21. Eigenvalues and Eigenvectors

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<p>MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: http://ocw.mit.edu/18-06S05
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<p>21. Eigenvalues and Eigenvectors</p>

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<p>00:07
OK.
00:08
So this is the first lecture on eigenvalues and eigenvectors,
00:16
and that's a big subject that will take up
00:19
most of the rest of the course.
00:23
It's, again, matrices are square and we're looking now
00:31
for some special numbers, the eigenvalues,
00:34
and some special vectors, the eigenvectors.
00:38
And so this lecture is mostly about what are these numbers,
00:45
and then the other lectures are about how do we use them,
00:48
why do we want them.
00:50
OK, so what's an eigenvector?
00:54
Maybe I'll start with eigenvector.
00:56
What's an eigenvector?
00:57
So I have a matrix A.
01:02
OK.
01:05
What does a matrix do?
01:07
It acts on vectors.
01:09
It multiplies vectors x.
01:12
So the way that matrix acts is in goes a vector x and out
01:21
comes a vector Ax.
01:22
It's like a function.
01:23
With a function in calculus, in goes
01:27
a number x, out comes f(x).
01:31
Here in linear algebra we're up in more dimensions.
01:34
In goes a vector x, out comes a vector Ax.
01:39
And the vectors I'm specially interested in
01:43
are the ones the come out in the same direction
01:47
that they went in.
01:50
That won't be typical.
01:52
Most vectors, Ax is in -- points in some different direction.
01:59
But there are certain vectors where Ax comes out parallel
02:04
to x.
02:07
And those are the eigenvectors.
02:09
So Ax parallel to x.
02:18
Those are the eigenvectors.
02:23
And what do I mean by parallel?
02:25
Oh, much easier to just state it in an equation.
02:29
Ax is some multiple -- and everybody calls that multiple
02:36
lambda -- of x.
02:38
That's our big equation.
02:43
We look for special vectors -- and remember most vectors
02:46
won't be eigenvectors --
02:49
that -- for which Ax is in the same direction as x,
02:53
and by same direction I allow it to be the very opposite
02:57
direction, I allow lambda to be negative or zero.
03:03
Well, I guess we've met the eigenvectors that
03:07
have eigenvalue zero.
03:10
Those are in the same direction, but they're --
03:13
in a kind of very special way.
03:18
So this -- the eigenvector x.
03:22
Lambda, whatever this multiplying factor
03:24
is, whether it's six or minus six or zero or even
03:31
some imaginary number, that's the eigenvalue.
03:34
So there's the eigenvalue, there's the eigenvector.
03:38
Let's just take a second on eigenvalue zero.
03:42
From the point of view of eigenvalues, that's no special
03:45
deal.
03:46
That's, we have an eigenvector.
03:48
If the eigenvalue happened to be zero,
03:51
that would mean that Ax was zero x, in other words zero.
03:56
So what would x, where would we look for -- what are the x-s?
04:00
What are the eigenvectors with eigenvalue zero?
04:04
They're the guys in the null space, Ax equals zero.
04:08
So if our matrix is singular, let me write this down.
04:12
If, if A is singular, then that -- what does singular mean?
04:23
It means that it takes some vector x into zero.
04:29
Some non-zero vector, that's why --
04:31
will be the eigenvector into zero.
04:33
Then lambda equals zero is an eigenvalue.
04:42
But we're interested in all eigenvalues now,
04:46
lambda equals zero is not, like, so special anymore.
04:50
OK.
04:51
So the question is, how do we find these x-s and lambdas?
04:57
And notice -- we don't have an equation Ax equal B anymore.
05:01
I can't use elimination.
05:05
I've got two unknowns, and in fact they're
05:07
multiplied together.
05:08
Lambda and x are both unknowns here.
05:12
So, we need to, we need a good idea of how to find them.
05:18
But before I, before I do that, and that's
05:23
where determinant will come in, can I just
05:26
give you some matrices?
05:29
Like here you go.
05:31
Take the matrix, a projection matrix.
05:35
OK.
05:35
So suppose we have a plane and our matrix P is --
05:42
what I've called A, now I'm going to call it P
05:45
for the moment, because it's --
05:47
I'm thinking OK, let's look at a then this,
05:51
this other new matrix, I just have an Ax, projection matrix.
