WEBVTT
00:00:07.380 --> 00:00:08.230
OK.
00:00:08.230 --> 00:00:16.010
So this is the first lecture on
eigenvalues and eigenvectors,
00:00:16.010 --> 00:00:19.250
and that's a big subject
that will take up
00:00:19.250 --> 00:00:23.400
most of the rest of the course.
00:00:23.400 --> 00:00:31.500
It's, again, matrices are
square and we're looking now
00:00:31.500 --> 00:00:34.820
for some special
numbers, the eigenvalues,
00:00:34.820 --> 00:00:38.620
and some special vectors,
the eigenvectors.
00:00:38.620 --> 00:00:45.360
And so this lecture is mostly
about what are these numbers,
00:00:45.360 --> 00:00:48.850
and then the other lectures
are about how do we use them,
00:00:48.850 --> 00:00:50.440
why do we want them.
00:00:50.440 --> 00:00:54.200
OK, so what's an eigenvector?
00:00:54.200 --> 00:00:56.480
Maybe I'll start
with eigenvector.
00:00:56.480 --> 00:00:57.720
What's an eigenvector?
00:00:57.720 --> 00:01:02.900
So I have a matrix A.
00:01:02.900 --> 00:01:05.570
OK.
00:01:05.570 --> 00:01:07.430
What does a matrix do?
00:01:07.430 --> 00:01:09.290
It acts on vectors.
00:01:09.290 --> 00:01:12.250
It multiplies vectors x.
00:01:12.250 --> 00:01:21.110
So the way that matrix acts
is in goes a vector x and out
00:01:21.110 --> 00:01:22.690
comes a vector Ax.
00:01:22.690 --> 00:01:23.930
It's like a function.
00:01:23.930 --> 00:01:27.870
With a function in
calculus, in goes
00:01:27.870 --> 00:01:31.360
a number x, out comes f(x).
00:01:31.360 --> 00:01:34.960
Here in linear algebra
we're up in more dimensions.
00:01:34.960 --> 00:01:39.350
In goes a vector x,
out comes a vector Ax.
00:01:39.350 --> 00:01:43.060
And the vectors I'm
specially interested in
00:01:43.060 --> 00:01:47.630
are the ones the come
out in the same direction
00:01:47.630 --> 00:01:50.050
that they went in.
00:01:50.050 --> 00:01:52.050
That won't be typical.
00:01:52.050 --> 00:01:59.270
Most vectors, Ax is in -- points
in some different direction.
00:01:59.270 --> 00:02:04.456
But there are certain vectors
where Ax comes out parallel
00:02:04.456 --> 00:02:04.955
to x.
00:02:07.460 --> 00:02:09.660
And those are the eigenvectors.
00:02:09.660 --> 00:02:12.400
So Ax parallel to x.
00:02:18.130 --> 00:02:19.465
Those are the eigenvectors.
00:02:23.800 --> 00:02:25.990
And what do I mean by parallel?
00:02:25.990 --> 00:02:29.620
Oh, much easier to just
state it in an equation.
00:02:29.620 --> 00:02:36.340
Ax is some multiple -- and
everybody calls that multiple
00:02:36.340 --> 00:02:38.900
lambda -- of x.
00:02:38.900 --> 00:02:40.170
That's our big equation.
00:02:43.890 --> 00:02:46.780
We look for special vectors
-- and remember most vectors
00:02:46.780 --> 00:02:49.780
won't be eigenvectors --
00:02:49.780 --> 00:02:53.740
that -- for which Ax is in
the same direction as x,
00:02:53.740 --> 00:02:57.380
and by same direction I allow
it to be the very opposite
00:02:57.380 --> 00:03:03.830
direction, I allow lambda
to be negative or zero.
00:03:03.830 --> 00:03:07.950
Well, I guess we've met
the eigenvectors that
00:03:07.950 --> 00:03:10.600
have eigenvalue zero.
00:03:10.600 --> 00:03:13.600
Those are in the same
direction, but they're --
00:03:13.600 --> 00:03:18.020
in a kind of very special way.
00:03:18.020 --> 00:03:22.050
So this -- the eigenvector x.
00:03:22.050 --> 00:03:24.830
Lambda, whatever this
multiplying factor
00:03:24.830 --> 00:03:31.750
is, whether it's six or
minus six or zero or even
00:03:31.750 --> 00:03:34.800
some imaginary number,
that's the eigenvalue.
00:03:34.800 --> 00:03:38.350
So there's the eigenvalue,
there's the eigenvector.
00:03:38.350 --> 00:03:42.960
Let's just take a second
on eigenvalue zero.
00:03:42.960 --> 00:03:45.630
From the point of view of
eigenvalues, that's no special
00:03:45.630 --> 00:03:46.140
deal.
00:03:46.140 --> 00:03:48.910
That's, we have an eigenvector.
00:03:48.910 --> 00:03:51.240
If the eigenvalue
happened to be zero,
00:03:51.240 --> 00:03:56.380
that would mean that Ax was
zero x, in other words zero.
00:03:56.380 --> 00:04:00.280
So what would x, where would we
look for -- what are the x-s?
00:04:00.280 --> 00:04:04.460
What are the eigenvectors
with eigenvalue zero?
00:04:04.460 --> 00:04:08.200
They're the guys in the
null space, Ax equals zero.
00:04:08.200 --> 00:04:12.540
So if our matrix is singular,
let me write this down.
00:04:12.540 --> 00:04:23.210
If, if A is singular, then that
-- what does singular mean?
00:04:23.210 --> 00:04:26.015
It means that it takes
some vector x into zero.
00:04:29.190 --> 00:04:31.570
Some non-zero
vector, that's why --
00:04:31.570 --> 00:04:33.930
will be the
eigenvector into zero.
00:04:33.930 --> 00:04:38.945
Then lambda equals
zero is an eigenvalue.
00:04:42.530 --> 00:04:46.860
But we're interested
in all eigenvalues now,
00:04:46.860 --> 00:04:50.480
lambda equals zero is not,
like, so special anymore.
00:04:50.480 --> 00:04:51.140
OK.
00:04:51.140 --> 00:04:57.530
So the question is, how do we
find these x-s and lambdas?
00:04:57.530 --> 00:05:01.830
And notice -- we don't have an
equation Ax equal B anymore.
00:05:01.830 --> 00:05:02.830
I can't use elimination.
00:05:05.780 --> 00:05:07.670
I've got two unknowns,
and in fact they're
00:05:07.670 --> 00:05:08.860
multiplied together.
00:05:08.860 --> 00:05:12.390
Lambda and x are
both unknowns here.
00:05:12.390 --> 00:05:18.680
So, we need to, we need a
good idea of how to find them.
00:05:18.680 --> 00:05:23.930
But before I, before
I do that, and that's
00:05:23.930 --> 00:05:26.590
where determinant will
come in, can I just
00:05:26.590 --> 00:05:29.330
give you some matrices?
00:05:29.330 --> 00:05:31.720
Like here you go.
00:05:31.720 --> 00:05:35.261
Take the matrix, a
projection matrix.
00:05:35.261 --> 00:05:35.760
OK.
00:05:35.760 --> 00:05:42.950
So suppose we have a plane
and our matrix P is --
00:05:42.950 --> 00:05:45.450
what I've called A, now
I'm going to call it P
00:05:45.450 --> 00:05:47.620
for the moment, because it's --
00:05:47.620 --> 00:05:51.800
I'm thinking OK, let's
look at a then this,
00:05:51.800 --> 00:05:57.270
this other new matrix, I just
have an Ax, projection matrix.
00:05:57.270 --> 00:06:01.250
What are the eigenvalues
of a projection matrix?
00:06:01.250 --> 00:06:03.240
So that's my question.
00:06:03.240 --> 00:06:07.720
What are the x-s, the
eigenvectors, and the lambdas,
00:06:07.720 --> 00:06:13.190
the eigenvalues, thing,4 but the
roots of that quadratic for --
00:06:13.190 --> 00:06:17.170
and now let me say
a projection matrix.
00:06:17.170 --> 00:06:19.940
My, my point is that we --
00:06:19.940 --> 00:06:23.300
before we get into
determinants and, and formulas
00:06:23.300 --> 00:06:26.280
and all that stuff,
let's take some matrices
00:06:26.280 --> 00:06:28.690
where we know what they do.
00:06:28.690 --> 00:06:34.290
We know that if we take a
vector b, what this matrix does
00:06:34.290 --> 00:06:38.480
is it projects it down to Pb.
