1
00:00:04.55 --> 00:00:04.89
OK.
2
00:00:04.89 --> 00:00:11
So this is the first lecture on
eigenvalues and eigenvectors,
3
00:00:11 --> 00:00:18
and that's a big subject that
will take up most of the rest of
4
00:00:18 --> 00:00:19
the course.
5
00:00:19 --> 00:00:24
It's, again,
matrices are square and we're
6
00:00:24 --> 00:00:29
looking now for some special
numbers,
7
00:00:29 --> 00:00:33
the eigenvalues,
and some special vectors,
8
00:00:33 --> 00:00:35
the eigenvectors.
9
00:00:35 --> 00:00:40
And so this lecture is mostly
about what are these numbers,
10
00:00:40 --> 00:00:46
and then the other lectures are
about how do we use them,
11
00:00:46 --> 00:00:48
why do we want them.
12
00:00:48 --> 00:00:50
OK, so what's an eigenvector?
13
00:00:50 --> 00:00:55
Maybe I'll start with
eigenvector.
14
00:00:55 --> 00:00:57
What's an eigenvector?
15
00:00:57 --> 00:01:00
So I have a matrix A.
16
00:01:00 --> 00:01:00
OK.
17
00:01:00 --> 00:01:03
What does a matrix do?
18
00:01:03 --> 00:01:06
It acts on vectors.
19
00:01:06 --> 00:01:09
It multiplies vectors x.
20
00:01:09 --> 00:01:16
So the way that matrix acts is
in goes a vector x and out comes
21
00:01:16 --> 00:01:18.32
a vector Ax.
22
00:01:18.32 --> 00:01:20
It's like a function.
23
00:01:20 --> 00:01:27
With a function in calculus,
in goes a number x,
24
00:01:27 --> 00:01:28
out comes f(x).
25
00:01:28 --> 00:01:33
Here in linear algebra we're up
in more dimensions.
26
00:01:33 --> 00:01:36
In goes a vector x,
out comes a vector Ax.
27
00:01:36 --> 00:01:42
And the vectors I'm specially
interested in are the ones the
28
00:01:42 --> 00:01:46.27
come out in the same direction
that they went in.
29
00:01:46.27 --> 00:01:48.21
That won't be typical.
30
00:01:48.21 --> 00:01:52
Most vectors,
Ax is in -- points in
31
00:01:52 --> 00:01:56
some different direction.
32
00:01:56 --> 00:02:04
But there are certain vectors
where Ax comes out parallel to
33
00:02:04 --> 00:02:05
x.
34
00:02:05 --> 00:02:09
And those are the eigenvectors.
35
00:02:09 --> 00:02:12
So Ax parallel to x.
36
00:02:12 --> 00:02:16
Those are the eigenvectors.
37
00:02:16 --> 00:02:21
And what do I mean by parallel?
38
00:02:21 --> 00:02:29
Oh, much easier to just state
it in an equation.
39
00:02:29 --> 00:02:34
Ax is some multiple -- and
everybody calls that multiple
40
00:02:34 --> 00:02:36
lambda -- of x.
41
00:02:36 --> 00:02:38
That's our big equation.
42
00:02:38 --> 00:02:44
We look for special vectors --
and remember most vectors won't
43
00:02:44 --> 00:02:49.01
be eigenvectors -- that -- for
which Ax is in the same
44
00:02:49.01 --> 00:02:53.08
direction as x,
and by same direction I allow
45
00:02:53.08 --> 00:02:57
it to
be the very opposite direction,
46
00:02:57 --> 00:03:01
I allow lambda to be negative
or zero.
47
00:03:01 --> 00:03:06
Well, I guess we've met the
eigenvectors that have
48
00:03:06 --> 00:03:08.02
eigenvalue zero.
49
00:03:08.02 --> 00:03:13
Those are in the same
direction, but they're -- in a
50
00:03:13 --> 00:03:15
kind of very special way.
51
00:03:15 --> 00:03:19
So this --
the eigenvector x.
52
00:03:19 --> 00:03:23
Lambda, whatever this
multiplying factor is,
53
00:03:23 --> 00:03:28
whether it's six or minus six
or zero or even some imaginary
54
00:03:28 --> 00:03:31
number, that's the eigenvalue.
55
00:03:31 --> 00:03:35
So there's the eigenvalue,
there's the eigenvector.
56
00:03:35 --> 00:03:40
Let's just take a second on
eigenvalue zero.
57
00:03:40 --> 00:03:44
From the point of view of
eigenvalues, that's no special
58
00:03:44 --> 00:03:44
deal.
59
00:03:44 --> 00:03:46
That's, we have an eigenvector.
60
00:03:46 --> 00:03:50
If the eigenvalue happened to
be zero, that would mean that Ax
61
00:03:50 --> 00:03:53
was zero x, in other words zero.
62
00:03:53 --> 00:03:56
So what would x,
where would we look for -- what
63
00:03:56 --> 00:03:57
are the x-s?
64
00:03:57 --> 00:04:01
What are the eigenvectors with
eigenvalue zero?
65
00:04:01 --> 00:04:07
They're the guys in the null
space, Ax equals zero.
66
00:04:07 --> 00:04:13
So if our matrix is singular,
let me write this down.
67
00:04:13 --> 00:04:19
If, if A is singular,
then that -- what does singular
68
00:04:19 --> 00:04:20
mean?
69
00:04:20 --> 00:04:25
It means that it takes some
vector x into zero.
70
00:04:25 --> 00:04:32
Some non-zero vector,
that's why -- will be the
71
00:04:32 --> 00:04:34
eigenvector into zero.
72
00:04:34 --> 00:04:38
Then lambda equals zero is an
eigenvalue.
73
00:04:38 --> 00:04:42
But we're interested in all
eigenvalues now,
74
00:04:42 --> 00:04:47
lambda equals zero is not,
like, so special anymore.
75
00:04:47 --> 00:04:47
OK.
76
00:04:47 --> 00:04:52
So the question is,
how do we find these x-s and
77
00:04:52 --> 00:04:53
lambdas?
78
00:04:53 --> 00:05:00
And notice -- we don't have
an equation Ax equal B anymore.
79
00:05:00 --> 00:05:02
I can't use elimination.
80
00:05:02 --> 00:05:07
I've got two unknowns,
and in fact they're multiplied
81
00:05:07 --> 00:05:08
together.
82
00:05:08 --> 00:05:11
Lambda and x are both unknowns
here.
83
00:05:11 --> 00:05:16
So, we need to,
we need a good idea of how to
84
00:05:16 --> 00:05:17.18
find them.
85
00:05:17.18 --> 00:05:20
But before I,
before I
86
00:05:20 --> 00:05:25
do that, and that's where
determinant will come in,
87
00:05:25 --> 00:05:28
can I just give you some
matrices?
88
00:05:28 --> 00:05:30
Like here you go.
89
00:05:30 --> 00:05:33
Take the matrix,
a projection matrix.
90
00:05:33 --> 00:05:34
OK.
91
00:05:34 --> 00:05:39
So suppose we have a plane and
our matrix P is -- what I've
92
00:05:39 --> 00:05:45.44
called A, now I'm going to call
it P for the
93
00:05:45.44 --> 00:05:51
moment, because it's -- I'm
thinking OK, let's look at a
94
00:05:51 --> 00:05:53
projection matrix.
95
00:05:53 --> 00:05:58
What are the eigenvalues of a
projection matrix?
96
00:05:58 --> 00:06:00
So that's my question.
97
00:06:00 --> 00:06:03
What are the x-s,
the eigenvectors,
98
00:06:03 --> 00:06:07
and the lambdas,
the eigenvalues,
99
00:06:07 --> 00:06:12
for -- and now let me say a
projection matrix.
100
00:06:12 --> 00:06:16
My,
my point is that we -- before
101
00:06:16 --> 00:06:22
we get into determinants and,
and formulas and all that
102
00:06:22 --> 00:06:28
stuff, let's take some matrices
where we know what they do.
103
00:06:28 --> 00:06:34.16
We know that if we take a
vector b, what this matrix does
104
00:06:34.16 --> 00:06:37
is it projects it down to Pb.
105
00:06:37 --> 00:06:42
So is b an eigenvector in,
in that picture?