05:57
What are the eigenvalues of a projection matrix?
06:01
So that's my question.
06:03
What are the x-s, the eigenvectors, and the lambdas,
06:07
the eigenvalues, thing,4 but the roots of that quadratic for --
06:13
and now let me say a projection matrix.
06:17
My, my point is that we --
06:19
before we get into determinants and, and formulas
06:23
and all that stuff, let's take some matrices
06:26
where we know what they do.
06:28
We know that if we take a vector b, what this matrix does
06:34
is it projects it down to Pb.
06:38
So is b an eigenvector in, in that picture?
06:43
Is that vector b an eigenvector?
06:48
No.
06:50
Not so, so b is not an eigenvector
06:53
c- because Pb, its projection, is in a different direction.
06:59
So now tell me what vectors are eigenvectors of P?
07:04
What vectors do get projected in the same direction that they
07:09
start?
07:09
So, so answer, tell me some x-s.
07:12
Do you see what3 so it's if Ax equals lambda x,
07:16
In this picture, where could I start with a vector b or x,
07:21
do its projection, and end up in the same direction?
07:27
Well, that would happen if the vector was right in that plane
07:33
already.
07:34
If the vector x was -- so let the vector x --
07:41
so any vector, any x in the plane will be an eigenvector.
07:49
And what will happen when I multiply by P,
07:53
when I project a vector x --
07:57
I called it b here, because this is our familiar picture,
08:00
but now I'm going to say that b was no good for, for the,
08:03
for our purposes.
08:05
I'm interested in a vector x that's actually in the plane,
08:09
and I project it, and what do I get back?
08:14
x, of course.
08:15
Doesn't move. can be complex numbers.
08:17
So any x in the plane is unchanged by P,
08:24
and what's that telling me?
08:26
That's telling me that x is an eigenvector,
08:28
and it's also telling me what's the eigenvalue, which is --
08:32
just compare it with that.
08:34
The eigenvalue, the multiplier, is just one.
08:39
Good.
08:41
So we have actually a whole plane of eigenvectors.
08:45
Now I ask, are there any other eigenvectors?
08:49
And I expect the answer to be yes,
08:51
because I would like to get three,
08:53
if I'm in three dimensions, I would
08:55
like to hope for three independent eigenvectors, two
08:59
of them in the plane and one not in the plane.
09:02
OK.
09:03
So this guy b that I drew there was not any good.
09:06
What's the right eigenvector that's not in the plane?
09:11
The, the good one is the one that's perpendicular to the
09:16
plane.
09:17
There's an, another good x, because what's the projection?
09:22
So these are eigenvectors.
09:24
Another guy here would be another eigenvector.
09:27
But now here is another one. two.
09:29
Any x that's perpendicular to the plane,
09:37
what's Px for that, for that, vector?
09:43
What's the projection of this guy perpendicular to the plane?
09:47
It is zero, of course.
09:50
So -- there's the null space.
09:53
Px and n- for those guys are zero, or zero x if we like,
09:58
and the eigenvalue is zero.
10:02
So my answer to the question is, what are the eigenvalues for
10:05
In our example, the one we worked out,
10:07
a projection matrix?
10:08
There they are.
10:09
One and zero.
10:11
OK.
10:13
We know projection matrices.
10:17
We can write them down as that A, A transpose, A inverse,
10:22
A transpose thing, but without doing that from the picture
10:28
we could see what are the eigenvectors.
10:31
OK.
10:32
Are there other matrices?
10:34
Let me take a second example.
10:36
How about a permutation matrix?
10:39
What about the matrix, I'll call it A now.
10:41
Zero one, one zero.
10:42
A equals zero one one zero, that had eigenvalue one and
10:49
Can you tell me a vector x --
10:53
see, we'll have a system soon enough,
10:55
so I, I would like to just do these e-
10:58
these couple of examples, just to see the picture before we,
11:03
before we let it all, go into a system
11:11
where that, matrix isn't anything special.
11:13
Because it is special.
11:15
And what, so what vector could I multiply by and end up
11:20
in the same direction?
11:21
Can you spot an eigenvector for this guy?
11:25
That's a matrix that permutes x1 and x2, right?
11:32
It switches the two components of x.
11:37
How could the vector with its x2 x1, with --
11:43
permuted turn out to be a multiple of x1 x2,
11:49
the vector we start with?
