350
00:21:02,660 --> 00:21:06,140
So playing with
these delta functions

351
00:21:06,140 --> 00:21:09,120
and coming up with
this solution,

352
00:21:09,120 --> 00:21:12,910
well, as I say,
different ways to do it.

353
00:21:12,910 --> 00:21:16,710
I worked through one
way in class last time.

354
00:21:16,710 --> 00:21:18,470
It takes practice.

355
00:21:18,470 --> 00:21:21,610
So that's what the
homework's really for.

356
00:21:21,610 --> 00:21:24,820
You can see me come
up with this thing,

357
00:21:24,820 --> 00:21:28,810
then you can, with leisure,
you can follow the steps,

358
00:21:28,810 --> 00:21:32,740
but you've gotta do
it yourself to see.

359
00:21:32,740 --> 00:21:36,460
Eigenvalues and, of
course, eigenvectors.

360
00:21:36,460 --> 00:21:46,320
We have to give
them a fair shot.

361
00:21:46,320 --> 00:21:49,950
Square matrix.

362
00:21:49,950 --> 00:21:53,470
So I'm talking
about general, what

363
00:21:53,470 --> 00:21:57,100
eigenvectors and eigenvalues
are and why do we want them.

364
00:21:57,100 --> 00:22:01,400
I'm always trying to say
what's the purpose, you know,

365
00:22:01,400 --> 00:22:07,320
not doing this just for
abstract linear algebra.

366
00:22:07,320 --> 00:22:10,430
We do this, we look
for these things

367
00:22:10,430 --> 00:22:13,510
because they tremendously
simplify a problem

368
00:22:13,510 --> 00:22:16,260
if we can find it.

369
00:22:16,260 --> 00:22:19,970
So what's an eigenvector?

370
00:22:19,970 --> 00:22:23,510
The eigenvalue is
this number, lambda,

371
00:22:23,510 --> 00:22:26,840
and the eigenvector
is this vector y.

372
00:22:26,840 --> 00:22:33,120
And now, how do I
think about those?

373
00:22:33,120 --> 00:22:37,850
Suppose I take a vector
and I multiply by A.

374
00:22:37,850 --> 00:22:42,250
So the vector is headed
off in some direction.

375
00:22:42,250 --> 00:22:45,240
Here's a vector
v. If I multiply,

376
00:22:45,240 --> 00:22:47,070
and I'm given this
matrix, so I'm

377
00:22:47,070 --> 00:22:51,030
given the matrix,
whatever my matrix is.

378
00:22:51,030 --> 00:22:54,280
Could be one of those
matrices, any other matrix.

379
00:22:54,280 --> 00:23:00,130
If I multiply that by v,
I get some result, Av.

380
00:23:00,130 --> 00:23:01,410
What do I do?

381
00:23:01,410 --> 00:23:06,490
I look at that and I say that
v was not an eigenvector.

382
00:23:06,490 --> 00:23:10,290
Eigenvectors are the
special vectors which

383
00:23:10,290 --> 00:23:12,970
come out in the same direction.

384
00:23:12,970 --> 00:23:18,750
Av comes out parallel to v. So
this was not an eigenvector.

385
00:23:18,750 --> 00:23:21,360
Very few vectors
are eigenvectors,

386
00:23:21,360 --> 00:23:22,890
they're very special.

387
00:23:22,890 --> 00:23:25,960
Most vectors, that'll
be a typical picture.

388
00:23:25,960 --> 00:23:33,180
But there's a few of them
where I've a vector y

389
00:23:33,180 --> 00:23:36,840
and I multiply by A. And
then what's the point?

390
00:23:36,840 --> 00:23:42,530
Ay is in the same direction.

391
00:23:42,530 --> 00:23:45,090
It's on that same line as y.

392
00:23:45,090 --> 00:23:48,890
It could be, it might
be twice as far out.

393
00:23:48,890 --> 00:23:51,880
That would be Ay=2y.

394
00:23:51,880 --> 00:23:53,710
It might go backwards.

395
00:23:53,710 --> 00:23:56,070
This would be a
possibility, Ay=-y.

396
00:23:56,070 --> 00:23:58,830


397
00:23:58,830 --> 00:24:02,200
It could be just halfway.

398
00:24:02,200 --> 00:24:05,020
It could be, not move at all.

399
00:24:05,020 --> 00:24:06,400
That's even a possibility.

400
00:24:06,400 --> 00:24:07,960
Ay=0y.

401
00:24:07,960 --> 00:24:10,910
Count that.

402
00:24:10,910 --> 00:24:16,870
Those y's are eigenvectors
and the eigenvalue is just,

403
00:24:16,870 --> 00:24:19,910
from this point of view, the
eigenvalue has come in second

404
00:24:19,910 --> 00:24:25,390
because it's-- So y was a
special vector that kept its

405
00:24:25,390 --> 00:24:26,290
direction.

406
00:24:26,290 --> 00:24:32,430
And then lambda is just the
number, the two, the zero,

407
00:24:32,430 --> 00:24:39,050
the minus one, the 1/2
that tells you stretching,

408
00:24:39,050 --> 00:24:41,260
shrinking, reversing, whatever.

409
00:24:41,260 --> 00:24:42,760
That's the number.

410
00:24:42,760 --> 00:24:45,830
But y is the vector.

411
00:24:45,830 --> 00:24:53,230
And notice that
if I knew y and I

412
00:24:53,230 --> 00:24:56,440
knew it was an eigenvector, then
of course if I multiply by A,

413
00:24:56,440 --> 00:24:59,360
I'll learn the eigenvalue.

414
00:24:59,360 --> 00:25:01,370
And if I knew an
eigenvalue, you'll

415
00:25:01,370 --> 00:25:03,790
see how I could find
the eigenvector.

416
00:25:03,790 --> 00:25:06,070
Problem is you have
to find them both.

417
00:25:06,070 --> 00:25:07,810
And they multiply each other.

418
00:25:07,810 --> 00:25:11,050
So we're not talking about
linear equations anymore.