106
00:06:42 --> 00:06:45
Is that vector b an
eigenvector?
107
00:06:45 --> 00:06:46
No.
108
00:06:46 --> 00:06:51
Not so, so b is not an
eigenvector c- because Pb,
109
00:06:51 --> 00:06:55
its projection,
is in a different direction.
110
00:06:55 --> 00:07:00
So now tell me what vectors are
eigenvectors of P?
111
00:07:00 --> 00:07:07
What vectors do get projected
in the same direction that they
112
00:07:07 --> 00:07:07
start?
113
00:07:07 --> 00:07:12
So, so answer,
tell me some x-s.
114
00:07:12 --> 00:07:17
In this picture,
where could I start with a
115
00:07:17 --> 00:07:21
vector b or x,
do its projection,
116
00:07:21 --> 00:07:25
and end up in the same
direction?
117
00:07:25 --> 00:07:33
Well, that would happen if the
vector was right in that plane
118
00:07:33 --> 00:07:34
already.
119
00:07:34 --> 00:07:42
If the vector x was -- so let
the vector x -- so any vector,
120
00:07:42 --> 00:07:46
any x in the plane will be an
eigenvector.
121
00:07:46 --> 00:07:50
And what will happen when I
multiply by P,
122
00:07:50 --> 00:07:54
when I project a vector x -- I
called it b here,
123
00:07:54 --> 00:07:59
because this is our familiar
picture, but now I'm going to
124
00:07:59 --> 00:08:05
say that b was no good for,
for the, for our purposes.
125
00:08:05 --> 00:08:09
I'm interested in a vector x
that's actually in the plane,
126
00:08:09 --> 00:08:13
and I project it,
and what do I get back?
127
00:08:13 --> 00:08:14
x, of course.
128
00:08:14 --> 00:08:15.39
Doesn't move.
129
00:08:15.39 --> 00:08:18
So any x in the plane is
unchanged by P,
130
00:08:18 --> 00:08:20
and what's that telling me?
131
00:08:20 --> 00:08:25
That's telling me that x is an
eigenvector,
132
00:08:25 --> 00:08:29
and it's also telling me what's
the eigenvalue,
133
00:08:29 --> 00:08:32.94
which is -- just compare it
with that.
134
00:08:32.94 --> 00:08:35
The eigenvalue,
the multiplier,
135
00:08:35 --> 00:08:36.85
is just one.
136
00:08:36.85 --> 00:08:37
Good.
137
00:08:37 --> 00:08:41
So we have actually a whole
plane of eigenvectors.
138
00:08:41 --> 00:08:47
Now I ask, are there any other
eigenvectors?
139
00:08:47 --> 00:08:51
And I expect the answer to be
yes, because I would like to get
140
00:08:51 --> 00:08:55
three, if I'm in three
dimensions, I would like to hope
141
00:08:55 --> 00:08:58
for three independent
eigenvectors,
142
00:08:58 --> 00:09:01
two of them in the plane and
one not in the plane.
143
00:09:01 --> 00:09:01
OK.
144
00:09:01 --> 00:09:05
So this guy b that I drew there
was
145
00:09:05 --> 00:09:06.35
not any good.
146
00:09:06.35 --> 00:09:10.71
What's the right eigenvector
that's not in the plane?
147
00:09:10.71 --> 00:09:15
The, the good one is the one
that's perpendicular to the
148
00:09:15 --> 00:09:15
plane.
149
00:09:15 --> 00:09:20
There's an, another good x,
because what's the projection?
150
00:09:20 --> 00:09:23
So these are eigenvectors.
151
00:09:23 --> 00:09:28
Another guy here would be
another eigenvector.
152
00:09:28 --> 00:09:31
But now here is another one.
153
00:09:31 --> 00:09:38
Any x that's perpendicular to
the plane, what's Px for that,
154
00:09:38 --> 00:09:40
for that, vector?
155
00:09:40 --> 00:09:44
What's the projection of this
guy
156
00:09:44 --> 00:09:47
perpendicular to the plane?
157
00:09:47 --> 00:09:49
It is zero, of course.
158
00:09:49 --> 00:09:52
So -- there's the null space.
159
00:09:52 --> 00:09:57
Px and n- for those guys are
zero, or zero x if we like,
160
00:09:57 --> 00:09:59
and the eigenvalue is zero.
161
00:09:59 --> 00:10:05
So my answer to the question
is, what are the eigenvalues for
162
00:10:05 --> 00:10:08
a
projection matrix?
163
00:10:08 --> 00:10:09
There they are.
164
00:10:09 --> 00:10:11
One and zero.
165
00:10:11 --> 00:10:11
OK.
166
00:10:11 --> 00:10:14
We know projection matrices.
167
00:10:14 --> 00:10:19
We can write them down as that
A, A transpose,
168
00:10:19 --> 00:10:25
A inverse, A transpose thing,
but without doing that
169
00:10:25 --> 00:10:30
from the picture we could see
what are the eigenvectors.
170
00:10:30 --> 00:10:31.26
OK.
171
00:10:31.26 --> 00:10:33
Are there other matrices?
172
00:10:33 --> 00:10:36
Let me take a second example.
173
00:10:36 --> 00:10:39
How about a permutation matrix?
174
00:10:39 --> 00:10:44
What about the matrix,
I'll call it A now.
175
00:10:44 --> 00:10:47
Zero one, one zero.
176
00:10:47 --> 00:10:52
Can you tell me a vector x --
see, we'll have a system soon
177
00:10:52 --> 00:10:57
enough, so I,
I would like to just do these
178
00:10:57 --> 00:11:03
e- these couple of examples,
just to see the picture before
179
00:11:03 --> 00:11:08
we, before we let it all,
go into a system where
180
00:11:08 --> 00:11:12
that, matrix isn't anything
special.
181
00:11:12 --> 00:11:14
Because it is special.
182
00:11:14 --> 00:11:20
And what, so what vector could
I multiply by and end up in the
183
00:11:20 --> 00:11:21
same direction?
184
00:11:21 --> 00:11:25
Can you spot an eigenvector for
this guy?
185
00:11:25 --> 00:11:30.5
That's a matrix that permutes
x1 and x2,
186
00:11:30.5 --> 00:11:31
right?
187
00:11:31 --> 00:11:34
It switches the two components
of x.
188
00:11:34 --> 00:11:40
How could the vector with its
x2 x1, with -- permuted turn out
189
00:11:40 --> 00:11:46
to be a multiple of x1 x2,
the vector we start with?
190
00:11:46 --> 00:11:51.18
Can you tell me an eigenvector
here for this guy?
191
00:11:51.18 --> 00:11:58
x equal -- what is -- actually,
can you tell me one vector that
192
00:11:58 --> 00:11:59
has eigenvalue one?
193
00:11:59 --> 00:12:04
So what, what vector would have
eigenvalue one,
194
00:12:04 --> 00:12:07
so that if I,
if I permute it it doesn't
195
00:12:07 --> 00:12:08.15
change?
196
00:12:08.15 --> 00:12:11
There, that could be one one,
thanks.
197
00:12:11 --> 00:12:13
One one.
198
00:12:13 --> 00:12:15
OK, take that vector one one.
199
00:12:15 --> 00:12:21
That will be an eigenvector,
because if I do Ax I get one
200
00:12:21 --> 00:12:22
one.
201
00:12:22 --> 00:12:25
So that's the eigenvalue is
one.
202
00:12:25 --> 00:12:25
Great.
203
00:12:25 --> 00:12:27
That's one eigenvalue.
204
00:12:27 --> 00:12:33
But I have here a two by two
matrix, and I figure there's
205
00:12:33 --> 00:12:38
going to be a second eigenvalue.
206
00:12:38 --> 00:12:39
And eigenvector.
207
00:12:39 --> 00:12:41
Now, what about that?
208
00:12:41 --> 00:12:45
What's a vector,
OK, maybe we can just,
209
00:12:45 --> 00:12:47
like, guess it.
210
00:12:47 --> 00:12:52
A vector that the other --
actually, this one that I'm
211
00:12:52 --> 00:12:59
thinking of is going to be a
vector that has eigenvalue minus
212
00:12:59 --> 00:13:00
one.
213
00:13:00 --> 00:13:04
That's going to be my other
eigenvalue for this matrix.