00:06:38.480 --> 00:06:43.340
So is b an eigenvector
in, in that picture?
00:06:43.340 --> 00:06:45.010
Is that vector b an eigenvector?
00:06:48.810 --> 00:06:50.270
No.
00:06:50.270 --> 00:06:53.850
Not so, so b is
not an eigenvector
00:06:53.850 --> 00:06:59.090
c- because Pb, its projection,
is in a different direction.
00:06:59.090 --> 00:07:04.030
So now tell me what vectors
are eigenvectors of P?
00:07:04.030 --> 00:07:09.080
What vectors do get projected
in the same direction that they
00:07:09.080 --> 00:07:09.580
start?
00:07:09.580 --> 00:07:12.310
So, so answer, tell me some x-s.
00:07:12.310 --> 00:07:16.590
Do you see what3 so it's
if Ax equals lambda x,
00:07:16.590 --> 00:07:21.230
In this picture, where could
I start with a vector b or x,
00:07:21.230 --> 00:07:27.070
do its projection, and end
up in the same direction?
00:07:27.070 --> 00:07:33.360
Well, that would happen if the
vector was right in that plane
00:07:33.360 --> 00:07:34.420
already.
00:07:34.420 --> 00:07:41.510
If the vector x was --
so let the vector x --
00:07:41.510 --> 00:07:49.810
so any vector, any x in the
plane will be an eigenvector.
00:07:49.810 --> 00:07:53.140
And what will happen
when I multiply by P,
00:07:53.140 --> 00:07:57.110
when I project a vector x --
00:07:57.110 --> 00:08:00.190
I called it b here, because
this is our familiar picture,
00:08:00.190 --> 00:08:03.650
but now I'm going to say that
b was no good for, for the,
00:08:03.650 --> 00:08:05.140
for our purposes.
00:08:05.140 --> 00:08:09.850
I'm interested in a vector x
that's actually in the plane,
00:08:09.850 --> 00:08:14.100
and I project it, and
what do I get back?
00:08:14.100 --> 00:08:15.370
x, of course.
00:08:15.370 --> 00:08:17.920
Doesn't move. can
be complex numbers.
00:08:17.920 --> 00:08:24.190
So any x in the plane
is unchanged by P,
00:08:24.190 --> 00:08:26.070
and what's that telling me?
00:08:26.070 --> 00:08:28.490
That's telling me that
x is an eigenvector,
00:08:28.490 --> 00:08:32.909
and it's also telling me what's
the eigenvalue, which is --
00:08:32.909 --> 00:08:34.470
just compare it with that.
00:08:34.470 --> 00:08:39.549
The eigenvalue, the
multiplier, is just one.
00:08:39.549 --> 00:08:41.510
Good.
00:08:41.510 --> 00:08:45.290
So we have actually a whole
plane of eigenvectors.
00:08:45.290 --> 00:08:49.050
Now I ask, are there
any other eigenvectors?
00:08:49.050 --> 00:08:51.020
And I expect the
answer to be yes,
00:08:51.020 --> 00:08:53.290
because I would
like to get three,
00:08:53.290 --> 00:08:55.470
if I'm in three
dimensions, I would
00:08:55.470 --> 00:08:59.490
like to hope for three
independent eigenvectors, two
00:08:59.490 --> 00:09:02.610
of them in the plane and
one not in the plane.
00:09:02.610 --> 00:09:03.300
OK.
00:09:03.300 --> 00:09:06.940
So this guy b that I drew
there was not any good.
00:09:06.940 --> 00:09:11.330
What's the right eigenvector
that's not in the plane?
00:09:11.330 --> 00:09:16.910
The, the good one is the one
that's perpendicular to the
00:09:16.910 --> 00:09:17.490
plane.
00:09:17.490 --> 00:09:22.170
There's an, another good x,
because what's the projection?
00:09:22.170 --> 00:09:24.300
So these are eigenvectors.
00:09:24.300 --> 00:09:27.000
Another guy here would
be another eigenvector.
00:09:27.000 --> 00:09:29.950
But now here is
another one. two.
00:09:29.950 --> 00:09:37.710
Any x that's perpendicular
to the plane,
00:09:37.710 --> 00:09:41.390
what's Px for that,
for that, vector?
00:09:43.950 --> 00:09:47.070
What's the projection of this
guy perpendicular to the plane?
00:09:47.070 --> 00:09:50.570
It is zero, of course.
00:09:50.570 --> 00:09:53.640
So -- there's the null space.
00:09:53.640 --> 00:09:58.800
Px and n- for those guys are
zero, or zero x if we like,
00:09:58.800 --> 00:10:02.480
and the eigenvalue is zero.
00:10:02.480 --> 00:10:05.697
So my answer to the question
is, what are the eigenvalues for
00:10:05.697 --> 00:10:07.280
In our example, the
one we worked out,
00:10:07.280 --> 00:10:08.610
a projection matrix?
00:10:08.610 --> 00:10:09.660
There they are.
00:10:09.660 --> 00:10:11.880
One and zero.
00:10:11.880 --> 00:10:13.060
OK.
00:10:13.060 --> 00:10:17.440
We know projection matrices.
00:10:17.440 --> 00:10:22.660
We can write them down as that
A, A transpose, A inverse,
00:10:22.660 --> 00:10:28.400
A transpose thing, but without
doing that from the picture
00:10:28.400 --> 00:10:31.420
we could see what
are the eigenvectors.
00:10:31.420 --> 00:10:32.280
OK.
00:10:32.280 --> 00:10:34.580
Are there other matrices?
00:10:34.580 --> 00:10:36.790
Let me take a second example.
00:10:36.790 --> 00:10:39.650
How about a permutation matrix?
00:10:39.650 --> 00:10:41.619
What about the matrix,
I'll call it A now.
00:10:41.619 --> 00:10:42.410
Zero one, one zero.
00:10:42.410 --> 00:10:49.120
A equals zero one one zero,
that had eigenvalue one and
00:10:49.120 --> 00:10:53.740
Can you tell me a vector x --
00:10:53.740 --> 00:10:55.680
see, we'll have a
system soon enough,
00:10:55.680 --> 00:10:58.040
so I, I would like
to just do these e-
00:10:58.040 --> 00:11:03.950
these couple of examples, just
to see the picture before we,
00:11:03.950 --> 00:11:11.300
before we let it
all, go into a system
00:11:11.300 --> 00:11:13.850
where that, matrix
isn't anything special.
00:11:13.850 --> 00:11:15.590
Because it is special.
00:11:15.590 --> 00:11:20.260
And what, so what vector
could I multiply by and end up
00:11:20.260 --> 00:11:21.430
in the same direction?
00:11:21.430 --> 00:11:25.800
Can you spot an
eigenvector for this guy?
00:11:25.800 --> 00:11:32.380
That's a matrix that
permutes x1 and x2, right?
00:11:32.380 --> 00:11:37.190
It switches the two
components of x.
00:11:37.190 --> 00:11:43.710
How could the vector
with its x2 x1, with --
00:11:43.710 --> 00:11:49.630
permuted turn out to
be a multiple of x1 x2,
00:11:49.630 --> 00:11:51.060
the vector we start with?
00:11:51.060 --> 00:11:53.300
Can you tell me an
eigenvector here for this guy?
00:11:53.300 --> 00:11:56.450
x equal -- what is -- actually,
can you tell me one vector that
00:11:56.450 --> 00:11:58.690
which is lambda x,
and I have a three x,
00:11:58.690 --> 00:12:01.160
And of course you -- everybody
knows that they're -- what,
00:12:01.160 --> 00:12:01.830
has eigenvalue one?
00:12:01.830 --> 00:12:03.670
So what, what vector
would have eigenvalue one,
00:12:03.670 --> 00:12:05.544
just above what we2
found here. so that if I,
00:12:05.544 --> 00:12:09.530
if I permute it it
doesn't change? right?
00:12:09.530 --> 00:12:13.450
There, that could
be one one, thanks.
00:12:13.450 --> 00:12:14.460
One one.
00:12:14.460 --> 00:12:16.780
OK, take that vector one one.
00:12:16.780 --> 00:12:20.470
That will be an eigenvector,
because if I do Ax
00:12:20.470 --> 00:12:23.120
I get one one.
00:12:23.120 --> 00:12:27.230
So that's the eigenvalue is one.
00:12:27.230 --> 00:12:28.710
Great.
00:12:28.710 --> 00:12:30.600
That's one eigenvalue.