11:51
Can you tell me an eigenvector here for this guy?
11:53
x equal -- what is -- actually, can you tell me one vector that
11:56
which is lambda x, and I have a three x,
11:58
And of course you -- everybody knows that they're -- what,
12:01
has eigenvalue one?
12:01
So what, what vector would have eigenvalue one,
12:03
just above what we2 found here. so that if I,
12:05
if I permute it it doesn't change? right?
12:09
There, that could be one one, thanks.
12:13
One one.
12:14
OK, take that vector one one.
12:16
That will be an eigenvector, because if I do Ax
12:20
I get one one.
12:23
So that's the eigenvalue is one.
12:27
Great.
12:28
That's one eigenvalue.
12:30
But I have here a two by two matrix,
12:33
and I figure there's going to be a second eigenvalue.
12:40
And eigenvector.
12:42
Now, what about that?
12:44
What's a vector, OK, maybe we can just, like, guess it.
12:50
A vector that the other -- actually,
12:55
this one that I'm thinking of is going to be a vector that has
12:58
eigenvalue minus one.
13:03
That's going to be my other eigenvalue for this matrix.
13:06
It's a -- notice the nice positive or not negative
13:09
matrix, but an eigenvalue is going to come out negative.
13:13
And can you guess, spot the x that will work for
13:15
Times x is supposed to give me zero, right? that?
13:18
So I want a, a vector.
13:20
When I multiply by A, which reverses the two components,
13:27
I want the thing to come out minus the original.
13:31
So what shall I send in in that case?
13:34
If I send in negative one one.
13:40
Then when I apply A, I get I do that multiplication,
13:47
and I get one negative one, so it reversed sign.
13:52
So Ax is -x.
13:55
Lambda is minus one.
13:57
Ax -- so Ax was x there and Ax is minus x here.
14:04
Can I just mention, like, jump ahead, have,
14:08
give a perfectly innocent-looking quadratic
14:10
and point out a special little fact about eigenvalues.
14:14
n by n matrices will have n eigenvalues.
14:18
And I get this matrix4 zero zero zero one,
14:23
And it's not like -- suppose n is three or four or more.
14:28
It's not so easy to find them.
14:32
We'd have a third degree or a fourth degree or an n-th degree
14:35
equation.
14:36
But here's one nice fact.
14:37
There, there's one pleasant fact. we --
14:40
the eigenvalues came out four and two.
14:42
That the sum of the eigenvalues equals the sum
14:45
down the diagonal.
14:46
That's called the trace, and I put that in the lecture
14:50
Now I add three I to that matrix. content specifically.
14:54
So this is a neat fact, the fact that sthe sum of the lambdas,
15:03
add up the lambdas, equals the sum --
15:08
what would you like me to, shall I write that down?
15:11
What I'm want to say in words is the sum down the diagonal of A.
15:17
Shall I write a11+a22+...+ ann.
15:18
That's add up the diagonal entries.
15:27
In this example, it's zero.
15:31
In other words, once I found this eigenvalue of one,
15:36
I knew the other one had to be minus one
15:39
in this two by two case, because in the two by two case, which
15:43
is a good one to, to, play with, the trace tells you
15:51
right away what the other eigenvalue is.
15:54
So if I tell you one eigenvalue, you can tell me the
15:56
other one.
15:57
We'll, we'll have that -- we'll, minus one and eigenvectors one
16:02
one and eigenvector minus one we'll see that again.
16:05
OK.
16:06
Now can I --
16:07
I could give more examples, but maybe it's
16:11
time to face the, the equation, Ax equal lambda x, and figure
16:16
how are we going to find x and lambda.
16:19
And that is lambda one times lambda3
16:22
OK.
16:23
So this, so the question now is how
16:25
to find eigenvalues and eigenvectors.
16:30
How to solve, how to solve Ax equal lambda x from the three
16:35
x, so it's just I mean, when we've got two unknowns
16:43
both in the equation.
16:47
OK.
16:48
Here's the trick.
16:51
Simple idea.
16:53
Bring this onto the same side.
16:58
Rewrite.
16:59
Bring this over as A minus lambda times the identity x
17:05
One. equals zero.
17:09
Right?
17:10
I have Ax minus lambda x, so I brought that over
17:14
and I've got zero left on the, on the right-hand side.