419
00:25:11,050 --> 00:25:13,540
Because one unknown is
multiplying another.

420
00:25:13,540 --> 00:25:19,780
But we'll find a way to look
to discover eigenvectors

421
00:25:19,780 --> 00:25:23,790
and eigenvalues.

422
00:25:23,790 --> 00:25:27,750
I said I would try to make
clear what's the purpose.

423
00:25:27,750 --> 00:25:36,460
The purpose is that in this
direction on this y line, line

424
00:25:36,460 --> 00:25:43,080
of multiples of y, A is
just acting like a number.

425
00:25:43,080 --> 00:25:48,050
A is some big n by n,
1,000 by 1,000 matrix.

426
00:25:48,050 --> 00:25:50,050
So a million numbers.

427
00:25:50,050 --> 00:25:58,110
But on this line, if we find
it, if we find an eigenline,

428
00:25:58,110 --> 00:26:03,010
you could say, an eigendirection
in that direction,

429
00:26:03,010 --> 00:26:06,070
all the complications
of A are gone.

430
00:26:06,070 --> 00:26:08,200
It's just acting like a number.

431
00:26:08,200 --> 00:26:14,410
So in particular we could solve
1,000 differential equations

432
00:26:14,410 --> 00:26:23,680
with 1,000 unknown u's with
this 1,000 by 1,000 matrix.

433
00:26:23,680 --> 00:26:26,620
We can find a
solution and this is

434
00:26:26,620 --> 00:26:29,550
where the eigenvector
and eigenvalue

435
00:26:29,550 --> 00:26:34,250
are going to pay off.

436
00:26:34,250 --> 00:26:35,250
You recognize this.

437
00:26:35,250 --> 00:26:38,100
Matrix A is of size 1,000.

438
00:26:38,100 --> 00:26:41,940
And u is a vector
of 1,000 unknowns.

439
00:26:41,940 --> 00:26:44,330
So that's a system
of 1,000 equations.

440
00:26:44,330 --> 00:26:50,870
But if we have found an
eigenvector and its eigenvalue

441
00:26:50,870 --> 00:26:56,600
then the equation will, if
it starts in that direction

442
00:26:56,600 --> 00:26:59,770
it'll stay in that direction
and the matrix will just

443
00:26:59,770 --> 00:27:01,200
be acting like a number.

444
00:27:01,200 --> 00:27:03,660
And we know how to
solve u'=lambda*u.

445
00:27:03,660 --> 00:27:06,850


446
00:27:06,850 --> 00:27:10,550
That one by one scalar
problem we know how to solve.

447
00:27:10,550 --> 00:27:13,770
The solution to that
is e to the lambda*t.

448
00:27:13,770 --> 00:27:17,430


449
00:27:17,430 --> 00:27:21,270
And of course it could
have a constant in it.

450
00:27:21,270 --> 00:27:25,950
Don't forget that these
equations are linear.

451
00:27:25,950 --> 00:27:29,450
If I multiply it, if
I take 2e^(lambda*t),

452
00:27:29,450 --> 00:27:32,180
I have a two here and a two
here and it's just as good.

453
00:27:32,180 --> 00:27:37,660
So I better allow that as well.

454
00:27:37,660 --> 00:27:41,600
A constant times
e^(lambda*t) times y.

455
00:27:41,600 --> 00:27:43,240
Notice this is a vector.

456
00:27:43,240 --> 00:27:47,490
It's a number times
a number, the growth.

457
00:27:47,490 --> 00:27:50,520
So the lambda is now, for
the differential equation,

458
00:27:50,520 --> 00:27:54,570
the lambda, this number
lambda is crucial.

459
00:27:54,570 --> 00:27:58,810
It's telling us whether the
solution grows, whether it

460
00:27:58,810 --> 00:28:01,330
decays, whether it oscillates.

461
00:28:01,330 --> 00:28:04,680
And we're just looking
at this one normal mode,

462
00:28:04,680 --> 00:28:09,560
you could say normal
mode, for eigenvector y.

463
00:28:09,560 --> 00:28:18,010
We certainly have not found
all possible solutions.

464
00:28:18,010 --> 00:28:25,070
If we have an eigenvector,
we found that one.

465
00:28:25,070 --> 00:28:30,340
And there's other uses
and then, let me think.

466
00:28:30,340 --> 00:28:31,510
Other uses, what?

467
00:28:31,510 --> 00:28:33,950
So let me write again
the fundamental equation,

468
00:28:33,950 --> 00:28:34,450
Ay=lambda*y.

469
00:28:34,450 --> 00:28:37,330


470
00:28:37,330 --> 00:28:41,150
So that was a
differential equation.

471
00:28:41,150 --> 00:28:43,200
Going forward in time.

472
00:28:43,200 --> 00:28:49,090
Now if we go forward in
steps we might multiply by A

473
00:28:49,090 --> 00:28:55,460
at every step.

474
00:28:55,460 --> 00:28:59,140
Tell me an eigenvector
of A squared.

475
00:28:59,140 --> 00:29:01,960
I'm looking for a vector
that doesn't change direction

476
00:29:01,960 --> 00:29:06,810
when I multiply twice by
A. You're going to tell me

477
00:29:06,810 --> 00:29:10,550
it's y. y will work.

478
00:29:10,550 --> 00:29:15,320
If I multiply once by
A I get lambda times y.

479
00:29:15,320 --> 00:29:20,620
When I multiply again by A I
get lambda squared times y.

480
00:29:20,620 --> 00:29:29,980
You see eigenvalues are
great for powers of a matrix,

481
00:29:29,980 --> 00:29:33,840
for differential equations.

482
00:29:33,840 --> 00:29:37,440
The nth power will just take
the eigenvalue to the nth.

483
00:29:37,440 --> 00:29:42,800
The nth power of A will just
have lambda to the nth there.

484
00:29:42,800 --> 00:29:47,350
You know, the pivots
of a matrix are all

485
00:29:47,350 --> 00:29:49,360
messed up when I square it.