214
00:13:04 --> 00:13:07
It's a -- notice the nice
positive or not negative matrix,
215
00:13:07 --> 00:13:11
but an eigenvalue is going to
come out negative.
216
00:13:11 --> 00:13:14
And can you guess,
spot the x that will work for
217
00:13:14 --> 00:13:15
that?
218
00:13:15 --> 00:13:17
So I want a,
a vector.
219
00:13:17 --> 00:13:22
When I multiply by A,
which reverses the two
220
00:13:22 --> 00:13:29
components, I want the thing to
come out minus the original.
221
00:13:29 --> 00:13:33
So what shall I send in in that
case?
222
00:13:33 --> 00:13:38
If I send in negative one one.
223
00:13:38 --> 00:13:45
Then when I apply A,
I get I do that multiplication,
224
00:13:45 --> 00:13:51
and I get one negative one,
so it reversed sign.
225
00:13:51 --> 00:13:53
So Ax is -x.
226
00:13:53 --> 00:13:56
Lambda is minus one.
227
00:13:56 --> 00:14:03
Ax -- so Ax was x there and Ax
is minus x here.
228
00:14:03 --> 00:14:09
Can I just mention,
like, jump ahead,
229
00:14:09 --> 00:14:15
and point out a special little
fact about eigenvalues.
230
00:14:15 --> 00:14:19
n by n matrices will have n
eigenvalues.
231
00:14:19 --> 00:14:25
And it's not like -- suppose n
is three or four or more.
232
00:14:25 --> 00:14:30
It's not so easy to find them.
233
00:14:30 --> 00:14:35
We'd have a third degree or a
fourth degree or an n-th degree
234
00:14:35 --> 00:14:36
equation.
235
00:14:36 --> 00:14:38
But here's one nice fact.
236
00:14:38 --> 00:14:41
There, there's one pleasant
fact.
237
00:14:41 --> 00:14:45
That the sum of the eigenvalues
equals the sum down the
238
00:14:45 --> 00:14:46
diagonal.
239
00:14:46 --> 00:14:52
That's called the trace,
and I put that in the lecture
240
00:14:52 --> 00:14:54
content specifically.
241
00:14:54 --> 00:14:59
So this is a neat fact,
the fact that sthe sum of the
242
00:14:59 --> 00:15:04
lambdas, add up the lambdas,
equals the sum -- what would
243
00:15:04 --> 00:15:08
you like me to,
shall I write that down?
244
00:15:08 --> 00:15:13
What I'm want to say in words
is the sum down
245
00:15:13 --> 00:15:16
the diagonal of A.
246
00:15:16 --> 00:15:19
Shall I write a11+a22+...+ ann.
247
00:15:19 --> 00:15:23
That's add up the diagonal
entries.
248
00:15:23 --> 00:15:26
In this example,
it's zero.
249
00:15:26 --> 00:15:31
In other words,
once I found this eigenvalue of
250
00:15:31 --> 00:15:37
one, I knew the other one had to
be minus
251
00:15:37 --> 00:15:43
one in this two by two case,
because in the two by two case,
252
00:15:43 --> 00:15:46
which is a good one to,
to, play with,
253
00:15:46 --> 00:15:52
the trace tells you right away
what the other eigenvalue is.
254
00:15:52 --> 00:15:57
So if I tell you one
eigenvalue, you can tell me the
255
00:15:57 --> 00:15:57
other one.
256
00:15:57 --> 00:16:02
We'll,
we'll have that -- we'll,
257
00:16:02 --> 00:16:04
we'll see that again.
258
00:16:04 --> 00:16:04
OK.
259
00:16:04 --> 00:16:11
Now can I -- I could give more
examples, but maybe it's time to
260
00:16:11 --> 00:16:16
face the, the equation,
Ax equal lambda x,
261
00:16:16 --> 00:16:21
and figure how are we going to
find x and lambda.
262
00:16:21 --> 00:16:23
OK.
263
00:16:23 --> 00:16:31
So this, so the question now is
how to find eigenvalues and
264
00:16:31 --> 00:16:33
eigenvectors.
265
00:16:33 --> 00:16:39
How to solve,
how to solve Ax equal lambda x
266
00:16:39 --> 00:16:45.79
when we've got two unknowns both
in the equation.
267
00:16:45.79 --> 00:16:46
OK.
268
00:16:46 --> 00:16:48.6
Here's the trick.
269
00:16:48.6 --> 00:16:50
Simple idea.
270
00:16:50 --> 00:16:55
Bring this onto the
same side.
271
00:16:55 --> 00:16:56
Rewrite.
272
00:16:56 --> 00:17:02
Bring this over as A minus
lambda times the identity x
273
00:17:02 --> 00:17:03
equals zero.
274
00:17:03 --> 00:17:03
Right?
275
00:17:03 --> 00:17:09
I have Ax minus lambda x,
so I brought that over and I've
276
00:17:09 --> 00:17:14
got zero left on the,
on the right-hand side.
277
00:17:14 --> 00:17:14
OK.
278
00:17:14 --> 00:17:22
I don't know lambda and I don't
know x, but I do know something
279
00:17:22 --> 00:17:22
here.
280
00:17:22 --> 00:17:26
What I know is if I,
if I'm going to be able to
281
00:17:26 --> 00:17:30
solve this thing,
for some x that's not the zero
282
00:17:30 --> 00:17:35
vector, that's not,
that's a useless eigenvector,
283
00:17:35 --> 00:17:36
doesn't count.
284
00:17:36 --> 00:17:41
What I know now is that
this matrix must be what?
285
00:17:41 --> 00:17:47
If I'm going to be -- if there
is an x -- I don't -- right now
286
00:17:47 --> 00:17:49
I don't know what it is.
287
00:17:49 --> 00:17:53
I'm going to find lambda first,
actually.
288
00:17:53 --> 00:17:59
And -- but if there is an x,
it tells me that this matrix,
289
00:17:59 --> 00:18:05
this special combination,
which is like the matrix A with
290
00:18:05 --> 00:18:10
lambda -- shifted by lambda,
shifted by lambda I,
291
00:18:10 --> 00:18:13
that it has to be singular.
292
00:18:13 --> 00:18:19
This matrix must be singular,
otherwise the only x would be
293
00:18:19 --> 00:18:23
the zero x, and zero matrix.OK.
294
00:18:23 --> 00:18:25
So this is singular.
295
00:18:25 --> 00:18:30
And what do I now know about
singular matrices?
296
00:18:30 --> 00:18:32
Their determinant is zero.
297
00:18:32 --> 00:18:38
So I've -- so from the fact
that that has to be singular,
298
00:18:38 --> 00:18:46
I know that the determinant of
A minus lambda I has to be zero.
299
00:18:46 --> 00:18:49
And that, now I've got x out of
it.
300
00:18:49 --> 00:18:54
I've got an equation for
lambda, that the key equation --
301
00:18:54 --> 00:19:00.35
it's called the characteristic
equation or the eigenvalue
302
00:19:00.35 --> 00:19:01
equation.
303
00:19:01 --> 00:19:06
And that -- in other words,
I'm now in a position to find
304
00:19:06 --> 00:19:09
lambda first.
305
00:19:09 --> 00:19:14
So -- the idea will be to find
lambda first.
306
00:19:14 --> 00:19:18.67
And actually,
I won't find one lambda,
307
00:19:18.67 --> 00:19:22
I'll find N different lambdas.
308
00:19:22 --> 00:19:27
Well, n lambdas,
maybe not n different
309
00:19:27 --> 00:19:28
ones.
310
00:19:28 --> 00:19:30
A lambda could be repeated.
311
00:19:30 --> 00:19:36
A repeated lambda is the source
of all trouble in 18.06.
312
00:19:36 --> 00:19:41
So, let's hope for the moment
that they're not repeated.
313
00:19:41 --> 00:19:45
There, there they were
different, right?
314
00:19:45 --> 00:19:51
One and minus one in that,
in that, for that permutation.
315
00:19:51 --> 00:19:51
OK.
316
00:19:51 --> 00:19:56
So and after I found this
lambda, can I just look ahead?
317
00:19:56 --> 00:19:59
How I going to find x?
318
00:19:59 --> 00:20:05
After I have found this lambda,
the lambda being this -- one of
319
00:20:05 --> 00:20:10
the numbers that makes this
matrix singular.