00:12:30.600 --> 00:12:33.420
But I have here a
two by two matrix,
00:12:33.420 --> 00:12:37.620
and I figure there's going
to be a second eigenvalue.
00:12:40.180 --> 00:12:42.740
And eigenvector.
00:12:42.740 --> 00:12:44.410
Now, what about that?
00:12:44.410 --> 00:12:50.280
What's a vector, OK, maybe
we can just, like, guess it.
00:12:50.280 --> 00:12:55.280
A vector that the
other -- actually,
00:12:55.280 --> 00:12:58.940
this one that I'm thinking of
is going to be a vector that has
00:12:58.940 --> 00:13:00.630
eigenvalue minus one.
00:13:03.240 --> 00:13:06.670
That's going to be my other
eigenvalue for this matrix.
00:13:06.670 --> 00:13:09.430
It's a -- notice the nice
positive or not negative
00:13:09.430 --> 00:13:13.240
matrix, but an eigenvalue is
going to come out negative.
00:13:13.240 --> 00:13:15.520
And can you guess, spot
the x that will work for
00:13:15.520 --> 00:13:18.910
Times x is supposed to
give me zero, right? that?
00:13:18.910 --> 00:13:20.960
So I want a, a vector.
00:13:20.960 --> 00:13:27.140
When I multiply by A, which
reverses the two components,
00:13:27.140 --> 00:13:31.120
I want the thing to come
out minus the original.
00:13:31.120 --> 00:13:34.480
So what shall I send
in in that case?
00:13:34.480 --> 00:13:37.650
If I send in negative one one.
00:13:40.550 --> 00:13:47.150
Then when I apply A, I get
I do that multiplication,
00:13:47.150 --> 00:13:52.440
and I get one negative
one, so it reversed sign.
00:13:52.440 --> 00:13:55.840
So Ax is -x.
00:13:55.840 --> 00:13:57.710
Lambda is minus one.
00:13:57.710 --> 00:14:04.130
Ax -- so Ax was x there
and Ax is minus x here.
00:14:04.130 --> 00:14:08.080
Can I just mention,
like, jump ahead, have,
00:14:08.080 --> 00:14:10.550
give a perfectly
innocent-looking quadratic
00:14:10.550 --> 00:14:14.990
and point out a special
little fact about eigenvalues.
00:14:14.990 --> 00:14:18.940
n by n matrices will
have n eigenvalues.
00:14:18.940 --> 00:14:23.390
And I get this matrix4
zero zero zero one,
00:14:23.390 --> 00:14:28.390
And it's not like -- suppose
n is three or four or more.
00:14:28.390 --> 00:14:32.120
It's not so easy to find them.
00:14:32.120 --> 00:14:35.570
We'd have a third degree or a
fourth degree or an n-th degree
00:14:35.570 --> 00:14:36.330
equation.
00:14:36.330 --> 00:14:37.960
But here's one nice fact.
00:14:37.960 --> 00:14:40.270
There, there's one
pleasant fact. we --
00:14:40.270 --> 00:14:42.570
the eigenvalues came
out four and two.
00:14:42.570 --> 00:14:45.540
That the sum of the
eigenvalues equals the sum
00:14:45.540 --> 00:14:46.530
down the diagonal.
00:14:46.530 --> 00:14:50.150
That's called the trace, and
I put that in the lecture
00:14:50.150 --> 00:14:54.590
Now I add three I to that
matrix. content specifically.
00:14:54.590 --> 00:15:03.390
So this is a neat fact, the fact
that sthe sum of the lambdas,
00:15:03.390 --> 00:15:08.390
add up the lambdas,
equals the sum --
00:15:08.390 --> 00:15:11.220
what would you like me to,
shall I write that down?
00:15:11.220 --> 00:15:17.330
What I'm want to say in words is
the sum down the diagonal of A.
00:15:17.330 --> 00:15:18.640
Shall I write a11+a22+...+ ann.
00:15:18.640 --> 00:15:22.570
That's add up the
diagonal entries.
00:15:27.470 --> 00:15:31.700
In this example, it's zero.
00:15:31.700 --> 00:15:36.560
In other words, once I found
this eigenvalue of one,
00:15:36.560 --> 00:15:39.110
I knew the other one
had to be minus one
00:15:39.110 --> 00:15:43.710
in this two by two case, because
in the two by two case, which
00:15:43.710 --> 00:15:51.970
is a good one to, to, play
with, the trace tells you
00:15:51.970 --> 00:15:54.804
right away what the
other eigenvalue is.
00:15:54.804 --> 00:15:56.970
So if I tell you one
eigenvalue, you can tell me the
00:15:56.970 --> 00:15:57.780
other one.
00:15:57.780 --> 00:16:02.270
We'll, we'll have that -- we'll,
minus one and eigenvectors one
00:16:02.270 --> 00:16:05.850
one and eigenvector minus
one we'll see that again.
00:16:05.850 --> 00:16:06.350
OK.
00:16:06.350 --> 00:16:07.980
Now can I --
00:16:07.980 --> 00:16:11.240
I could give more
examples, but maybe it's
00:16:11.240 --> 00:16:16.140
time to face the, the equation,
Ax equal lambda x, and figure
00:16:16.140 --> 00:16:19.810
how are we going to
find x and lambda.
00:16:19.810 --> 00:16:22.571
And that is lambda
one times lambda3
00:16:22.571 --> 00:16:23.070
OK.
00:16:23.070 --> 00:16:25.590
So this, so the
question now is how
00:16:25.590 --> 00:16:30.630
to find eigenvalues
and eigenvectors.
00:16:30.630 --> 00:16:35.430
How to solve, how to solve Ax
equal lambda x from the three
00:16:35.430 --> 00:16:43.670
x, so it's just I mean,
when we've got two unknowns
00:16:43.670 --> 00:16:47.500
both in the equation.
00:16:47.500 --> 00:16:48.460
OK.
00:16:48.460 --> 00:16:51.330
Here's the trick.
00:16:51.330 --> 00:16:53.240
Simple idea.
00:16:53.240 --> 00:16:58.980
Bring this onto the same side.
00:16:58.980 --> 00:16:59.950
Rewrite.
00:16:59.950 --> 00:17:05.589
Bring this over as A minus
lambda times the identity x
00:17:05.589 --> 00:17:09.770
One. equals zero.
00:17:09.770 --> 00:17:10.740
Right?
00:17:10.740 --> 00:17:14.150
I have Ax minus lambda
x, so I brought that over
00:17:14.150 --> 00:17:14.260
and I've got zero left on
the, on the right-hand side.
00:17:14.260 --> 00:17:16.740
What's the relation
between that problem and --
00:17:16.740 --> 00:17:19.170
let me write
00:17:19.170 --> 00:17:19.980
OK.
00:17:19.980 --> 00:17:23.200
I don't know lambda and I don't
know x, but I do know something
00:17:23.200 --> 00:17:23.700
here.
00:17:26.490 --> 00:17:28.460
What I know is if
I, if I'm going
00:17:28.460 --> 00:17:31.870
to be able to solve this thing,
for some x that's not the zero
00:17:31.870 --> 00:17:35.760
vector, that's not, that's
a useless eigenvector,
00:17:35.760 --> 00:17:37.410
doesn't count.
00:17:37.410 --> 00:17:44.860
What I know now is that
this matrix must be what?
00:17:44.860 --> 00:17:48.350
If I'm going to be --
if there is an x --
00:17:48.350 --> 00:17:51.520
I don't -- right now I
don't know what it is.
00:17:51.520 --> 00:17:54.490
I'm going to find
lambda first, actually.
00:17:54.490 --> 00:18:00.300
And -- but if there is an x,
it tells me that this matrix,
00:18:00.300 --> 00:18:05.160
this special combination,
which is like the matrix A with
00:18:05.160 --> 00:18:10.430
lambda -- shifted by
lambda, shifted by lambda I,
00:18:10.430 --> 00:18:14.550
that it has to be singular.
00:18:14.550 --> 00:18:17.030
This matrix must be
singular, otherwise
00:18:17.030 --> 00:18:21.510
the only x would be the
zero x, and zero matrix.OK.
00:18:21.510 --> 00:18:22.700
So this is singular.
00:18:22.700 --> 00:18:30.320
And what do I now know
about singular matrices?
00:18:30.320 --> 00:18:31.520
So, so take three away.
00:18:31.520 --> 00:18:35.140
Their determinant is zero.
00:18:35.140 --> 00:18:38.790
So I've -- so from the fact
that that has to be singular,
00:18:38.790 --> 00:18:44.750
I know that the determinant of
A minus lambda I has to be zero.