486
00:29:49,360 --> 00:29:52,480
I can't see what's
happening with the pivots.

487
00:29:52,480 --> 00:29:56,870
The eigenvalues are a different
way to look at a matrix.

488
00:29:56,870 --> 00:30:01,770
The pivots are critical numbers
for steady-state problems.

489
00:30:01,770 --> 00:30:04,930
The eigenvalues are
the critical numbers

490
00:30:04,930 --> 00:30:09,250
for moving problems,
dynamic problems,

491
00:30:09,250 --> 00:30:13,520
things are oscillating
or growing or decaying.

492
00:30:13,520 --> 00:30:19,370
And by the way, let's just
recognize since this is

493
00:30:19,370 --> 00:30:24,990
the only thing that's
changing in time,

494
00:30:24,990 --> 00:30:31,400
what would be the-- I'll just
go down here, e^(lambda*t).

495
00:30:31,400 --> 00:30:32,850
Let's just look and see.

496
00:30:32,850 --> 00:30:35,630
When would I have decay?

497
00:30:35,630 --> 00:30:38,160
Which you might want
to call stability.

498
00:30:38,160 --> 00:30:40,800
A stable problem.

499
00:30:40,800 --> 00:30:42,840
What would be the
condition on lambda

500
00:30:42,840 --> 00:30:46,980
to get-- for this to decay.

501
00:30:46,980 --> 00:30:49,000
Lambda less than zero.

502
00:30:49,000 --> 00:30:52,600
Now there's one little
bit of bad news.

503
00:30:52,600 --> 00:30:55,180
Lambda could be complex.

504
00:30:55,180 --> 00:30:58,000
Lambda could be 3+4i.

505
00:30:58,000 --> 00:31:00,850


506
00:31:00,850 --> 00:31:03,860
It can be a complex
number, these eigenvalues,

507
00:31:03,860 --> 00:31:09,200
even if A is real.

508
00:31:09,200 --> 00:31:11,370
You'll say, how did
that happen, let me see?

509
00:31:11,370 --> 00:31:13,200
I didn't think.

510
00:31:13,200 --> 00:31:14,980
Well, let me finish
this thought.

511
00:31:14,980 --> 00:31:18,040
Suppose lambda was 3+4i.

512
00:31:18,040 --> 00:31:22,090


513
00:31:22,090 --> 00:31:26,610
So I'm thinking about what would
e to the lambda*t do in that

514
00:31:26,610 --> 00:31:27,900
case?

515
00:31:27,900 --> 00:31:30,820
So this is small example.

516
00:31:30,820 --> 00:31:32,510
If I had lambda is (3+4i), t.

517
00:31:32,510 --> 00:31:35,860


518
00:31:35,860 --> 00:31:40,520
What does that do as time grows?

519
00:31:40,520 --> 00:31:42,410
It's going to grow
and oscillate.

520
00:31:42,410 --> 00:31:45,160
And what decides the growth?

521
00:31:45,160 --> 00:31:46,680
The real part.

522
00:31:46,680 --> 00:31:49,130
So it's really the
decay or growth

523
00:31:49,130 --> 00:31:51,850
is decided by the real part.

524
00:31:51,850 --> 00:31:55,810
The three, e to the 3t,
that would be a growth.

525
00:31:55,810 --> 00:31:58,430
Let me put growth.

526
00:31:58,430 --> 00:32:01,190
And that would be,
of course, unstable.

527
00:32:01,190 --> 00:32:05,000
And that's a problem
when I have a real part

528
00:32:05,000 --> 00:32:07,910
of lambda bigger than zero.

529
00:32:07,910 --> 00:32:12,340
And then if lambda
has a zero real part,

530
00:32:12,340 --> 00:32:17,640
so it's pure oscillation, let
me just take a case like that.

531
00:32:17,640 --> 00:32:18,250
So e^(4it).

532
00:32:18,250 --> 00:32:20,990


533
00:32:20,990 --> 00:32:24,580
So that would be,
oscillating, right?

534
00:32:24,580 --> 00:32:31,570
It's cos(4t) + i*sin(4t),
it's just oscillating.

535
00:32:31,570 --> 00:32:39,430
So in this discussion we've
seen growth and decay.

536
00:32:39,430 --> 00:32:41,490
Tell me the parallels
because I'm always

537
00:32:41,490 --> 00:32:43,150
shooting for the parallels.

538
00:32:43,150 --> 00:32:45,930
What about the growth of A?

539
00:32:45,930 --> 00:32:51,300
What matrices, how
can I recognize

540
00:32:51,300 --> 00:32:56,500
a matrix whose powers grow?

541
00:32:56,500 --> 00:33:02,950
How can I recognize a matrix
whose powers go to zero?

542
00:33:02,950 --> 00:33:05,630
I'm asking you for
powers here, over there

543
00:33:05,630 --> 00:33:08,940
for exponentials somehow.

544
00:33:08,940 --> 00:33:15,400
So here would be A to
higher and higher powers

545
00:33:15,400 --> 00:33:18,670
goes to zero, the zero matrix.

546
00:33:18,670 --> 00:33:20,740
In other words, when
I multiply, multiply,

547
00:33:20,740 --> 00:33:24,440
multiply by that matrix I get
smaller and smaller and smaller

548
00:33:24,440 --> 00:33:26,760
matrices, zero in the limit.

549
00:33:26,760 --> 00:33:33,730
What do you think's the
test on the lambda now?

550
00:33:33,730 --> 00:33:37,500
So what are the
eigenvalues of A to the k?

551
00:33:37,500 --> 00:33:38,000
Let's see.

552
00:33:38,000 --> 00:33:40,980
If A had eigenvalues
lambda, A squared

553
00:33:40,980 --> 00:33:43,000
will have eigenvalues
lambda squared,

554
00:33:43,000 --> 00:33:45,510
A cubed will have
eigenvalues lambda cubed,

555
00:33:45,510 --> 00:33:48,165
A to the thousandth will
have eigenvalues lambda

556
00:33:48,165 --> 00:33:49,780
to the thousandth.