320
00:20:10 --> 00:20:13
Then of course finding x is
just by elimination.
321
00:20:13 --> 00:20:13
Right?
322
00:20:13 --> 00:20:17.03
It's just -- now I've got a
singular matrix,
323
00:20:17.03 --> 00:20:19
I'm looking for the null space.
324
00:20:19 --> 00:20:22
We're experts at finding the
null space.
325
00:20:22 --> 00:20:25
You know, you do elimination,
you identify the,
326
00:20:25 --> 00:20:28
the, the pivot columns and so
on,
327
00:20:28 --> 00:20:32
you're -- and,
give values to the free
328
00:20:32 --> 00:20:33
variables.
329
00:20:33 --> 00:20:37.83
Probably there'll only be one
free variable.
330
00:20:37.83 --> 00:20:41
We'll give it the value one,
like there.
331
00:20:41 --> 00:20:44
And we find the other variable.
332
00:20:44 --> 00:20:45
OK.
333
00:20:45 --> 00:20:51.21
So let's -- find the x second
will be a doable job.
334
00:20:51.21 --> 00:20:56.65
Let's go, let's look at the
first job of finding lambda.
335
00:20:56.65 --> 00:20:56
OK.
336
00:20:56 --> 00:20:59
Can I take another example?
337
00:20:59 --> 00:21:02
And let's, let's work that one
out.
338
00:21:02 --> 00:21:03
OK.
339
00:21:03 --> 00:21:08
So let me take the example,
say, let me make it easy.
340
00:21:08 --> 00:21:11
Three
three one and one.
341
00:21:11 --> 00:21:14
So I've made it easy.
342
00:21:14 --> 00:21:16
I've made it two by two.
343
00:21:16 --> 00:21:18
I've made it symmetric.
344
00:21:18 --> 00:21:23
And I even made it constant
down the diagonal.
345
00:21:23 --> 00:21:28
So that -- so the more,
like, special properties I
346
00:21:28 --> 00:21:34
stick into the matrix,
the more special outcome I
347
00:21:34 --> 00:21:36
get for the eigenvalues.
348
00:21:36 --> 00:21:40
For example,
this symmetric matrix,
349
00:21:40 --> 00:21:44
I know that it'll come out with
real eigenvalues.
350
00:21:44 --> 00:21:49
The eigenvalues will turn out
to be nice real numbers.
351
00:21:49 --> 00:21:56
And up in our previous example,
that was a symmetric matrix.
352
00:21:56 --> 00:22:00
Actually, while we're at it,
that was a symmetric matrix.
353
00:22:00 --> 00:22:05
Its eigenvalues were nice real
numbers, one and minus one.
354
00:22:05 --> 00:22:08
And do you notice anything
about its eigenvectors?
355
00:22:08 --> 00:22:13
Anything particular about those
two vectors, one one and minus
356
00:22:13 --> 00:22:15
one one?
357
00:22:15 --> 00:22:19
They just happen to be -- no,
I can't say they just happen to
358
00:22:19 --> 00:22:23
be, because that's the whole
point, is that they had to be --
359
00:22:23 --> 00:22:23.46
what?
360
00:22:23.46 --> 00:22:24
What are they?
361
00:22:24 --> 00:22:25
They're perpendicular.
362
00:22:25 --> 00:22:30.5
The vector, when I -- if I see
a vector one one and a one --
363
00:22:30.5 --> 00:22:34
and a minus one one,
my mind immediately takes that
364
00:22:34 --> 00:22:34
dot product.
365
00:22:34 --> 00:22:35
It's zero.
366
00:22:35 --> 00:22:37
Those vectors are
perpendicular.
367
00:22:37 --> 00:22:39
That'll happen here too.
368
00:22:39 --> 00:22:41
Well, let's find the
eigenvalues.
369
00:22:41 --> 00:22:45
Actually, oh,
my example's too easy.
370
00:22:45 --> 00:22:47
My example is too easy.
371
00:22:47 --> 00:22:51
Let me tell you in advance
what's going to happen.
372
00:22:51 --> 00:22:52.13
May I?
373
00:22:52.13 --> 00:22:56
Or shall I do the determinant
of A minus lambda,
374
00:22:56 --> 00:22:59
and then point out at the end?
375
00:22:59 --> 00:23:04
Will you remind me at the --
after I've found the
376
00:23:04 --> 00:23:11.05
eigenvalues to say why they
were -- why they were easy from
377
00:23:11.05 --> 00:23:14
the, from the example we did?
378
00:23:14 --> 00:23:17
OK, let's do the job here.
379
00:23:17 --> 00:23:22
Let's compute determinant of A
minus lambda I.
380
00:23:22 --> 00:23:25
So that's a determinant.
381
00:23:25 --> 00:23:29
And what's,
what is this thing?
382
00:23:29 --> 00:23:34
It's the matrix A with lambda
removed from the diagonal.
383
00:23:34 --> 00:23:40
So the diagonal matrix is
shifted, and then I'm taking the
384
00:23:40 --> 00:23:41
determinant.
385
00:23:41 --> 00:23:42
OK.
386
00:23:42 --> 00:23:44.45
So I multiply this out.
387
00:23:44.45 --> 00:23:47
So what is that determinant?
388
00:23:47 --> 00:23:52
Do you notice,
I didn't take lambda away from
389
00:23:52 --> 00:23:54
all the entries.
390
00:23:54 --> 00:23:57
It's lambda I,
so it's lambda along the
391
00:23:57 --> 00:23:58
diagonal.
392
00:23:58 --> 00:24:03
So I get three minus lambda
squared and then minus one,
393
00:24:03 --> 00:24:04
right?
394
00:24:04 --> 00:24:08
And I want that to be zero.
395
00:24:08 --> 00:24:13
Well, I'm going to simplify it.
396
00:24:13 --> 00:24:16
And what will I get?
397
00:24:16 --> 00:24:25.3
So if I multiply this out,
I get lambda squared minus six
398
00:24:25.3 --> 00:24:28
lambda plus what?
399
00:24:28 --> 00:24:29
Plus eight.
400
00:24:29 --> 00:24:35
And that I'm going to set to
zero.
401
00:24:35 --> 00:24:39
And I'm
going to solve it.
402
00:24:39 --> 00:24:42
So and it's,
it's a quadratic equation.
403
00:24:42 --> 00:24:45
I can use factorization,
I can use the quadratic
404
00:24:45 --> 00:24:46
formula.
405
00:24:46 --> 00:24:48
I'll get two lambdas.
406
00:24:48 --> 00:24:51
Before I do it,
tell me what's that number six
407
00:24:51 --> 00:24:55
that's showing up in this
equation?
408
00:24:55 --> 00:24:56
It's the trace.
409
00:24:56 --> 00:24:59
That number six is three plus
three.
410
00:24:59 --> 00:25:03
And while we're at it,
what's the number eight that's
411
00:25:03 --> 00:25:06
showing up in this equation?
412
00:25:06 --> 00:25:08
It's the determinant.
413
00:25:08 --> 00:25:11.41
That our matrix has determinant
eight.
414
00:25:11.41 --> 00:25:15.95
So in a two by two case,
it's really nice.
415
00:25:15.95 --> 00:25:20
It's lambda squared minus the
trace times lambda -- the trace
416
00:25:20 --> 00:25:25
is the linear coefficient -- and
plus the determinant,
417
00:25:25 --> 00:25:26
the constant term.
418
00:25:26 --> 00:25:26
OK.
419
00:25:26 --> 00:25:29
So let's -- can,
can we find the roots?
420
00:25:29 --> 00:25:34
I guess the easy way is to
factor that as something times
421
00:25:34 --> 00:25:36
something.
422
00:25:36 --> 00:25:41
If we couldn't factor it,
then we'd have to use the old
423
00:25:41 --> 00:25:45
b^2-4ac formula,
but I, I think we can factor
424
00:25:45 --> 00:25:50
that into lambda minus what
times lambda minus what?
425
00:25:50 --> 00:25:53
Can you do that factorization?
426
00:25:53 --> 00:25:54
Four and two?
427
00:25:54 --> 00:25:59
Lambda minus four times lambda
minus two.