00:18:48.590 --> 00:18:52.430
And that, now I've
got x out of it.
00:18:52.430 --> 00:18:55.680
I've got an equation for
lambda, that the key equation --
00:18:55.680 --> 00:19:01.020
it's called the characteristic
equation or the eigenvalue
00:19:01.020 --> 00:19:01.520
equation.
00:19:04.180 --> 00:19:08.370
And that -- in other words,
I'm now in a position to find
00:19:08.370 --> 00:19:10.480
lambda first.
00:19:10.480 --> 00:19:15.940
So -- the idea will be
to find lambda first.
00:19:18.890 --> 00:19:21.660
And actually, I won't
find one lambda,
00:19:21.660 --> 00:19:24.120
I'll find N different lambdas.
00:19:24.120 --> 00:19:28.440
Well, n lambdas, maybe
not n different ones.
00:19:28.440 --> 00:19:31.430
A lambda could be repeated.
00:19:31.430 --> 00:19:38.040
A repeated lambda is the
source of all trouble in 18.06.
00:19:38.040 --> 00:19:43.620
So, let's hope for the moment
that they're not repeated.
00:19:43.620 --> 00:19:48.310
There, there they
were different, right?
00:19:48.310 --> 00:19:51.920
One and minus one in that, in
that, for that permutation.
00:19:51.920 --> 00:19:52.800
OK.
00:19:52.800 --> 00:19:59.460
So and after I found this
lambda, can I just look ahead?
00:19:59.460 --> 00:20:02.200
How I going to find x?
00:20:02.200 --> 00:20:06.670
After I have found this lambda,
the lambda being this --
00:20:06.670 --> 00:20:10.551
one of the numbers that
makes this matrix singular.
00:20:10.551 --> 00:20:11.550
Their product was eight.
00:20:11.550 --> 00:20:15.110
Then of course finding x
is just by elimination.
00:20:15.110 --> 00:20:15.610
Right?
00:20:15.610 --> 00:20:18.630
It's just -- now I've
got a singular matrix,
00:20:18.630 --> 00:20:20.640
I'm looking for the null space.
00:20:20.640 --> 00:20:24.020
We're experts at
finding the null space.
00:20:24.020 --> 00:20:26.570
You know, you do
elimination, you identify
00:20:26.570 --> 00:20:31.780
the, the, the pivot columns
and so on, you're --
00:20:31.780 --> 00:20:36.900
and, give values to
the free variables.
00:20:36.900 --> 00:20:39.300
Probably there'll only
be one free variable.
00:20:39.300 --> 00:20:42.480
We'll give it the
value one, like there.
00:20:42.480 --> 00:20:44.550
And we find the other variable.
00:20:44.550 --> 00:20:45.390
OK.
00:20:45.390 --> 00:20:52.610
So let's -- find the x
second will be a doable job.
00:20:52.610 --> 00:20:54.540
That's my big equation for x.
00:20:54.540 --> 00:20:58.070
Let's go, let's look at the
first job of finding lambda.
00:20:58.070 --> 00:20:59.500
Can I take another example?
00:20:59.500 --> 00:21:00.000
OK.
00:21:00.000 --> 00:21:02.130
And let's, let's
work that one out.
00:21:02.130 --> 00:21:02.630
OK.
00:21:02.630 --> 00:21:07.590
So let me take the example,
say, let me make it easy.
00:21:07.590 --> 00:21:09.240
it's just sitting there.
00:21:09.240 --> 00:21:12.550
Three three one and
one. what do you
00:21:12.550 --> 00:21:14.610
know about the complex numbers?
00:21:14.610 --> 00:21:16.680
So I've made it easy.
00:21:16.680 --> 00:21:19.160
I've made it two by two.
00:21:19.160 --> 00:21:20.810
I've made it symmetric.
00:21:20.810 --> 00:21:24.530
And I even made it
constant down the diagonal.
00:21:24.530 --> 00:21:27.830
That a matrix, a perfectly
real matrix could
00:21:27.830 --> 00:21:32.380
So that -- so the more, like,
special properties I stick
00:21:32.380 --> 00:21:36.270
into the matrix, the more
special outcome I get
00:21:36.270 --> 00:21:38.370
for the eigenvalues.
00:21:38.370 --> 00:21:42.050
For example, this
symmetric matrix,
00:21:42.050 --> 00:21:48.130
I know that it'll come out
with real eigenvalues. one.
00:21:48.130 --> 00:21:52.530
The eigenvalues will turn
out to be nice real numbers.
00:21:52.530 --> 00:21:57.760
And up in our previous example,
that was a symmetric matrix.
00:21:57.760 --> 00:22:02.130
Actually, while we're at it,
that was a symmetric matrix.
00:22:02.130 --> 00:22:05.500
Its eigenvalues were nice real
numbers, one and minus one.
00:22:05.500 --> 00:22:07.630
And do you notice anything
about its eigenvectors?
00:22:07.630 --> 00:22:08.970
And what do you notice?
00:22:08.970 --> 00:22:11.640
Anything particular about those
two vectors, one one and minus
00:22:11.640 --> 00:22:14.840
And now comes that thing that
I wanted to be reminded of.
00:22:14.840 --> 00:22:15.380
one one?
00:22:15.380 --> 00:22:18.850
They just happen to be -- no,
I can't say they just happen
00:22:18.850 --> 00:22:20.610
to be, because that's
the whole point,
00:22:20.610 --> 00:22:23.390
is that they had to be -- what?
00:22:23.390 --> 00:22:25.610
What are they?
00:22:25.610 --> 00:22:27.380
They're perpendicular.
00:22:27.380 --> 00:22:30.180
The vector, when I -- if I see
a vector one one and a one --
00:22:30.180 --> 00:22:33.121
and a minus one one, my
mind immediately takes that
00:22:33.121 --> 00:22:33.620
dot product.
00:22:33.620 --> 00:22:36.570
It's zero. what's the
determinant of that matrix?
00:22:36.570 --> 00:22:38.140
Those vectors are perpendicular.
00:22:38.140 --> 00:22:40.090
That'll happen here too.
00:22:40.090 --> 00:22:42.670
Well, let's find
the eigenvalues.
00:22:42.670 --> 00:22:45.770
Actually, oh, my
example's too easy.
00:22:45.770 --> 00:22:48.350
My example is too easy.
00:22:48.350 --> 00:22:53.550
Let me tell you in advance
what's going to happen.
00:22:53.550 --> 00:22:56.210
May I?
00:22:56.210 --> 00:22:58.510
Or shall I do the determinant
of A minus lambda,
00:22:58.510 --> 00:23:00.720
and then point out at the end?
00:23:00.720 --> 00:23:03.180
Will you remind me at
the -- after I've found
00:23:03.180 --> 00:23:10.830
the eigenvalues to say why they
were -- why they were easy from
00:23:10.830 --> 00:23:17.050
That -- it had to be eight,
because we factored into lambda
00:23:17.050 --> 00:23:20.160
the, from the example we did?
00:23:20.160 --> 00:23:23.270
OK, let's do the job here.
00:23:23.270 --> 00:23:27.410
Let's compute determinant
of A minus lambda I.
00:23:27.410 --> 00:23:29.490
So that's a determinant.
00:23:29.490 --> 00:23:32.600
And what's, what is this thing?
00:23:32.600 --> 00:23:36.090
It's the matrix A with lambda
removed from the diagonal.
00:23:36.090 --> 00:23:38.570
for this matrix?
00:23:38.570 --> 00:23:40.770
So the diagonal
matrix is shifted,
00:23:40.770 --> 00:23:43.800
and then I'm taking
the determinant.
00:23:43.800 --> 00:23:44.370
OK.
00:23:44.370 --> 00:23:47.390
So I multiply this out.
00:23:47.390 --> 00:23:49.340
So what is that determinant?
00:23:49.340 --> 00:23:52.520
Do you notice, I
didn't take lambda away
00:23:52.520 --> 00:23:55.090
from all the entries.
00:23:55.090 --> 00:23:56.790
It's lambda I, so
it's lambda along the
00:23:56.790 --> 00:23:58.040
Lambda plus three x. diagonal.
00:23:58.040 --> 00:24:04.040
So I get three minus lambda
squared and then minus one,
00:24:04.040 --> 00:24:06.380
right?
00:24:06.380 --> 00:24:08.000
And I want that to be zero.
00:24:08.000 --> 00:24:09.709
And what is A minus lambda I x?
00:24:09.709 --> 00:24:11.000
Well, I'm going to simplify it.