557
00:33:49,780 --> 00:33:54,820
And what's the test for
that to be getting small?

558
00:33:54,820 --> 00:33:58,640
Lambda less than one.

559
00:33:58,640 --> 00:34:03,880
So the test for stability will
be-- In the discrete case,

560
00:34:03,880 --> 00:34:07,530
it won't be the
real part of lambda,

561
00:34:07,530 --> 00:34:10,880
it'll be the size of
lambda less than one.

562
00:34:10,880 --> 00:34:16,430
And growth would be the size
of lambda greater than one.

563
00:34:16,430 --> 00:34:19,400
And again, there'd be
this borderline case

564
00:34:19,400 --> 00:34:24,310
when the eigenvalue has
magnitude exactly one.

565
00:34:24,310 --> 00:34:30,000
So you're seeing here
and also here the idea

566
00:34:30,000 --> 00:34:34,850
that we may have to deal
with complex numbers here.

567
00:34:34,850 --> 00:34:37,190
We don't have to deal
with the whole world

568
00:34:37,190 --> 00:34:40,620
of complex functions
and everything

569
00:34:40,620 --> 00:34:45,710
but it's possible for
complex numbers to come in.

570
00:34:45,710 --> 00:34:49,720
Well while I'm saying that,
why don't I give an example

571
00:34:49,720 --> 00:34:58,550
where it would come in.

572
00:34:58,550 --> 00:35:03,130
This is going to
be a real matrix

573
00:35:03,130 --> 00:35:07,360
with complex eigenvalues.

574
00:35:07,360 --> 00:35:11,030
Complex lambdas.

575
00:35:11,030 --> 00:35:19,770
It'll be an example.

576
00:35:19,770 --> 00:35:22,530
So I guess I'm
looking for a matrix

577
00:35:22,530 --> 00:35:29,320
where y and Ay never come
out in the same direction.

578
00:35:29,320 --> 00:35:34,240
For real y's I know, okay,
here's a good matrix.

579
00:35:34,240 --> 00:35:41,690
Take the matrix that rotates
every vector by 90 degrees.

580
00:35:41,690 --> 00:35:43,570
Or by theta.

581
00:35:43,570 --> 00:35:48,450
But let's say here's a matrix
that rotates every vector

582
00:35:48,450 --> 00:35:55,500
by 90 degrees.

583
00:35:55,500 --> 00:35:57,350
I'm going to raise
this board and hide it

584
00:35:57,350 --> 00:35:58,970
behind there in a minute.

585
00:35:58,970 --> 00:36:05,290
I just wanted to-- just to open
up this thought that we will

586
00:36:05,290 --> 00:36:09,960
have to face complex numbers.

587
00:36:09,960 --> 00:36:15,610
If you know how to multiply two
complex numbers and add them,

588
00:36:15,610 --> 00:36:16,800
you're okay.

589
00:36:16,800 --> 00:36:20,480
This isn't going to
turn into a big deal.

590
00:36:20,480 --> 00:36:25,020
But let's just realize
that-- Suppose that matrix,

591
00:36:25,020 --> 00:36:30,040
if I put in a vector y and
I multiply by that matrix,

592
00:36:30,040 --> 00:36:33,580
it'll turn it
through 90 degrees.

593
00:36:33,580 --> 00:36:35,420
So y couldn't be an eigenvector.

594
00:36:35,420 --> 00:36:37,070
That's the point
I'm trying to make.

595
00:36:37,070 --> 00:36:41,860
No real vector could
be the eigenvector

596
00:36:41,860 --> 00:36:46,720
of a rotation matrix because
every vector gets turned.

597
00:36:46,720 --> 00:36:53,230
So that's an example where you'd
have to go to complex vectors.

598
00:36:53,230 --> 00:36:58,960
and I think if I tried
the vector [1, i],

599
00:36:58,960 --> 00:37:02,730
so I'm letting the square
root of minus one into here,

600
00:37:02,730 --> 00:37:05,710
then I think it would come out.

601
00:37:05,710 --> 00:37:09,040
If I do that multiplication
I get minus i.

602
00:37:09,040 --> 00:37:10,670
And I get one.

603
00:37:10,670 --> 00:37:15,710
And I think that
this is, what is it?

604
00:37:15,710 --> 00:37:17,930
This is probably
minus i times that.

605
00:37:17,930 --> 00:37:32,230
So this is minus
i times the input.

606
00:37:32,230 --> 00:37:34,120
No big deal.

607
00:37:34,120 --> 00:37:36,940
That was like, you
can forget that.

608
00:37:36,940 --> 00:37:43,310
It's just complex
numbers can come in.

609
00:37:43,310 --> 00:37:52,650
Now let me come back to the
main point about eigenvectors.

610
00:37:52,650 --> 00:37:57,710
Things can be complex.

611
00:37:57,710 --> 00:38:02,400
So the main point is
how do we use them?

612
00:38:02,400 --> 00:38:08,690
And how many are there?

613
00:38:08,690 --> 00:38:10,610
Here's the key.

614
00:38:10,610 --> 00:38:13,520
A typical, good
matrix, which includes

615
00:38:13,520 --> 00:38:15,810
every symmetric
matrix, so it includes

616
00:38:15,810 --> 00:38:20,870
all of our examples and
more, if it's of size 1,000,

617
00:38:20,870 --> 00:38:24,810
it will have 1,000
different eigenvectors.

618
00:38:24,810 --> 00:38:29,770
And let me just say for
our symmetric matrices

619
00:38:29,770 --> 00:38:33,250
those eigenvectors
will all be real.

620
00:38:33,250 --> 00:38:37,170
They're great, the eigenvectors
of symmetric matrices.

621
00:38:37,170 --> 00:38:40,980
Oh, let me find them for one
particular symmetric matrix.

622
00:38:40,980 --> 00:38:47,040
Say this guy.