428
00:25:59 --> 00:26:02.61
So the, the eigenvalues are
four and two.
429
00:26:02.61 --> 00:26:07
So the eigenvalues are -- one
eigenvalue, lambda one,
430
00:26:07 --> 00:26:09
let's say, is four.
431
00:26:09 --> 00:26:12
Lambda two, the other
eigenvalue, is two.
432
00:26:12 --> 00:26:15
The eigenvalues are four and
two.
433
00:26:15 --> 00:26:19
And then I can go for the
eigenvectors.
434
00:26:19 --> 00:26:23
You see I got the eigenvalues
first.
435
00:26:23 --> 00:26:24
Four and two.
436
00:26:24 --> 00:26:26
Now for the eigenvectors.
437
00:26:26 --> 00:26:29
So what are the eigenvectors?
438
00:26:29 --> 00:26:34
They're these guys in the null
space when I take away,
439
00:26:34 --> 00:26:38
when I make the matrix singular
by
440
00:26:38 --> 00:26:41
taking four I or two I away.
441
00:26:41 --> 00:26:45
So we're -- we got to do those
separately.
442
00:26:45 --> 00:26:51
I'll -- let me find the
eigenvector for four first.
443
00:26:51 --> 00:26:56
So I'll subtract four,
so A minus four I is -- so
444
00:26:56 --> 00:27:00
taking four away will put minus
ones there.
445
00:27:00 --> 00:27:05
And what's the point
about that matrix?
446
00:27:05 --> 00:27:09.84
If four is an eigenvalue,
then A minus four I had better
447
00:27:09.84 --> 00:27:11
be a what kind of matrix?
448
00:27:11 --> 00:27:12
Singular.
449
00:27:12 --> 00:27:16
If that matrix isn't singular,
the four wasn't correct.
450
00:27:16 --> 00:27:19
But we're OK,
that matrix is singular.
451
00:27:19 --> 00:27:21
And what's the x now?
452
00:27:21 --> 00:27:24
The x is in the null
space.
453
00:27:24 --> 00:27:30.08
So what's the x1 that goes
with, with the lambda one?
454
00:27:30.08 --> 00:27:36
So that A -- so this is -- now
I'm doing A x1 is lambda one x1.
455
00:27:36 --> 00:27:42
So I took A minus lambda one I,
that's this matrix,
456
00:27:42 --> 00:27:47
and now I'm looking for the x1
in its null space,
457
00:27:47 --> 00:27:48
and who is he?
458
00:27:48 --> 00:27:51
What's the vector x in the null
space?
459
00:27:51 --> 00:27:53
Of course it's one one.
460
00:27:53 --> 00:27:57
So that's the eigenvector that
goes with that eigenvalue.
461
00:27:57 --> 00:28:01
Now how about the eigenvector
that goes with the other
462
00:28:01 --> 00:28:02.7
eigenvalue?
463
00:28:02.7 --> 00:28:05
Can I do that with,
with erasing?
464
00:28:05 --> 00:28:07
I take A minus two I.
465
00:28:07 --> 00:28:11.94
So now I take two away from the
diagonal, and that leaves me
466
00:28:11.94 --> 00:28:13
with a one and a one.
467
00:28:13 --> 00:28:18
So A minus two I has again
produced a singular matrix,
468
00:28:18 --> 00:28:19
as it had to.
469
00:28:19 --> 00:28:23
I'm looking for the null space
of that guy.
470
00:28:23 --> 00:28:25
What vector is in its null
space?
471
00:28:25 --> 00:28:28
Well, of course,
a whole line of vectors.
472
00:28:28 --> 00:28:32
So when I say the eigenvector,
I'm not speaking correctly.
473
00:28:32 --> 00:28:36
There's a whole line of
eigenvectors,
474
00:28:36 --> 00:28:39
and you just -- I just want a
basis.
475
00:28:39 --> 00:28:41
And for a line I just want one
vector.
476
00:28:41 --> 00:28:45
But -- You could,
you're -- there's some freedom
477
00:28:45 --> 00:28:49.14
in choosing that one,
but choose a reasonable one.
478
00:28:49.14 --> 00:28:53
What's a vector in the null
space of that?
479
00:28:53 --> 00:28:58
Well, the natural vector to
pick as the eigenvector with,
480
00:28:58 --> 00:29:00
with lambda two is minus one
one.
481
00:29:00 --> 00:29:05
If I did elimination on that
vector and set that,
482
00:29:05 --> 00:29:10.53
the free variable to be one,
I would get minus one and get
483
00:29:10.53 --> 00:29:12
that eigenvector.
484
00:29:12 --> 00:27:24
So you see then that
I've got eigenvector,
485
00:27:24 --> 00:23:32
eigenvalue, eigenvector,
eigenvalue for this,
486
00:23:32 --> 00:22:08
for this matrix?
487
00:22:08 --> 00:17:13
And now comes that thing that I
wanted to be reminded of.
488
00:17:13 --> 00:12:02.24
What's the relation between
that problem and -- let me write
489
00:12:02.24 --> 00:10:44
just above what we2
found here.
490
00:10:44 --> 00:15:59
A equals zero one one zero,
that had eigenvalue one and
491
00:15:59 --> 00:21:43
minus one and eigenvectors one
one and eigenvector minus one
492
00:21:43 --> 00:22:06
one.
493
00:22:06 --> 00:24:20
And what do you notice?
494
00:24:20 --> 00:30:11
What's -- how is this matrix
related to that matrix?
495
00:30:11 --> 00:30:14
How are those two matrices
related?
496
00:30:14 --> 00:30:18
Well, one is just three I more
than the other one,
497
00:30:18 --> 00:30:19
right?
498
00:30:19 --> 00:30:24
I just took that matrix and I
-- I took this matrix and I
499
00:30:24 --> 00:30:25
added three I.
500
00:30:25 --> 00:30:28
So my question is,
what happened to the
501
00:30:28 --> 00:30:32
eigenvalues and
what happened to the
502
00:30:32 --> 00:30:33
eigenvectors?
503
00:30:33 --> 00:30:36
That's the, that's like the
question we keep asking now in
504
00:30:36 --> 00:30:37
this chapter.
505
00:30:37 --> 00:30:41
If I, if I do something to the
matrix, what happens if I -- or
506
00:30:41 --> 00:30:45
I know something about the
matrix, what's the what's the
507
00:30:45 --> 00:30:48
conclusion for its eigenvectors
and
508
00:30:48 --> 00:30:49
eigenvalues?
509
00:30:49 --> 00:30:53
Because -- those eigenvalues
and eigenvectors are going to
510
00:30:53 --> 00:30:57.28
tell us important information
about the matrix.
511
00:30:57.28 --> 00:30:59
And here what are we seeing?
512
00:30:59 --> 00:31:03
What's happening to these
eigenvalues, one and minus one,
513
00:31:03 --> 00:31:06
when I add three
I?
514
00:31:06 --> 00:31:09
It just added three to the
eigenvalues.
515
00:31:09 --> 00:31:13
I got four and two,
three more than one and minus
516
00:31:13 --> 00:31:13
one.
517
00:31:13 --> 00:31:16
What happened to the
eigenvectors?
518
00:31:16 --> 00:31:17
Nothing at all.
519
00:31:17 --> 00:31:22.75
One one is -- and minus -- and
one -- and minus
520
00:31:22.75 --> 00:31:26
one one are -- is still the
eigenvectors.
521
00:31:26 --> 00:31:30
In other words,
simple but useful observation.
522
00:31:30 --> 00:31:36
If I add three I to a matrix,
its eigenvectors don't change
523
00:31:36 --> 00:31:39.92
and its eigenvalues are three
bigger.
524
00:31:39.92 --> 00:31:42
Let's, let's just see why.
525
00:31:42 --> 00:26:15
Let me keep all
this on the same board.
526
00:26:15 --> 00:14:48
Suppose I have a matrix A,
and Ax equal lambda x.
527
00:14:48 --> 00:07:10
Now I add three I to that
matrix.
528
00:07:10 --> 00:05:46
Do you see what3
so it's if Ax equals lambda x,
529
00:05:46 --> 00:11:54
then this, this other new
matrix, I just have an Ax,
530
00:11:54 --> 00:16:36
which is lambda x,
and I have a three x,
531
00:16:36 --> 00:21:03
from the three x,
so it's just I mean,
532
00:21:03 --> 00:23:56
it's just sitting there.