00:24:11.000 --> 00:24:11.050
And what will I get?
00:24:11.050 --> 00:24:11.190
So if I multiply this out, I
get lambda squared minus six
00:24:11.190 --> 00:24:11.290
What's -- how is this matrix
related to that matrix?
00:24:11.290 --> 00:24:11.330
lambda plus what?
00:24:11.330 --> 00:24:11.350
Plus eight.
00:24:11.350 --> 00:24:11.400
But it's out there.
00:24:11.400 --> 00:24:11.490
And that I'm going
to set to zero.
00:24:11.490 --> 00:24:11.560
And I'm going to solve it.
00:24:11.560 --> 00:24:11.640
So and it's, it's a
quadratic equation.
00:24:11.640 --> 00:24:11.750
I can use factorization, I
can use the quadratic formula.
00:24:11.750 --> 00:24:12.625
I'll get two lambdas.
00:24:12.625 --> 00:24:30.690
Before I do it, tell me
what's that number six that's
00:24:30.690 --> 00:24:41.560
showing up in this equation?
00:24:41.560 --> 00:24:48.080
It's the trace.
00:24:48.080 --> 00:25:03.300
That number six is
three plus three.
00:25:03.300 --> 00:25:05.820
And while we're at it, what's
the number eight that's
00:25:05.820 --> 00:25:08.590
showing up in this equation?
00:25:08.590 --> 00:25:10.340
It's the determinant.
00:25:10.340 --> 00:25:13.290
That our matrix has
determinant eight.
00:25:13.290 --> 00:25:16.820
So in a two by two
case, it's really nice.
00:25:16.820 --> 00:25:21.630
It's lambda squared minus
the trace times lambda --
00:25:21.630 --> 00:25:24.430
the trace is the
linear coefficient --
00:25:24.430 --> 00:25:27.880
and plus the determinant,
the constant term.
00:25:27.880 --> 00:25:28.540
OK.
00:25:28.540 --> 00:25:32.730
So let's -- can, can
we find the roots?
00:25:32.730 --> 00:25:36.310
I guess the easy way is to
factor that as something times
00:25:36.310 --> 00:25:37.980
something.
00:25:37.980 --> 00:25:41.280
If we couldn't factor it,
then we'd have to use the old
00:25:41.280 --> 00:25:46.590
b^2-4ac formula, but I, I think
we can factor that into lambda
00:25:46.590 --> 00:25:50.870
minus what times
lambda minus what?
00:25:50.870 --> 00:25:54.850
Can you do that factorization?
00:25:54.850 --> 00:25:57.240
Four and two?
00:25:57.240 --> 00:25:59.172
Lambda minus four
times lambda minus two.
00:25:59.172 --> 00:26:00.880
So the, the eigenvalues
are four and two.
00:26:00.880 --> 00:26:01.210
So the eigenvalues are --
one eigenvalue, lambda one,
00:26:01.210 --> 00:26:01.300
Now I'm looking for x, the
eigenvector. let's say,
00:26:01.300 --> 00:26:01.320
is four.
00:26:01.320 --> 00:26:01.490
Lambda two, the other
eigenvalue, is two.
00:26:01.490 --> 00:26:03.570
The eigenvalues
are four and two.
00:26:03.570 --> 00:26:08.540
And then I can go
for the eigenvectors.
00:26:08.540 --> 00:26:15.380
Suppose I have a matrix
A, and Ax equal lambda x.
00:26:15.380 --> 00:26:16.620
equals zero.
00:26:16.620 --> 00:26:20.980
You see I got the
eigenvalues first.
00:26:20.980 --> 00:26:25.950
So if they, if this
had eigenvalue lambda,
00:26:25.950 --> 00:26:26.860
Four and two.
00:26:26.860 --> 00:26:29.030
Now for the eigenvectors.
00:26:29.030 --> 00:26:31.300
So what are the eigenvectors?
00:26:31.300 --> 00:26:34.880
They're these guys in
the null space when
00:26:34.880 --> 00:26:40.090
I take away, when I make the
matrix singular by taking
00:26:40.090 --> 00:26:42.740
four I or two I away.
00:26:42.740 --> 00:26:46.470
So we're -- we got to
do those separately.
00:26:46.470 --> 00:26:50.100
I'll -- let me find the
eigenvector for four first.
00:26:50.100 --> 00:26:57.420
So I'll subtract four,
so A minus four I is --
00:26:57.420 --> 00:27:00.590
so taking four away will
put minus ones there.
00:27:04.070 --> 00:27:07.000
And what's the point
about that matrix?
00:27:07.000 --> 00:27:10.260
If four is an eigenvalue,
then A minus four
00:27:10.260 --> 00:27:13.220
I had better be a
what kind of matrix?
00:27:13.220 --> 00:27:14.800
Singular.
00:27:14.800 --> 00:27:17.800
If that matrix isn't singular,
the four wasn't correct.
00:27:17.800 --> 00:27:21.280
But we're OK, that
matrix is singular.
00:27:21.280 --> 00:27:23.110
And what's the x now?
00:27:23.110 --> 00:27:25.290
The x is in the null space.
00:27:25.290 --> 00:27:28.910
So what's the x1 that goes
with, with the lambda one?
00:27:28.910 --> 00:27:32.580
eigenvalue, eigenvector,
eigenvalue for this,
00:27:32.580 --> 00:27:38.294
So that A -- so this is -- now
I'm doing A x1 is lambda one
00:27:38.294 --> 00:27:38.794
x1.
00:27:41.640 --> 00:27:45.160
So I took A minus lambda
one I, that's this matrix,
00:27:45.160 --> 00:27:48.630
and now I'm looking for
the x1 in its null space,
00:27:48.630 --> 00:27:49.510
and who is he?
00:27:49.510 --> 00:27:51.760
What's the vector x
in the null space?
00:27:51.760 --> 00:27:53.380
Of course it's one one.
00:27:53.380 --> 00:27:56.310
So that's the eigenvector that
goes with that eigenvalue.
00:27:56.310 --> 00:27:57.610
So, so now --
00:27:57.610 --> 00:28:00.540
Let's just spend one
more minute on this bad
00:28:00.540 --> 00:28:02.490
Now how about the
eigenvector that
00:28:02.490 --> 00:28:04.110
goes with the other eigenvalue?
00:28:04.110 --> 00:28:06.040
Can I do that
with, with erasing?
00:28:06.040 --> 00:28:08.940
I take A minus two I.
00:28:08.940 --> 00:28:11.380
So now I take two away
from the diagonal,
00:28:11.380 --> 00:28:15.670
and that leaves me
with a one and a one.
00:28:15.670 --> 00:28:19.490
So A minus two I has again
produced a singular matrix,
00:28:19.490 --> 00:28:21.710
as it had to.
00:28:21.710 --> 00:28:24.990
I'm looking for the
null space of that guy.
00:28:24.990 --> 00:28:27.210
What vector is in
its null space?
00:28:27.210 --> 00:28:29.320
Well, of course, a
whole line of vectors.
00:28:32.040 --> 00:28:35.990
So when I say the eigenvector,
I'm not speaking correctly.
00:28:35.990 --> 00:28:38.490
There's a whole line of
eigenvectors, and you just --
00:28:38.490 --> 00:28:40.760
I just want a basis.
00:28:40.760 --> 00:28:43.360
And for a line I
just want one vector.
00:28:43.360 --> 00:28:48.130
But -- You could, you're
-- there's some freedom
00:28:48.130 --> 00:28:50.860
in choosing that one, but
choose a reasonable one.
00:28:50.860 --> 00:28:54.200
What's a vector in the
null space of that?
00:28:54.200 --> 00:28:58.800
Well, the natural vector
to pick as the eigenvector
00:28:58.800 --> 00:29:01.540
with, with lambda
two is minus one one.
00:29:05.130 --> 00:29:07.760
If I did elimination
on that vector
00:29:07.760 --> 00:29:10.520
and set that, the free
variable to be one,
00:29:10.520 --> 00:29:15.010
I would get minus one
and get that eigenvector.
00:29:15.010 --> 00:29:22.250
So you see then that
I've got eigenvector,
00:29:22.250 --> 00:29:32.890
Now the other neat fact
is that the determinant,
00:29:32.890 --> 00:29:36.906
How are those two
matrices related?
00:29:40.530 --> 00:29:46.710
Well, one is just three I more
than the other one, right? two.
00:29:46.710 --> 00:29:52.410
I just took that matrix and I --
00:29:52.410 --> 00:30:29.276
I took this matrix
and I added three I.