623
00:38:47,040 --> 00:38:49,950
So that's a matrix, two by two.

624
00:38:49,950 --> 00:38:53,850
How many eigenvectors
am I now looking for?

625
00:38:53,850 --> 00:38:55,960
Two.

626
00:38:55,960 --> 00:38:59,980
You could say, how
do I find them?

627
00:38:59,980 --> 00:39:07,740
Maybe with a two by two,
I can even just wing it.

628
00:39:07,740 --> 00:39:13,350
We can come up with a vector
that is an eigenvector.

629
00:39:13,350 --> 00:39:15,840
Actually that's what
we're going to do

630
00:39:15,840 --> 00:39:20,070
here is we're going to
guess the eigenvectors

631
00:39:20,070 --> 00:39:22,330
and then we're going to
show that they really

632
00:39:22,330 --> 00:39:24,990
are eigenvectors and then
we'll know the eigenvalues

633
00:39:24,990 --> 00:39:27,010
and it's fantastic.

634
00:39:27,010 --> 00:39:31,140
So like let's start here
with the two by two case.

635
00:39:31,140 --> 00:39:33,050
Anybody spot an eigenvector?

636
00:39:33,050 --> 00:39:35,360
Is [1, 0] an eigenvector?

637
00:39:35,360 --> 00:39:36,310
Try [1, 0].

638
00:39:36,310 --> 00:39:39,530
What comes out of [1, 0]?

639
00:39:39,530 --> 00:39:43,530
Well that picks the
first column, right?

640
00:39:43,530 --> 00:39:45,860
That's how I see,
multiplying by [1, 0],

641
00:39:45,860 --> 00:39:48,460
that says take one
of the first column.

642
00:39:48,460 --> 00:39:52,450
And is it an eigenvector?

643
00:39:52,450 --> 00:39:53,960
Yes, no?

644
00:39:53,960 --> 00:39:55,760
No.

645
00:39:55,760 --> 00:39:59,300
This vector is not in the
same direction as that one.

646
00:39:59,300 --> 00:40:00,870
No good.

647
00:40:00,870 --> 00:40:09,360
Now can you tell me one that is?

648
00:40:09,360 --> 00:40:12,200
You're going to
guess it. [1,  1].

649
00:40:12,200 --> 00:40:15,000
Try [1, 1].

650
00:40:15,000 --> 00:40:21,080
Do the multiplication
and what do you get?

651
00:40:21,080 --> 00:40:23,780
Right?

652
00:40:23,780 --> 00:40:29,000
If I input this vector
y, what do I get out?

653
00:40:29,000 --> 00:40:33,390
Actually I get y itself.

654
00:40:33,390 --> 00:40:36,510
Right?

655
00:40:36,510 --> 00:40:38,620
The point is it didn't
change direction,

656
00:40:38,620 --> 00:40:40,780
and it didn't even
change length.

657
00:40:40,780 --> 00:40:42,840
So what's the
eigenvalue for that?

658
00:40:42,840 --> 00:40:47,840
So I've got one eigenvalue
now, one eigenvector. [1, 1].

659
00:40:47,840 --> 00:40:51,120
And I've got the eigenvalue.

660
00:40:51,120 --> 00:40:53,980
So here are the
vectors, the y's.

661
00:40:53,980 --> 00:40:55,690
And here are the lambdas.

662
00:40:55,690 --> 00:41:01,440
And I've got one of them
and it's one, right?

663
00:41:01,440 --> 00:41:03,390
Would you like to
guess the other one?

664
00:41:03,390 --> 00:41:06,560
I'm only looking for two because
it's a two by two matrix.

665
00:41:06,560 --> 00:41:10,020
So let me erase here,
hope that you'll

666
00:41:10,020 --> 00:41:17,350
come up with another one. [1,
 -1] is certainly worth a try.

667
00:41:17,350 --> 00:41:19,220
Let's test it.

668
00:41:19,220 --> 00:41:21,430
If it's an eigenvector,
then it should come out

669
00:41:21,430 --> 00:41:22,430
in the same direction.

670
00:41:22,430 --> 00:41:26,290
What do I get when I do that?

671
00:41:26,290 --> 00:41:28,600
So I do that multiplication.

672
00:41:28,600 --> 00:41:32,740
Three and I get three
and minus three,

673
00:41:32,740 --> 00:41:35,790
so have we got an eigenvector?

674
00:41:35,790 --> 00:41:37,390
Yep.

675
00:41:37,390 --> 00:41:42,390
And what's, so if this was
y, what is this vector?

676
00:41:42,390 --> 00:41:44,220
3y.

677
00:41:44,220 --> 00:41:47,660
So there's the other
eigenvector, is [1, -1],

678
00:41:47,660 --> 00:41:56,150
and the other
eigenvalue is three.

679
00:41:56,150 --> 00:42:01,370
So we did it by
spotting it here.

680
00:42:01,370 --> 00:42:03,420
MATLAB can't do it that way.

681
00:42:03,420 --> 00:42:06,310
It's got to figure it out.

682
00:42:06,310 --> 00:42:12,120
But we're ahead of
MATLAB this time.

683
00:42:12,120 --> 00:42:15,650
So what do I notice?

684
00:42:15,650 --> 00:42:17,550
What do I notice
about this matrix?

685
00:42:17,550 --> 00:42:20,690
It was symmetric.

686
00:42:20,690 --> 00:42:25,150
And what do I notice
about the eigenvectors?

687
00:42:25,150 --> 00:42:29,390
If I show you those two
vectors, [1, 1] and [1, -1],

688
00:42:29,390 --> 00:42:32,710
what do you see there?

689
00:42:32,710 --> 00:42:38,270
They're orthogonal. [1, 1]
is orthogonal to [1, -1],

690
00:42:38,270 --> 00:42:40,910
perpendicular is the
same as orthogonal.

691
00:42:40,910 --> 00:42:49,890
These are orthogonal,
perpendicular.