533
00:23:56 --> 00:26:20
Lambda plus three x.
534
00:26:20 --> 00:32:36
So if they, if this had
eigenvalue lambda,
535
00:32:36 --> 00:32:40
this has eigenvalue lambda plus
three.
536
00:32:40 --> 00:32:45
And x, the eigenvector,
is the same x for both
537
00:32:45 --> 00:32:46.84
matrices.
538
00:32:46.84 --> 00:32:47
OK.
539
00:32:47 --> 00:32:49
So that's, great.
540
00:32:49 --> 00:32:52
Of course, it's special.
541
00:32:52 --> 00:32:57.75
We got the new matrix by adding
three I.
542
00:32:57.75 --> 00:33:01
Suppose I had added another
matrix.
543
00:33:01 --> 00:33:07
Suppose I know the eigenvalues
and eigenvectors of A.
544
00:33:07 --> 00:33:13
So this is, this,
this little board here is going
545
00:33:13 --> 00:33:15
to be not so great.
546
00:33:15 --> 00:33:23
Suppose I have a matrix A and
it has an eigenvector x with
547
00:33:23 --> 00:33:25
an eigenvalue lambda.
548
00:33:25 --> 00:33:28
And now I add on some other
matrix.
549
00:33:28 --> 00:33:33
So, so what I'm asking you is,
if you know the eigenvalues of
550
00:33:33 --> 00:33:39.59
A and you know the eigenvalues
of B, let me say suppose B -- so
551
00:33:39.59 --> 00:33:42
this is if -- let me put an if
here.
552
00:33:42 --> 00:33:46
If Ax equals lambda x,
fine, and B has,
553
00:33:46 --> 00:33:52
eigenvalues,
has eigenvalues -- what shall
554
00:33:52 --> 00:33:54
we call them?
555
00:33:54 --> 00:34:02
Alpha, alpha one and alpha --
let's say -- I'll use alpha for
556
00:34:02 --> 00:34:08
the eigenvalues of B for no good
reason.
557
00:34:08 --> 00:34:18
What a- you see what I'm going
to ask is, how about A plus B?
558
00:34:18 --> 00:34:22
Let me, let me give you the,
let me give you,
559
00:34:22 --> 00:34:25
what you might think first.
560
00:34:25 --> 00:34:26
OK.
561
00:34:26 --> 00:34:32.31
If Ax equals lambda x and if B
has an eigenvalue alpha,
562
00:34:32.31 --> 00:34:39.13
then I allowed to say -- what's
the matter with this argument?
563
00:34:39.13 --> 00:34:41
It's wrong.
564
00:34:41 --> 00:34:44.42
What I'm going to write up is
wrong.
565
00:34:44.42 --> 00:34:47
I'm going to say Bx is alpha x.
566
00:34:47 --> 00:34:50
Add those up,
and you get A plus B x equals
567
00:34:50 --> 00:34:52
lambda plus alpha x.
568
00:34:52 --> 00:34:57
So you would think that if you
know the eigenvalues of A and
569
00:34:57 --> 00:35:02.66
you knew the eigenvalues of B,
then if you added you would
570
00:35:02.66 --> 00:35:06.5
know the eigenvalues of A plus
B.
571
00:35:06.5 --> 00:35:08
But that's false.
572
00:35:08 --> 00:35:13
A plus B -- well,
when B was three I,
573
00:35:13 --> 00:35:15
that worked great.
574
00:35:15 --> 00:35:19
But this is not so great.
575
00:35:19 --> 00:35:25
And what's the matter with that
argument there?
576
00:35:25 --> 00:35:33.05
We have no reason to believe
that x is also an eigenvector of
577
00:35:33.05 --> 00:35:33
B.
578
00:35:33 --> 00:35:36.94
B has some
eigenvalues,
579
00:35:36.94 --> 00:35:40.48
but it's got some different
eigenvectors normally.
580
00:35:40.48 --> 00:35:42
It's a different matrix.
581
00:35:42 --> 00:35:44
I don't know anything special.
582
00:35:44 --> 00:35:48
If I don't know anything
special, then as far as I know,
583
00:35:48 --> 00:35:51
it's got some different
eigenvector y,
584
00:35:51 --> 00:35:54
and when I add I
get just rubbish.
585
00:35:54 --> 00:36:00
I mean, I get -- I can add,
but I don't learn anything.
586
00:36:00 --> 00:36:03
So not so great is A plus B.
587
00:36:03 --> 00:36:05
Or A times B.
588
00:36:05 --> 00:36:11
Normally the eigenvalues of A
plus B or A times B are not
589
00:36:11 --> 00:36:16.32
eigenvalues of A plus
eigenvalues of B.
590
00:36:16.32 --> 00:36:19
Ei- eigenvalues are not,
like, linear.
591
00:36:19 --> 00:36:22
Or -- and they don't multiply.
592
00:36:22 --> 00:36:27
Because, eigenvectors are
usually different and,
593
00:36:27 --> 00:36:32
and there's just no way to find
out what A plus B does to affect
594
00:36:32 --> 00:36:33
it.
595
00:36:33 --> 00:36:33
OK.
596
00:36:33 --> 00:36:37
So that's, like,
a caution.
597
00:36:37 --> 00:36:40
Don't, if B is a multiple of
the identity,
598
00:36:40 --> 00:36:47
great, but if B is some general
matrix, then for A plus B you've
599
00:36:47 --> 00:36:52.51
got to find -- you've got to
solve the eigenvalue problem.
600
00:36:52.51 --> 00:36:52
OK.
601
00:36:52 --> 00:36:58
Now I want to do another
example that brings out a,
602
00:36:58 --> 00:37:02
another point about
eigenvalues.
603
00:37:02 --> 00:37:08.44
Let me make this example a
rotation matrix.
604
00:37:08.44 --> 00:37:08.85
OK.
605
00:37:08.85 --> 00:37:12
So here's another example.
606
00:37:12 --> 00:37:17
So a rotate -- oh,
I'd better call it Q.
607
00:37:17 --> 00:37:23.99
I often use Q for,
for, rotations because those
608
00:37:23.99 --> 00:37:29
are the, like,
very important examples of
609
00:37:29 --> 00:37:31
orthogonal matrices.
610
00:37:31 --> 00:37:35
Let me make it a ninety degree
rotation.
611
00:37:35 --> 00:37:41
So -- my matrix is going to be
the one that rotates every
612
00:37:41 --> 00:37:43
vector by ninety degrees.
613
00:37:43 --> 00:37:47
So do you remember
that matrix?
614
00:37:47 --> 00:37:52
It's the cosine of ninety
degrees, which is zero,
615
00:37:52 --> 00:37:57
the sine of ninety degrees,
which is one,
616
00:37:57 --> 00:38:02
minus the sine of ninety,
the cosine of ninety.
617
00:38:02 --> 00:38:06
So that matrix deserves the
letter Q.
618
00:38:06 --> 00:38:12
It's an orthogonal matrix,
very, very orthogonal matrix.
619
00:38:12 --> 00:38:17
Now I'm interested in its
eigenvalues and eigenvectors.
620
00:38:17 --> 00:38:21
Two by two, it can't be that
tough.
621
00:38:21 --> 00:38:24
We know that the eigenvalues
add to zero.
622
00:38:24 --> 00:38:28
Actually, we know something
already here.
623
00:38:28 --> 00:38:33
The eigen- what's the sum of
the two eigenvalues?
624
00:38:33 --> 00:38:36
Just tell
me what I just said.
625
00:38:36 --> 00:38:37
Zero, right.
626
00:38:37 --> 00:38:39
From that trace business.
627
00:38:39 --> 00:38:43
The sum of the eigenvalues is,
is going to come out zero.
628
00:38:43 --> 00:38:47
And the product of the
eigenvalues, did I tell you
629
00:38:47 --> 00:38:51.39
about the determinant
being the product of the
630
00:38:51.39 --> 00:38:52
eigenvalues?
631
00:38:52 --> 00:38:52
No.
632
00:38:52 --> 00:38:55
But that's a good thing to
know.
633
00:38:55 --> 00:38:59
We pointed out how that eight
appeared in the,
634
00:38:59 --> 00:39:02
in the quadratic equation.