00:30:29.276 --> 00:30:31.900
So my question is, what happened
to the minus four times lambda
00:30:31.900 --> 00:30:34.720
minus two. eigenvalues and what
happened to the eigenvectors?
00:30:34.720 --> 00:30:37.530
That's the, that's like the
question we keep asking now
00:30:37.530 --> 00:30:39.500
in this chapter.
00:30:39.500 --> 00:30:42.690
If I, if I do something to the
matrix, what happens if I --
00:30:42.690 --> 00:30:44.730
or I know something
about the matrix,
00:30:44.730 --> 00:30:48.550
what's the what's the
conclusion for its eigenvectors
00:30:48.550 --> 00:30:49.280
and eigenvalues?
00:30:49.280 --> 00:30:54.180
Because -- those eigenvalues and
eigenvectors are going to tell
00:30:54.180 --> 00:30:57.280
us important information
about the matrix.
00:30:57.280 --> 00:31:00.970
And here what are we seeing?
00:31:00.970 --> 00:31:04.330
What's happening to these
eigenvalues, one and minus
00:31:04.330 --> 00:31:06.460
one, when I add three I?
00:31:10.040 --> 00:31:13.160
It just added three
to the eigenvalues.
00:31:13.160 --> 00:31:16.640
I got four and two, three
more than one and minus
00:31:16.640 --> 00:31:18.220
one.
00:31:18.220 --> 00:31:19.960
What happened to
the eigenvectors?
00:31:19.960 --> 00:31:21.490
Nothing at all.
00:31:21.490 --> 00:31:25.170
One one is -- and minus -- and
one -- and minus one one are --
00:31:25.170 --> 00:31:27.050
is still the eigenvectors.
00:31:27.050 --> 00:31:34.200
In other words, simple
but useful observation.
00:31:34.200 --> 00:31:40.320
If I add three I to a matrix,
its eigenvectors don't change
00:31:40.320 --> 00:31:42.870
and its eigenvalues
are three bigger.
00:31:42.870 --> 00:31:44.130
Let's, let's just see why.
00:31:44.130 --> 00:32:02.150
Let me keep all this on the same
board. but just so you see --
00:32:02.150 --> 00:32:08.410
so I'll try to do that.
00:32:08.410 --> 00:32:39.970
this has eigenvalue
lambda plus three.
00:32:39.970 --> 00:32:46.590
And x, the eigenvector, is
the same x for both matrices.
00:32:46.590 --> 00:32:48.790
OK.
00:32:48.790 --> 00:32:50.195
So that's, great.
00:32:53.440 --> 00:32:54.670
Of course, it's special.
00:32:54.670 --> 00:32:57.570
We got the new matrix
by adding three I.
00:32:57.570 --> 00:32:59.500
Suppose I had added
another matrix.
00:32:59.500 --> 00:33:02.410
Suppose I know the eigenvalues
and eigenvectors of A.
00:33:02.410 --> 00:33:07.250
So I took A minus lambda I x,
and what kind of a matrix I
00:33:07.250 --> 00:33:09.510
So this is, this,
this little board
00:33:09.510 --> 00:33:11.875
here is going to
be not so great.
00:33:17.730 --> 00:33:21.270
Suppose I have a matrix A and
it has an eigenvector x with
00:33:21.270 --> 00:33:23.260
an eigenvalue lambda.
00:33:23.260 --> 00:33:28.730
You remember, I solve
A minus lambda I x
00:33:28.730 --> 00:33:31.020
And now I add on
some other matrix.
00:33:31.020 --> 00:33:34.000
So, so what I'm asking you is,
if you know the eigenvalues
00:33:34.000 --> 00:33:38.430
of A and you know
the eigenvalues of B,
00:33:38.430 --> 00:33:43.990
let me say suppose B -- so this
is if -- let me put an if here.
00:33:43.990 --> 00:33:49.230
If Ax equals lambda x, fine,
and B has, eigenvalues,
00:33:49.230 --> 00:33:52.240
has eigenvalues --
00:33:52.240 --> 00:33:56.800
what shall we call them?
00:33:56.800 --> 00:34:02.270
Alpha, alpha one and alpha --
00:34:02.270 --> 00:34:05.020
let's say --
00:34:05.020 --> 00:34:07.270
I'll use alpha for
the eigenvalues of B
00:34:07.270 --> 00:34:08.199
for no good reason.
00:34:11.489 --> 00:34:16.389
What a- you see what I'm going
to ask is, how about A plus B?
00:34:20.949 --> 00:34:24.070
Let me, let me give you
the, let me give you,
00:34:24.070 --> 00:34:27.770
what you might think first.
00:34:27.770 --> 00:34:28.730
OK.
00:34:28.730 --> 00:34:35.460
If Ax equals lambda x and if
B has an eigenvalue alpha,
00:34:35.460 --> 00:34:40.630
then I allowed to say -- what's
the matter with this argument?
00:34:40.630 --> 00:34:43.300
That gave us the
constant term eight.
00:34:43.300 --> 00:34:44.070
It's wrong.
00:34:44.070 --> 00:34:47.130
What I'm going to
write up is wrong.
00:34:47.130 --> 00:34:50.179
I'm going to say Bx is alpha x.
00:34:50.179 --> 00:34:55.909
Add those up, and you get A plus
B x equals lambda plus alpha x.
00:34:55.909 --> 00:35:00.500
So you would think that if
you know the eigenvalues of A
00:35:00.500 --> 00:35:03.490
and you knew the
eigenvalues of B,
00:35:03.490 --> 00:35:07.900
then if you added you would know
the eigenvalues of A plus B.
00:35:07.900 --> 00:35:11.310
But that's false.
00:35:11.310 --> 00:35:17.840
A plus B -- well, when B was
three I, that worked great.
00:35:17.840 --> 00:35:21.290
But this is not so great.
00:35:21.290 --> 00:35:25.720
And what's the matter
with that argument there?
00:35:25.720 --> 00:35:32.610
We have no reason to
believe that x is also
00:35:32.610 --> 00:35:33.500
an eigenvector of
00:35:33.500 --> 00:35:38.930
B has some eigenvalues,
B. but it's
00:35:38.930 --> 00:35:43.600
got some different
eigenvectors normally.
00:35:43.600 --> 00:35:45.520
It's a different matrix.
00:35:45.520 --> 00:35:47.250
I don't know anything special.
00:35:47.250 --> 00:35:50.150
If I don't know anything
special, then as far as I know,
00:35:50.150 --> 00:35:52.670
it's got some different
eigenvector y,
00:35:52.670 --> 00:35:55.790
and when I add I
get just rubbish.
00:35:55.790 --> 00:35:57.480
I mean, I get --
00:35:57.480 --> 00:35:59.970
I can add, but I
don't learn anything.
00:35:59.970 --> 00:36:05.380
So not so great is A plus B.
00:36:05.380 --> 00:36:09.460
Or A times B.
00:36:09.460 --> 00:36:12.670
Normally the
eigenvalues of A plus B
00:36:12.670 --> 00:36:18.630
or A times B are not eigenvalues
of A plus eigenvalues of B.
00:36:18.630 --> 00:36:22.500
Ei- eigenvalues are
not, like, linear.
00:36:22.500 --> 00:36:24.610
Or -- and they don't multiply.
00:36:24.610 --> 00:36:27.580
Because, eigenvectors
are usually different
00:36:27.580 --> 00:36:31.600
and, and there's just
no way to find out
00:36:31.600 --> 00:36:33.520
what A plus B does to affect
00:36:33.520 --> 00:36:34.960
What do I do now? it.
00:36:34.960 --> 00:36:35.470
OK.
00:36:35.470 --> 00:36:41.660
So that's, like, a caution.
00:36:41.660 --> 00:36:44.610
Don't, if B is a multiple
of the identity, great,
00:36:44.610 --> 00:36:50.340
but if B is some general matrix,
then for A plus B you've got
00:36:50.340 --> 00:36:54.580
to find -- you've got to
solve the eigenvalue problem.
00:36:54.580 --> 00:36:59.720
Now I want to do another
example that brings out a,
00:36:59.720 --> 00:37:02.200
OK. another point
about eigenvalues.
00:37:02.200 --> 00:37:06.280
Let me make this example
a rotation matrix.
00:37:06.280 --> 00:37:09.390
possibility of complex numbers.
00:37:09.390 --> 00:37:10.170
OK.
00:37:10.170 --> 00:37:13.280
So here's another example.