692
00:42:49,890 --> 00:42:53,090
I can draw them, of course,
and see that. [1, 1]

693
00:42:53,090 --> 00:42:58,260
will go, if this is
one, it'll go here.

694
00:42:58,260 --> 00:43:00,510
So that's [1, 1].

695
00:43:00,510 --> 00:43:03,420
And [1, -1] will go
there, it'll go down,

696
00:43:03,420 --> 00:43:06,030
this would be the
other one. [1,  -1].

697
00:43:06,030 --> 00:43:07,820
So there's y_1.

698
00:43:07,820 --> 00:43:08,780
There's y_2.

699
00:43:08,780 --> 00:43:11,260
And they are perpendicular.

700
00:43:11,260 --> 00:43:17,300
But of course I don't draw
pictures all the time.

701
00:43:17,300 --> 00:43:21,920
What's the test for two
vectors being orthogonal?

702
00:43:21,920 --> 00:43:23,440
The dot product.

703
00:43:23,440 --> 00:43:24,530
The dot product.

704
00:43:24,530 --> 00:43:31,540
The inner product. y
transpose-- y_1 transpose * y_2.

705
00:43:31,540 --> 00:43:35,510
Do you prefer to write it
as y_1 with a dot, y_2?

706
00:43:35,510 --> 00:43:38,550


707
00:43:38,550 --> 00:43:42,660
This is maybe better because
it's matrix notation.

708
00:43:42,660 --> 00:43:51,420
And the point is orthogonal,
the dot product is zero.

709
00:43:51,420 --> 00:43:53,160
So that's good.

710
00:43:53,160 --> 00:43:56,240
Very good, in fact.

711
00:43:56,240 --> 00:43:59,180
So here's a very important fact.

712
00:43:59,180 --> 00:44:06,730
Symmetric matrices have
orthogonal eigenvectors.

713
00:44:06,730 --> 00:44:09,470
What I'm trying to say is
eigenvectors and eigenvalues

714
00:44:09,470 --> 00:44:13,390
are like a new way
to look at a matrix.

715
00:44:13,390 --> 00:44:16,820
A new way to see into it.

716
00:44:16,820 --> 00:44:21,270
And when the matrix is
symmetric, what we see

717
00:44:21,270 --> 00:44:25,040
is perpendicular eigenvectors.

718
00:44:25,040 --> 00:44:28,080
And what comment do you
have about the eigenvalues

719
00:44:28,080 --> 00:44:32,260
of this symmetric matrix?

720
00:44:32,260 --> 00:44:36,290
Remembering what
was on the board

721
00:44:36,290 --> 00:44:40,330
for this anti-symmetric matrix.

722
00:44:40,330 --> 00:44:44,290
What was the point about
that anti-symmetric matrix?

723
00:44:44,290 --> 00:44:51,330
Its eigenvalues were imaginary
actually, an i there.

724
00:44:51,330 --> 00:44:53,240
Here it's the opposite.

725
00:44:53,240 --> 00:44:56,620
What's the property
of the eigenvalues

726
00:44:56,620 --> 00:45:00,940
for a symmetric matrix
that you would just guess?

727
00:45:00,940 --> 00:45:02,080
They're real.

728
00:45:02,080 --> 00:45:03,670
They're real.

729
00:45:03,670 --> 00:45:06,280
Symmetric matrices
are great because they

730
00:45:06,280 --> 00:45:18,960
have real eigenvalues and they
have perpendicular eigenvectors

731
00:45:18,960 --> 00:45:22,380
and actually, probably if a
matrix has real eigenvalues

732
00:45:22,380 --> 00:45:27,090
and perpendicular eigenvectors,
it's going to be symmetric.

733
00:45:27,090 --> 00:45:32,230
So symmetry is a great property
and it shows up in a great way

734
00:45:32,230 --> 00:45:38,580
in this real eigenvalue, real
lambdas, and orthogonal y's.

735
00:45:38,580 --> 00:45:48,270
Shows up perfectly
in the eigenpicture.

736
00:45:48,270 --> 00:45:53,660
Here's a handy little
check on the eigenvalues

737
00:45:53,660 --> 00:45:55,580
to see if we got it right.

738
00:45:55,580 --> 00:45:56,700
Course we did.

739
00:45:56,700 --> 00:45:59,460
That's one and three we can get.

740
00:45:59,460 --> 00:46:03,950
But let me just show you two
useful checks if you haven't

741
00:46:03,950 --> 00:46:06,230
seen eigenvalues before.

742
00:46:06,230 --> 00:46:10,520
If I add the eigenvalues,
what do I get?

743
00:46:10,520 --> 00:46:12,140
Four.

744
00:46:12,140 --> 00:46:15,060
And I compare that
with adding down

745
00:46:15,060 --> 00:46:17,340
the diagonal of the matrix.

746
00:46:17,340 --> 00:46:19,350
Two and two, four.

747
00:46:19,350 --> 00:46:21,800
And that check always works.

748
00:46:21,800 --> 00:46:25,160
The sum of the eigenvalues
matches the sum

749
00:46:25,160 --> 00:46:26,220
down the diagonal.

750
00:46:26,220 --> 00:46:30,810
So that's like, if you got all
the eigenvalues but one, that

751
00:46:30,810 --> 00:46:32,150
would tell you the last one.

752
00:46:32,150 --> 00:46:36,060
Because the sum
of the eigenvalues

753
00:46:36,060 --> 00:46:39,990
matches the sum
down the diagonal.

754
00:46:39,990 --> 00:46:45,130
You have no clue where that
comes from but it's true.

755
00:46:45,130 --> 00:46:48,060
And another useful fact.

756
00:46:48,060 --> 00:46:52,340
If I multiply the
eigenvalues what do I get?

757
00:46:52,340 --> 00:46:53,620
Three?

758
00:46:53,620 --> 00:46:58,390
And now, where do you
see a three over here?

759
00:46:58,390 --> 00:47:00,730
The determinant.

760
00:47:00,730 --> 00:47:03,370
4-1=3.