635
00:39:02 --> 00:39:04
So let me just say this.
636
00:39:04 --> 00:37:36
The trace is zero plus zero,
obviously.
637
00:37:36 --> 00:30:18
And that's the sum,
that's lambda one plus lambda
638
00:30:18 --> 00:29:42
two.
639
00:29:42 --> 00:22:34
Now the other neat fact is that
the determinant,
640
00:22:34 --> 00:16:56.71
what's the determinant of that
matrix?
641
00:16:56.71 --> 00:16:20
One.
642
00:16:20 --> 00:09:30
And that is lambda one times
lambda3
643
00:09:30 --> 00:10:00
two.
644
00:10:00 --> 00:14:40
In our example,
the one we worked out,
645
00:14:40 --> 00:20:06.03
we -- the eigenvalues came out
four and two.
646
00:20:06.03 --> 00:23:07
Their product was eight.
647
00:23:07 --> 00:30:26
That -- it had to be eight,
because we factored into lambda
648
00:30:26 --> 00:34:36
minus four times lambda minus
two.
649
00:34:36 --> 00:39:48
That gave us the constant term
eight.
650
00:39:48 --> 00:39:51
And that was the determinant.
651
00:39:51 --> 00:39:52
OK.
652
00:39:52 --> 00:40:00
What I'm leading up to with
this example is that something's
653
00:40:00 --> 00:40:02
going to go wrong.
654
00:40:02 --> 00:40:09
Something goes wrong for
rotation because what vector can
655
00:40:09 --> 00:40:17
come out parallel to itself
after a rotation?
656
00:40:17 --> 00:40:22.23
If this matrix rotates every
vector by ninety degrees,
657
00:40:22.23 --> 00:40:25
what could be an eigenvector?
658
00:40:25 --> 00:40:29
Do you see we're,
we're, we're going to have
659
00:40:29 --> 00:40:32
trouble.
eigenvectors are -- Well.
660
00:40:32 --> 00:40:35.75
Our, our picture of
eigenvectors,
661
00:40:35.75 --> 00:40:41
of, of coming out in the same
direction that they went in,
662
00:40:41 --> 00:40:43.41
there won't be it.
663
00:40:43.41 --> 00:40:48
And with, and with eigenvalues
we're going to have trouble.
664
00:40:48 --> 00:40:49
From these equations.
665
00:40:49 --> 00:40:50
Let's see.
666
00:40:50 --> 00:40:52
Why I expecting trouble?
667
00:40:52 --> 00:40:56.5
The, the first equation says
that the
668
00:40:56.5 --> 00:40:58
eigenvalues add to zero.
669
00:40:58 --> 00:41:01.91
So there's a plus and a minus.
670
00:41:01.91 --> 00:41:07
But then the second equation
says that the product is plus
671
00:41:07 --> 00:41:08
one.
672
00:41:08 --> 00:41:09
We're in trouble.
673
00:41:09 --> 00:41:11
But there's a way out.
674
00:41:11 --> 00:41:15
So how -- let's do the usual
stuff.
675
00:41:15 --> 00:41:20
Look at determinant of Q minus
lambda I.
676
00:41:20 --> 00:41:25
So I'll just follow the rules,
take the determinant,
677
00:41:25 --> 00:41:29
subtract lambda from the
diagonal, where I had zeros,
678
00:41:29 --> 00:41:31
the rest is the same.
679
00:41:31 --> 00:41:34
Rest of Q is just copied.
680
00:41:34 --> 00:41:36
Compute that determinant.
681
00:41:36 --> 00:41:40
OK, so what does that
determinant equal?
682
00:41:40 --> 00:41:45
Lambda squared minus minus one
plus what?
683
00:41:45 --> 00:41:46
What's up?
684
00:41:46 --> 00:41:49
There's my equation.
685
00:41:49 --> 00:41:56
My equation for the eigenvalues
is lambda squared plus one
686
00:41:56 --> 00:41:57
equals zero.
687
00:41:57 --> 00:42:03
What are the eigenvalues lambda
one
688
00:42:03 --> 00:42:07
and lambda two?
689
00:42:07 --> 00:42:19
They're I, whatever that is,
and minus it,
690
00:42:19 --> 00:42:21
right.
691
00:42:21 --> 00:42:30.75
Those are the right numbers.
692
00:42:30.75 --> 00:42:36
To be real numbers even though
the matrix was perfectly real.
693
00:42:36 --> 00:42:38
So this can happen.
694
00:42:38 --> 00:42:45
Complex numbers are going to --
have to enter eighteen oh six at
695
00:42:45 --> 00:42:46
this moment.
696
00:42:46 --> 00:42:47
Boo, right.
697
00:42:47 --> 00:42:48
All right.
698
00:42:48 --> 00:39:01.15
If I just choose good
matrices that have real
699
00:39:01.15 --> 00:31:59
eigenvalues, we can postpone
that evil day,
700
00:31:59 --> 00:24:28
but just so you see -- so I'll
try to do that.
701
00:24:28 --> 00:21:18
But it's out there.
702
00:21:18 --> 00:14:06
That a matrix,
a perfectly real matrix could
703
00:14:06 --> 00:06:05
have, give a perfectly
innocent-looking quadratic
704
00:06:05 --> 00:08:14.67
thing,4
but the roots of that quadratic
705
00:08:14.67 --> 00:11:57
can be complex numbers.
706
00:11:57 --> 00:21:08
And of course you -- everybody
knows that they're -- what,
707
00:21:08 --> 00:27:54.02
what do you know about the
complex numbers?
708
00:27:54.02 --> 00:37:05
So, so now -- Let's just spend
one more minute on this bad
709
00:37:05 --> 00:43:31
possibility of complex numbers.
710
00:43:31 --> 00:43:37
We do know a little information
about the, the two complex
711
00:43:37 --> 00:43:38.08
numbers.
712
00:43:38.08 --> 00:43:42
They're complex conjugates of
each other.
713
00:43:42 --> 00:43:46
If, if lambda is an eigenvalue,
then when I change,
714
00:43:46 --> 00:43:53
when I go -- you remember what
complex conjugates are?
715
00:43:53 --> 00:43:56
You switch the sign of the
imaginary part.
716
00:43:56 --> 00:43:59
Well, this was only imaginary,
had no real part,
717
00:43:59 --> 00:44:02
so we just switched its sign.
718
00:44:02 --> 00:44:05
So that eigenvalues come in
pairs like that,
719
00:44:05 --> 00:44:06
but they're complex.
720
00:44:06 --> 00:44:08
A complex conjugate pair.
721
00:44:08 --> 00:44:13
And that can happen
with a perfectly real matrix.
722
00:44:13 --> 00:44:17
And as a matter of fact -- so
that was my, my point earlier,
723
00:44:17 --> 00:44:21
that if a matrix was symmetric,
it wouldn't happen.
724
00:44:21 --> 00:44:25.57
So if we stick to matrices that
are symmetric or,
725
00:44:25.57 --> 00:44:29
like, close to symmetric,
then the eigenvalues will stay
726
00:44:29 --> 00:44:30
real.
727
00:44:30 --> 00:44:35
But if we move far away
from symmetric -- and that's as
728
00:44:35 --> 00:44:39
far as you can move,
because that matrix is -- how
729
00:44:39 --> 00:44:42
is Q transpose related to Q for
that matrix?
730
00:44:42 --> 00:44:45
That matrix is anti-symmetric.
731
00:44:45 --> 00:44:46
Q transpose is minus Q.
732
00:44:46 --> 00:44:50.5
That's the very opposite of
symmetry.
733
00:44:50.5 --> 00:44:54
When I flip across the diagonal
I get -- I reverse all the
734
00:44:54 --> 00:44:55
signs.
735
00:44:55 --> 00:44:59
Those are the guys that have
pure imaginary eigenvalues.
736
00:44:59 --> 00:45:01
So they're the extreme case.
737
00:45:01 --> 00:45:05
And in between are,
are matrices that are not
738
00:45:05 --> 00:45:09
symmetric or anti-symmetric but,
but they have partly a
739
00:45:09 --> 00:45:13
symmetric
part and an anti-symmetric
740
00:45:13 --> 00:45:13
part.