00:37:13.280 --> 00:37:16.390
So a rotate --
00:37:16.390 --> 00:37:21.060
oh, I'd better call it Q.
00:37:21.060 --> 00:37:26.500
I often use Q for,
for, rotations
00:37:26.500 --> 00:37:32.600
because those are the, like,
very important examples
00:37:32.600 --> 00:37:34.580
of orthogonal matrices.
00:37:34.580 --> 00:37:38.290
Let me make it a
ninety degree rotation.
00:37:38.290 --> 00:37:41.709
So -- my matrix is going to
be the one that rotates every
00:37:41.709 --> 00:37:43.750
And that's the sum, that's
lambda one plus lambda
00:37:43.750 --> 00:37:46.640
vector by ninety degrees.
00:37:46.640 --> 00:37:49.480
So do you remember that matrix?
00:37:49.480 --> 00:37:51.950
It's the cosine of
ninety degrees, which
00:37:51.950 --> 00:37:54.790
is zero, the sine
of ninety degrees,
00:37:54.790 --> 00:38:03.990
which is one, minus the sine of
ninety, the cosine of ninety.
00:38:03.990 --> 00:38:09.360
So that matrix
deserves the letter Q.
00:38:09.360 --> 00:38:15.500
It's an orthogonal matrix,
very, very orthogonal matrix.
00:38:15.500 --> 00:38:21.420
Now I'm interested in its
eigenvalues and eigenvectors.
00:38:21.420 --> 00:38:24.500
Two by two, it
can't be that tough.
00:38:24.500 --> 00:38:27.200
We know that the
eigenvalues add to zero.
00:38:30.780 --> 00:38:33.310
Actually, we know
something already here.
00:38:33.310 --> 00:38:35.750
The eigen- what's the sum
of the two eigenvalues?
00:38:35.750 --> 00:38:38.880
Just tell me what I just said.
00:38:38.880 --> 00:38:40.430
Zero, right.
00:38:40.430 --> 00:38:42.310
From that trace business.
00:38:42.310 --> 00:38:46.550
The sum of the eigenvalues
is, is going to come out zero.
00:38:46.550 --> 00:38:48.440
And the product of
the eigenvalues,
00:38:48.440 --> 00:38:50.190
did I tell you about
the determinant being
00:38:50.190 --> 00:38:50.870
the product of the eigenvalues?
00:38:50.870 --> 00:38:50.880
No.
00:38:50.880 --> 00:38:50.950
But that's a good thing to know.
00:38:50.950 --> 00:38:51.030
We pointed out how
that eight appeared in
00:38:51.030 --> 00:38:52.863
the, in the quadratic
equation. eigenvalues,
00:38:52.863 --> 00:39:07.700
we can postpone that evil day,
00:39:07.700 --> 00:39:23.700
So let me just say this.
00:39:23.700 --> 00:39:49.930
The trace is zero
plus zero, obviously.
00:39:49.930 --> 00:39:52.675
And that was the determinant.
00:39:52.675 --> 00:39:53.175
OK.
00:39:56.230 --> 00:39:58.390
What I'm leading up
to with this example
00:39:58.390 --> 00:40:03.610
is that something's
going to go wrong.
00:40:03.610 --> 00:40:09.080
Something goes
wrong for rotation
00:40:09.080 --> 00:40:16.410
because what vector can
come out parallel to itself
00:40:16.410 --> 00:40:18.820
after a rotation?
00:40:18.820 --> 00:40:23.790
If this matrix rotates every
vector by ninety degrees,
00:40:23.790 --> 00:40:26.880
what could be an eigenvector?
00:40:26.880 --> 00:40:31.190
Do you see we're, we're,
we're going to have trouble.
00:40:31.190 --> 00:40:33.780
eigenvectors are --
00:40:36.060 --> 00:40:36.560
Well.
00:40:36.560 --> 00:40:39.220
Our, our picture
of eigenvectors,
00:40:39.220 --> 00:40:42.410
of, of coming out in the same
direction that they went in,
00:40:42.410 --> 00:40:45.140
there won't be it.
00:40:45.140 --> 00:40:48.820
And with, and with eigenvalues
we're going to have trouble.
00:40:48.820 --> 00:40:50.440
From these equations.
00:40:50.440 --> 00:40:51.710
Let's see.
00:40:51.710 --> 00:40:53.780
Why I expecting trouble?
00:40:53.780 --> 00:40:56.640
The, the first equation
says that the eigenvalues
00:40:56.640 --> 00:40:57.320
add to zero.
00:40:59.920 --> 00:41:01.170
So there's a plus and a minus.
00:41:01.170 --> 00:41:02.211
So I take the eigenvalue.
00:41:04.360 --> 00:41:06.490
But then the second
equation says
00:41:06.490 --> 00:41:09.450
that the product is plus one.
00:41:09.450 --> 00:41:10.880
We're in trouble.
00:41:10.880 --> 00:41:14.810
But there's a way out.
00:41:14.810 --> 00:41:17.860
So how -- let's do
the usual stuff.
00:41:17.860 --> 00:41:21.700
Look at determinant
of Q minus lambda I.
00:41:21.700 --> 00:41:27.230
So I'll just follow the
rules, take the determinant,
00:41:27.230 --> 00:41:31.940
subtract lambda from the
diagonal, where I had zeros,
00:41:31.940 --> 00:41:34.400
the rest is the same.
00:41:34.400 --> 00:41:37.180
Rest of Q is just copied.
00:41:37.180 --> 00:41:38.540
Compute that determinant.
00:41:38.540 --> 00:41:42.720
OK, so what does that
determinant equal?
00:41:42.720 --> 00:41:47.740
Lambda squared minus
minus one plus what?
00:41:51.930 --> 00:41:54.060
What's up?
00:41:54.060 --> 00:41:56.020
There's my equation.
00:41:56.020 --> 00:41:59.050
My equation for the
eigenvalues is lambda
00:41:59.050 --> 00:42:00.950
squared plus one equals zero.
00:42:00.950 --> 00:42:04.620
What are the eigenvalues
lambda one and lambda two?
00:42:04.620 --> 00:42:27.340
They're I, whatever that
is, and minus it, right.
00:42:27.340 --> 00:42:30.240
Those are the right numbers.
00:42:30.240 --> 00:42:36.270
To be real numbers even though
the matrix was perfectly real.
00:42:36.270 --> 00:42:38.190
So this can happen.
00:42:41.060 --> 00:42:44.950
Complex numbers are going to --
have to enter eighteen oh six
00:42:44.950 --> 00:42:51.500
at this moment.
00:42:51.500 --> 00:42:54.600
Boo, right.
00:42:54.600 --> 00:42:56.950
All right.
00:42:56.950 --> 00:43:02.970
If I just choose good
matrices that have real
00:43:02.970 --> 00:43:08.500
supposed to have here?
00:43:24.360 --> 00:43:35.590
We do know a little
information about the,
00:43:35.590 --> 00:43:38.672
the two complex numbers.
00:43:38.672 --> 00:43:40.380
They're complex
conjugates of each other.
00:43:45.850 --> 00:43:51.590
If, if lambda is an
eigenvalue, then when I change,
00:43:51.590 --> 00:43:54.360
when I go -- you remember
what complex conjugates are?
00:43:54.360 --> 00:43:57.170
You switch the sign
of the imaginary part.
00:43:57.170 --> 00:43:59.910
Well, this was only
imaginary, had no real part,
00:43:59.910 --> 00:44:02.690
so we just switched its sign.
00:44:02.690 --> 00:44:06.720
So that eigenvalues
come in pairs like that,
00:44:06.720 --> 00:44:08.440
but they're complex.
00:44:08.440 --> 00:44:11.040
A complex conjugate pair.
00:44:11.040 --> 00:44:14.530
And that can happen with
a perfectly real matrix.
00:44:14.530 --> 00:44:17.060
And as a matter of fact --
00:44:17.060 --> 00:44:18.930
so that was my,
my point earlier,
00:44:18.930 --> 00:44:23.240
that if a matrix was
symmetric, it wouldn't happen.
00:44:23.240 --> 00:44:27.090
So if we stick to matrices that
are symmetric or, like, close
00:44:27.090 --> 00:44:32.610
to symmetric, then the
eigenvalues will stay real.
00:44:32.610 --> 00:44:35.400
But if we move far
away from symmetric --
00:44:35.400 --> 00:44:39.520
and that's as far as you can
move, because that matrix is --
00:44:39.520 --> 00:44:44.470
how is Q transpose related
to Q for that matrix?