761
00:47:03,370 --> 00:47:07,860
Can I just write those two
facts with no idea of proof.

762
00:47:07,860 --> 00:47:12,290
The sum of the lambdas,
I could write "sum".

763
00:47:12,290 --> 00:47:16,000


764
00:47:16,000 --> 00:47:21,680
This is for any matrix, the sum
of the lambdas is equal to the,

765
00:47:21,680 --> 00:47:25,170
it's called the
trace, of the matrix.

766
00:47:25,170 --> 00:47:29,190
The trace of the matrix is
the sum down the diagonal.

767
00:47:29,190 --> 00:47:36,360
And the product of the lambdas,
lambda_1 times lambda_2

768
00:47:36,360 --> 00:47:40,300
is the determinant
of the matrix.

769
00:47:40,300 --> 00:47:42,830
Or if I had ten eigenvalues,
I would multiply all ten

770
00:47:42,830 --> 00:47:47,220
and I'd get the determinant.

771
00:47:47,220 --> 00:47:51,500
So that's some facts
about eigenvalues.

772
00:47:51,500 --> 00:47:55,960
There's more, of
course, in section 1.5

773
00:47:55,960 --> 00:47:58,260
about how you would
find eigenvalues

774
00:47:58,260 --> 00:48:04,760
and how you would use them.

775
00:48:04,760 --> 00:48:09,770
That's of course the key point,
is how would we use them.

776
00:48:09,770 --> 00:48:15,770
Let me say something more about
that, how to use eigenvalues.

777
00:48:15,770 --> 00:48:22,480
Suppose I have this system of
1,000 differential equations.

778
00:48:22,480 --> 00:48:27,600
Linear, constant coefficients,
starts from some u(0).

779
00:48:27,600 --> 00:48:34,220


780
00:48:34,220 --> 00:48:37,380
How do eigenvalues
and eigenvectors help?

781
00:48:37,380 --> 00:48:40,060
Well, first I have to
find them, that's the job.

782
00:48:40,060 --> 00:48:44,200
So suppose I find 1,000
eigenvalues and eigenvectors.

783
00:48:44,200 --> 00:48:50,680
A times eigenvector number
i is eigenvalue number i

784
00:48:50,680 --> 00:48:52,470
times eigenvector number i.

785
00:48:52,470 --> 00:49:00,230
So these, y_1 to y_1000, so y_1
to y_1000 are the eigenvectors.

786
00:49:00,230 --> 00:49:02,920
And each one has
its own eigenvalue,

787
00:49:02,920 --> 00:49:05,830
lambda_1 to lambda_1000.

788
00:49:05,830 --> 00:49:11,260
And now if I did that work,
sort of like, in advance,

789
00:49:11,260 --> 00:49:14,310
now I come to the
differential equation.

790
00:49:14,310 --> 00:49:21,250
How could I use this?

791
00:49:21,250 --> 00:49:26,770
This is now going to
be the most-- it's

792
00:49:26,770 --> 00:49:29,460
three steps to use
it, three steps

793
00:49:29,460 --> 00:49:33,600
to use these to get the answer.

794
00:49:33,600 --> 00:49:37,470
Ready for step one.

795
00:49:37,470 --> 00:49:44,280
Step one is break u
nought into eigenvectors.

796
00:49:44,280 --> 00:49:48,440
Split, separate,
write, express u(0)

797
00:49:48,440 --> 00:50:02,040
as a combination
of eigenvectors.

798
00:50:02,040 --> 00:50:05,360
Now step two.

799
00:50:05,360 --> 00:50:08,150
What happens to
each eigenvector?

800
00:50:08,150 --> 00:50:10,550
So this is where the
differential equation

801
00:50:10,550 --> 00:50:11,200
starts from.

802
00:50:11,200 --> 00:50:13,480
This is the initial condition.

803
00:50:13,480 --> 00:50:17,490
1,000 components of u
at the start and it's

804
00:50:17,490 --> 00:50:23,390
separated into 1,000
eigenvector pieces.

805
00:50:23,390 --> 00:50:28,580
Now step two is watch
each piece separately.

806
00:50:28,580 --> 00:50:41,620
So step two will be multiply
say c_1 by e^(lambda_1*t),

807
00:50:41,620 --> 00:50:44,150
by its growth.

808
00:50:44,150 --> 00:50:47,680
This is following
eigenvector number one.

809
00:50:47,680 --> 00:50:53,250
And in general, I would multiply
every one of the c's by e

810
00:50:53,250 --> 00:50:55,140
to those guys.

811
00:50:55,140 --> 00:50:59,550
So what would I have now?

812
00:50:59,550 --> 00:51:01,860
This is one piece of the start.

813
00:51:01,860 --> 00:51:05,300
And that gives me one
piece of the finish.

814
00:51:05,300 --> 00:51:14,220
So the finish is, the answer
is to add up the 1,000 pieces.

815
00:51:14,220 --> 00:51:18,900
And if you're with me, you see
what those 1,000 pieces are.

816
00:51:18,900 --> 00:51:23,480
Here's a piece, some multiple
of the first eigenvector.

817
00:51:23,480 --> 00:51:26,370
Now if we only were
working with that piece,

818
00:51:26,370 --> 00:51:29,470
we follow it in time
by multiplying it

819
00:51:29,470 --> 00:51:31,640
by e to the lambda_1
* t, and what do we

820
00:51:31,640 --> 00:51:36,034
have at a later time?

821
00:51:36,034 --> 00:51:36,950
c_1*e^(lambda_1*t)y_1.

822
00:51:36,950 --> 00:51:41,990


823
00:51:41,990 --> 00:51:45,520
This piece has grown into that.

824
00:51:45,520 --> 00:51:48,460
And other pieces have
grown into other things.

825
00:51:48,460 --> 00:51:50,840
And what about the last piece?

826
00:51:50,840 --> 00:51:57,100
So what is it that
I have to add up?

827
00:51:57,100 --> 00:51:59,690
Tell me what to write here.