741
00:45:13 --> 00:45:14.2
OK.
742
00:45:14.2 --> 00:45:19
So I'm doing a bunch of
examples here to show the
743
00:45:19 --> 00:45:20
possibilities.
744
00:45:20 --> 00:45:26
The good possibilities being
perpendicular eigenvectors,
745
00:45:26 --> 00:45:28
real eigenvalues.
746
00:45:28 --> 00:45:34
The bad possibilities being
complex eigenvalues.
747
00:45:34 --> 00:45:37
We could say that's bad.
748
00:45:37 --> 00:45:40
There's another even worse.
749
00:45:40 --> 00:45:46
I'm getting through the,
the bad things here today.
750
00:45:46 --> 00:45:53
Then, then the next lecture
can, can, can be like pure
751
00:45:53 --> 00:45:54
happiness.
752
00:45:54 --> 00:45:54
OK.
753
00:45:54 --> 00:46:00
Here's one more bad thing
that could happen.
754
00:46:00 --> 00:46:04
So I, again,
I'll do it with an example.
755
00:46:04 --> 00:46:09
Suppose my matrix is,
suppose I take this three three
756
00:46:09 --> 00:46:13
one and I change that guy to
zero.
757
00:46:13 --> 00:46:17.18
What are the eigenvalues of
that matrix?
758
00:46:17.18 --> 00:46:19
What are the eigenvectors?
759
00:46:19 --> 00:46:23
This is
always our question.
760
00:46:23 --> 00:46:27.54
Of course, the next section
we're going to show why are,
761
00:46:27.54 --> 00:46:28
why do we care.
762
00:46:28 --> 00:46:32
But for the moment,
this lecture is introducing
763
00:46:32 --> 00:46:32
them.
764
00:46:32 --> 00:46:34
And let's just find them.
765
00:46:34 --> 00:46:34
OK.
766
00:46:34 --> 00:46:38
What are the eigenvalues of
that matrix?
767
00:46:38 --> 00:46:43
Let me tell you -- at a glance
we could answer that question.
768
00:46:43 --> 00:46:46
Because the matrix is
triangular.
769
00:46:46 --> 00:46:51
It's really useful to know --
if you've got properties like a
770
00:46:51 --> 00:46:52
triangular matrix.
771
00:46:52 --> 00:46:57
It's very useful to know you
can read the eigenvalues
772
00:46:57 --> 00:46:57
off.
773
00:46:57 --> 00:47:00
They're right on the diagonal.
774
00:47:00 --> 00:47:03
So the eigenvalue is three and
also three.
775
00:47:03 --> 00:47:05.43
Three is a repeated eigenvalue.
776
00:47:05.43 --> 00:47:07
But let's see that happen.
777
00:47:07 --> 00:47:08
Let me do it right.
778
00:47:08 --> 00:47:12
The determinant of A minus
lambda I, what I always have to
779
00:47:12 --> 00:47:15
do is this
determinant.
780
00:47:15 --> 00:47:19
I take away lambda from the
diagonal.
781
00:47:19 --> 00:47:21
I leave the rest.
782
00:47:21 --> 00:47:27
I compute the determinant,
so I get a three minus lambda
783
00:47:27 --> 00:47:30
times a three minus lambda.
784
00:47:30 --> 00:47:31
And nothing.
785
00:47:31 --> 00:47:37
So that's where the triangular
part came in.
786
00:47:37 --> 00:47:40
Triangular part,
the one thing we know about
787
00:47:40 --> 00:47:44
triangular matrices is the
determinant is just the product
788
00:47:44 --> 00:47:46
down the diagonal.
789
00:47:46 --> 00:47:48
And in this case,
it's this same,
790
00:47:48 --> 00:47:51
repeated -- so lambda one is
one -- sorry,
791
00:47:51 --> 00:47:56
lambda one is three and lambda
two is three.
792
00:47:56 --> 00:47:57
That was easy.
793
00:47:57 --> 00:48:02
I mean, no -- why should I be
pessimistic about a matrix whose
794
00:48:02 --> 00:48:06.07
eigenvalues can be read off
right away?
795
00:48:06.07 --> 00:48:10
The problem with this matrix is
in the eigenvectors.
796
00:48:10 --> 00:48:13
So let's go to the
eigenvectors.
797
00:48:13 --> 00:48:17.3
So how do I find the
eigenvectors?
798
00:48:17.3 --> 00:41:01
I'm looking for a couple of
eigenvectors.
799
00:41:01 --> 00:36:28
So I take the eigenvalue.
800
00:36:28 --> 00:33:23
What do I do now?
801
00:33:23 --> 00:26:17
You remember,
I solve A minus lambda I x
802
00:26:17 --> 00:24:06
equals zero.
803
00:24:06 --> 00:18:28
And what is A minus lambda I x?
804
00:18:28 --> 00:14:18
So, so take three away.
805
00:14:18 --> 00:12:07
And I get this matrix4
zero zero zero one,
806
00:12:07 --> 00:13:12
right?
807
00:13:12 --> 00:20:50
Times x is supposed to give me
zero, right?
808
00:20:50 --> 00:26:06
That's my big equation for x.
809
00:26:06 --> 00:33:01.01
Now I'm looking for x,
the eigenvector.
810
00:33:01.01 --> 00:43:11
So I took A minus lambda I x,
and what kind of a matrix I
811
00:43:11 --> 00:49:00.48
supposed to have here?
812
00:49:00.48 --> 00:49:01
Singular, right?
813
00:49:01 --> 00:49:03
It's supposed to be singular.
814
00:49:03 --> 00:49:07
And then it's got some vectors
-- which it is.
815
00:49:07 --> 00:49:10
So it's got some vector x in
the null space.
816
00:49:10 --> 00:49:14
And what, what's the,
what's -- give me a basis for
817
00:49:14 --> 00:49:17
the null space for that guy.
818
00:49:17 --> 00:49:21
Tell me, what's a vector x in
the null space,
819
00:49:21 --> 00:49:26
so that'll be the,
the eigenvector that goes with
820
00:49:26 --> 00:49:28
lambda one equals three.
821
00:49:28 --> 00:49:33
The eigenvector is -- so what's
in the null space?
822
00:49:33 --> 00:49:35
One zero, is it?
823
00:49:35 --> 00:49:37
Great.
824
00:49:37 --> 00:49:39
Now, what's the other
eigenvector?
825
00:49:39 --> 00:49:44
What's, what's the eigenvector
that goes with lambda two?
826
00:49:44 --> 00:49:47
Well, lambda two is three
again.
827
00:49:47 --> 00:49:50
So I get the same thing again.
828
00:49:50 --> 00:49:54
Give me another vector -- I
want it to be independent.
829
00:49:54 --> 00:50:00.44
If I'm going to write
down an x2, I'm never going to
830
00:50:00.44 --> 00:50:03
let it be dependent on x1.
831
00:50:03 --> 00:50:07
I'm looking for independent
eigenvectors,
832
00:50:07 --> 00:50:10
and what's the conclusion?
833
00:50:10 --> 00:50:11
There isn't one.
834
00:50:11 --> 00:50:14.66
This is a degenerate matrix.
835
00:50:14.66 --> 00:50:21
It's only got one line of
eigenvectors instead of two.
836
00:50:21 --> 00:50:28
It's this possibility of a
repeated eigenvalue opens this
837
00:50:28 --> 00:50:34
further possibility of a
shortage of eigenvectors.
838
00:50:34 --> 00:50:40
And so there's no second
independent eigenvector x2.
839
00:50:40 --> 00:50:46.09
So it's a matrix,
it's a two by two matrix,
840
00:50:46.09 --> 00:50:49
but with
only one independent
841
00:50:49 --> 00:50:50
eigenvector.
842
00:50:50 --> 00:50:56
So that will be -- the matrices
that -- where eigenvectors are
843
00:50:56 --> 00:50:58
-- don't give the complete
story.
844
00:50:58 --> 00:50:59
OK.
845
00:50:59 --> 00:51:04
My lecture on Monday will give
the complete story for all the
846
00:51:04 --> 00:51:06
other matrices.
847
00:51:06 --> 00:51:07
Thanks.
848
00:51:07 --> 00:51:08.96
Have a good weekend.
849
00:51:08.96 --> 00:51:11
A real New England weekend.