00:44:44.470 --> 00:44:46.760
That matrix is anti-symmetric.
00:44:46.760 --> 00:44:49.520
Q transpose is minus Q.
00:44:49.520 --> 00:44:52.070
That's the very
opposite of symmetry.
00:44:52.070 --> 00:44:54.570
When I flip across
the diagonal I get --
00:44:54.570 --> 00:44:56.350
I reverse all the signs.
00:44:56.350 --> 00:45:00.840
Those are the guys that have
pure imaginary eigenvalues.
00:45:00.840 --> 00:45:03.320
So they're the extreme case.
00:45:03.320 --> 00:45:07.380
And in between are,
are matrices that
00:45:07.380 --> 00:45:10.990
are not symmetric or
anti-symmetric but,
00:45:10.990 --> 00:45:13.710
but they have partly
a symmetric part
00:45:13.710 --> 00:45:15.110
and an anti-symmetric part.
00:45:15.110 --> 00:45:16.660
OK.
00:45:16.660 --> 00:45:23.220
So I'm doing a bunch of examples
here to show the possibilities.
00:45:23.220 --> 00:45:28.320
The good possibilities being
perpendicular eigenvectors,
00:45:28.320 --> 00:45:30.360
real eigenvalues.
00:45:30.360 --> 00:45:33.910
The bad possibilities
being complex eigenvalues.
00:45:33.910 --> 00:45:37.360
We could say that's bad.
00:45:37.360 --> 00:45:39.455
There's another even worse.
00:45:42.380 --> 00:45:45.740
I'm getting through the,
the bad things here today.
00:45:45.740 --> 00:45:51.750
Then, then the next
lecture can, can,
00:45:51.750 --> 00:45:56.750
can be like pure happiness.
00:45:56.750 --> 00:45:57.720
OK.
00:45:57.720 --> 00:46:03.260
Here's one more bad
thing that could happen.
00:46:03.260 --> 00:46:05.940
So I, again, I'll do
it with an example.
00:46:05.940 --> 00:46:10.740
Suppose my matrix is, suppose
I take this three three one
00:46:10.740 --> 00:46:13.140
and I change that guy to zero.
00:46:18.150 --> 00:46:21.280
What are the eigenvalues
of that matrix?
00:46:21.280 --> 00:46:22.740
What are the eigenvectors?
00:46:22.740 --> 00:46:25.380
This is always our question.
00:46:25.380 --> 00:46:26.860
Of course, the
next section we're
00:46:26.860 --> 00:46:29.580
going to show why
are, why do we care.
00:46:29.580 --> 00:46:33.121
But for the moment, this
lecture is introducing
00:46:33.121 --> 00:46:33.620
them.
00:46:33.620 --> 00:46:35.990
And let's just find them.
00:46:35.990 --> 00:46:36.640
OK.
00:46:36.640 --> 00:46:40.530
What are the eigenvalues
of that matrix?
00:46:40.530 --> 00:46:45.910
Let me tell you -- at a glance
we could answer that question.
00:46:45.910 --> 00:46:49.490
Because the matrix
is triangular.
00:46:49.490 --> 00:46:53.540
It's really useful to know --
if you've got properties like
00:46:53.540 --> 00:46:55.320
a triangular matrix.
00:46:55.320 --> 00:46:57.980
It's very useful to know
you can read the eigenvalues
00:46:57.980 --> 00:46:58.610
off.
00:46:58.610 --> 00:47:01.920
They're right on the diagonal.
00:47:01.920 --> 00:47:05.470
So the eigenvalue is
three and also three.
00:47:05.470 --> 00:47:07.290
Three is a repeated eigenvalue.
00:47:07.290 --> 00:47:09.330
But let's see that happen.
00:47:09.330 --> 00:47:10.660
Let me do it right.
00:47:10.660 --> 00:47:15.370
The determinant of A minus
lambda I, what I always
00:47:15.370 --> 00:47:17.119
have to do is this determinant.
00:47:17.119 --> 00:47:18.660
I take away lambda
from the diagonal.
00:47:21.600 --> 00:47:24.070
I leave the rest.
00:47:24.070 --> 00:47:28.450
I compute the determinant,
so I get a three minus lambda
00:47:28.450 --> 00:47:32.090
times a three minus lambda.
00:47:32.090 --> 00:47:35.680
And nothing.
00:47:35.680 --> 00:47:38.860
So that's where the
triangular part came in.
00:47:38.860 --> 00:47:41.260
Triangular part, the one
thing we know about triangular
00:47:41.260 --> 00:47:44.660
matrices is the determinant
is just the product down
00:47:44.660 --> 00:47:45.890
the diagonal.
00:47:45.890 --> 00:47:48.990
And in this case, it's
this same, repeated --
00:47:48.990 --> 00:47:51.775
so lambda one is one --
00:47:51.775 --> 00:47:53.900
sorry, lambda one is three
and lambda two is three.
00:47:53.900 --> 00:47:58.540
That was easy.
00:47:58.540 --> 00:48:05.900
I mean, no -- why should I
be pessimistic about a matrix
00:48:05.900 --> 00:48:10.650
whose eigenvalues can
be read off right away?
00:48:10.650 --> 00:48:14.820
The problem with this matrix
is in the eigenvectors.
00:48:14.820 --> 00:48:16.450
So let's go to the eigenvectors.
00:48:16.450 --> 00:48:18.960
So how do I find
the eigenvectors?
00:48:18.960 --> 00:48:22.010
I'm looking for a
couple of eigenvectors.
00:48:22.010 --> 00:48:22.800
Singular, right?
00:48:22.800 --> 00:48:31.590
It's supposed to be singular.
00:48:31.590 --> 00:48:40.530
And then it's got some
vectors -- which it is.
00:48:40.530 --> 00:48:59.410
So it's got some vector
x in the null space.
00:49:12.890 --> 00:49:15.670
And what, what's the, what's
-- give me a basis for the null
00:49:15.670 --> 00:49:18.360
space for that guy.
00:49:18.360 --> 00:49:21.860
Tell me, what's a vector x
in the null space, so that'll
00:49:21.860 --> 00:49:25.710
be the, the eigenvector
that goes with lambda one
00:49:25.710 --> 00:49:27.030
equals three.
00:49:27.030 --> 00:49:31.030
The eigenvector is -- so
what's in the null space?
00:49:31.030 --> 00:49:32.180
One zero, is it?
00:49:34.700 --> 00:49:35.200
Great.
00:49:38.730 --> 00:49:40.600
Now, what's the
other eigenvector?
00:49:40.600 --> 00:49:47.960
What's, what's the eigenvector
that goes with lambda two?
00:49:47.960 --> 00:49:51.620
Well, lambda two is three again.
00:49:51.620 --> 00:49:53.150
So I get the same thing again.
00:49:53.150 --> 00:49:55.500
Give me another vector --
00:49:55.500 --> 00:49:57.750
I want it to be independent.
00:49:57.750 --> 00:49:59.340
If I'm going to
write down an x2,
00:49:59.340 --> 00:50:02.440
I'm never going to let
it be dependent on x1.
00:50:02.440 --> 00:50:05.240
I'm looking for
independent eigenvectors,
00:50:05.240 --> 00:50:08.720
and what's the conclusion?
00:50:08.720 --> 00:50:11.010
There isn't one.
00:50:11.010 --> 00:50:17.050
This is a degenerate matrix.
00:50:17.050 --> 00:50:22.960
It's only got one line of
eigenvectors instead of two.
00:50:22.960 --> 00:50:27.250
It's this possibility
of a repeated eigenvalue
00:50:27.250 --> 00:50:34.390
opens this further possibility
of a shortage of eigenvectors.
00:50:34.390 --> 00:50:43.310
And so there's no second
independent eigenvector x2.
00:50:43.310 --> 00:50:48.010
So it's a matrix, it's
a two by two matrix,
00:50:48.010 --> 00:50:51.850
but with only one
independent eigenvector.
00:50:51.850 --> 00:50:56.910
So that will be --
the matrices that --
00:50:56.910 --> 00:51:01.830
where eigenvectors are --
don't give the complete story.
00:51:01.830 --> 00:51:02.330
OK.
00:51:02.330 --> 00:51:05.930
My lecture on Monday will
give the complete story
00:51:05.930 --> 00:51:11.290
for all the other matrices.
00:51:11.290 --> 00:51:12.360
Thanks.
00:51:12.360 --> 00:51:16.640
Have a good weekend.
00:51:16.640 --> 00:51:22.480
A real New England weekend.