828
00:51:59,690 --> 00:52:05,190
c_1000, however much
of eigenvector 1,000

829
00:52:05,190 --> 00:52:10,700
was in there, and
then finally, never

830
00:52:10,700 --> 00:52:20,090
written left-handed
before, e to the who?

831
00:52:20,090 --> 00:52:23,310
Lambda number 1,000,
not 1,000 itself,

832
00:52:23,310 --> 00:52:31,500
but its eigenvalue, 1,000, t.

833
00:52:31,500 --> 00:52:36,900
This is just splitting, this
is constantly, constantly

834
00:52:36,900 --> 00:52:41,970
the method, the way to use
eigenvalues and eigenvectors.

835
00:52:41,970 --> 00:52:45,490
Split the problem
into the pieces that

836
00:52:45,490 --> 00:52:48,380
go-- that are eigenvectors.

837
00:52:48,380 --> 00:52:53,790
Watch each piece,
add up the pieces.

838
00:52:53,790 --> 00:52:56,560
That's why eigenvectors
are so important.

839
00:52:56,560 --> 00:52:59,800
Yeah?

840
00:52:59,800 --> 00:53:02,600
Yes, right.

841
00:53:02,600 --> 00:53:08,760
Well, now, very good question.

842
00:53:08,760 --> 00:53:09,979
Let's see.

843
00:53:09,979 --> 00:53:11,520
Well, the first
thing we have to know

844
00:53:11,520 --> 00:53:14,760
is that we do find
1,000 eigenvectors.

845
00:53:14,760 --> 00:53:19,940
And so my answer is going to
be for symmetric matrices,

846
00:53:19,940 --> 00:53:21,850
everything always works.

847
00:53:21,850 --> 00:53:25,060
For symmetric matrices,
if size is 1,000,

848
00:53:25,060 --> 00:53:28,360
they have 1,000 eigenvectors,
and next time we'll

849
00:53:28,360 --> 00:53:30,330
have a shot at some of these.

850
00:53:30,330 --> 00:53:33,920
What some of them are for
these special matrices.

851
00:53:33,920 --> 00:53:36,890
So this method
always works if I've

852
00:53:36,890 --> 00:53:42,660
got a full family of
independent eigenvectors.

853
00:53:42,660 --> 00:53:47,780
If it's of size 1,000, I need,
you're right, exactly right.

854
00:53:47,780 --> 00:53:52,690
To see that this was
the questionable step.

855
00:53:52,690 --> 00:53:55,270
If I haven't got
1,000 eigenvectors,

856
00:53:55,270 --> 00:53:57,270
I'm not going to be
able to take that step.

857
00:53:57,270 --> 00:53:59,890
And it happens.

858
00:53:59,890 --> 00:54:05,340
I am sad to report that
some matrices haven't

859
00:54:05,340 --> 00:54:07,650
got enough eigenvectors.

860
00:54:07,650 --> 00:54:11,800
Some matrices, they collapse.

861
00:54:11,800 --> 00:54:15,300
This always happens
in math, somehow.

862
00:54:15,300 --> 00:54:19,020
Two eigenvectors collapse
into one and the matrix

863
00:54:19,020 --> 00:54:23,950
is defective, like it's a loser.

864
00:54:23,950 --> 00:54:28,890
So now you have to, of
course, the equation

865
00:54:28,890 --> 00:54:31,720
still has a solution.

866
00:54:31,720 --> 00:54:34,320
So there has to be
something there,

867
00:54:34,320 --> 00:54:37,890
but the pure eigenvector
method is not

868
00:54:37,890 --> 00:54:40,890
going to make it on
those special matrices.

869
00:54:40,890 --> 00:54:43,530
I could write down
one but why should we

870
00:54:43,530 --> 00:54:46,760
give space to a loser?

871
00:54:46,760 --> 00:54:51,460
But what happens in that case?

872
00:54:51,460 --> 00:54:54,310
You might remember from
differential equations

873
00:54:54,310 --> 00:54:58,260
when two of these roots,
these are like roots,

874
00:54:58,260 --> 00:55:00,720
these lambdas are
like roots that you

875
00:55:00,720 --> 00:55:04,910
found in solving a
differential equation.

876
00:55:04,910 --> 00:55:09,290
When two of them come together,
that's when the danger is.

877
00:55:09,290 --> 00:55:11,680
When I have a double
eigenvalue, then there's

878
00:55:11,680 --> 00:55:15,090
a high risk that I've
only got one eigenvector.

879
00:55:15,090 --> 00:55:20,730
And I'll just put in this
little thing what the other,

880
00:55:20,730 --> 00:55:23,910
so the e^(lambda_1*t) is fine.

881
00:55:23,910 --> 00:55:29,130
But if that y_1 is like, if
the lambda_1's in there twice,

882
00:55:29,130 --> 00:55:30,570
I need something new.

883
00:55:30,570 --> 00:55:34,760
And the new thing turns
out to be t*e^(lambda* t).

884
00:55:34,760 --> 00:55:38,220


885
00:55:38,220 --> 00:55:40,320
I don't know if
anybody remembers.

886
00:55:40,320 --> 00:55:44,570
This was probably hammered back
in differential equations that

887
00:55:44,570 --> 00:55:49,660
if you had repeated
something or other then this,

888
00:55:49,660 --> 00:55:53,180
you didn't get pure
e^(lambda*t)'s, you got also

889
00:55:53,180 --> 00:55:54,650
a t*e^(lambda*t).

890
00:55:54,650 --> 00:55:56,660
Anyway that's the answer.

891
00:55:56,660 --> 00:55:58,870
That if we're
short eigenvectors,

892
00:55:58,870 --> 00:56:02,380
and it can happen, but it
won't for our good matrices.

893
00:56:02,380 --> 00:56:07,580
Okay, so Monday
I've got lots to do.

894
00:56:07,580 --> 00:56:11,403
Special eigenvalues and vectors
and then positive definite.

895
00:56:11,403 --> 00:56:11,903