17:14
What's the relation between that problem and --
17:16
let me write
17:19
OK.
17:19
I don't know lambda and I don't know x, but I do know something
17:23
here.
17:26
What I know is if I, if I'm going
17:28
to be able to solve this thing, for some x that's not the zero
17:31
vector, that's not, that's a useless eigenvector,
17:35
doesn't count.
17:37
What I know now is that this matrix must be what?
17:44
If I'm going to be -- if there is an x --
17:48
I don't -- right now I don't know what it is.
17:51
I'm going to find lambda first, actually.
17:54
And -- but if there is an x, it tells me that this matrix,
18:00
this special combination, which is like the matrix A with
18:05
lambda -- shifted by lambda, shifted by lambda I,
18:10
that it has to be singular.
18:14
This matrix must be singular, otherwise
18:17
the only x would be the zero x, and zero matrix.OK.
18:21
So this is singular.
18:22
And what do I now know about singular matrices?
18:30
So, so take three away.
18:31
Their determinant is zero.
18:35
So I've -- so from the fact that that has to be singular,
18:38
I know that the determinant of A minus lambda I has to be zero.
18:48
And that, now I've got x out of it.
18:52
I've got an equation for lambda, that the key equation --
18:55
it's called the characteristic equation or the eigenvalue
19:01
equation.
19:04
And that -- in other words, I'm now in a position to find
19:08
lambda first.
19:10
So -- the idea will be to find lambda first.
19:18
And actually, I won't find one lambda,
19:21
I'll find N different lambdas.
19:24
Well, n lambdas, maybe not n different ones.
19:28
A lambda could be repeated.
19:31
A repeated lambda is the source of all trouble in 18.06.
19:38
So, let's hope for the moment that they're not repeated.
19:43
There, there they were different, right?
19:48
One and minus one in that, in that, for that permutation.
19:51
OK.
19:52
So and after I found this lambda, can I just look ahead?
19:59
How I going to find x?
20:02
After I have found this lambda, the lambda being this --
20:06
one of the numbers that makes this matrix singular.
20:10
Their product was eight.
20:11
Then of course finding x is just by elimination.
20:15
Right?
20:15
It's just -- now I've got a singular matrix,
20:18
I'm looking for the null space.
20:20
We're experts at finding the null space.
20:24
You know, you do elimination, you identify
20:26
the, the, the pivot columns and so on, you're --
20:31
and, give values to the free variables.
20:36
Probably there'll only be one free variable.
20:39
We'll give it the value one, like there.
20:42
And we find the other variable.
20:44
OK.
20:45
So let's -- find the x second will be a doable job.
20:52
That's my big equation for x.
20:54
Let's go, let's look at the first job of finding lambda.
20:58
Can I take another example?
20:59
OK.
21:00
And let's, let's work that one out.
21:02
OK.
21:02
So let me take the example, say, let me make it easy.
21:07
it's just sitting there.
21:09
Three three one and one. what do you
21:12
know about the complex numbers?
21:14
So I've made it easy.
21:16
I've made it two by two.
21:19
I've made it symmetric.
21:20
And I even made it constant down the diagonal.
21:24
That a matrix, a perfectly real matrix could
21:27
So that -- so the more, like, special properties I stick
21:32
into the matrix, the more special outcome I get
21:36
for the eigenvalues.
21:38
For example, this symmetric matrix,
21:42
I know that it'll come out with real eigenvalues. one.
21:48
The eigenvalues will turn out to be nice real numbers.
21:52
And up in our previous example, that was a symmetric matrix.
21:57
Actually, while we're at it, that was a symmetric matrix.
22:02
Its eigenvalues were nice real numbers, one and minus one.
22:05
And do you notice anything about its eigenvectors?
22:07
And what do you notice?
22:08
Anything particular about those two vectors, one one and minus
22:11
And now comes that thing that I wanted to be reminded of.
22:14
one one?
22:15
They just happen to be -- no, I can't say they just happen
22:18
to be, because that's the whole point,
22:20
is that they had to be -- what?
22:23
What are they?
22:25
They're perpendicular.
22:27
The vector, when I -- if I see a vector one one and a one --
22:30
and a minus one one, my mind immediately takes that
22:33
dot product.
22:33
It's zero. what's the determinant of that matrix?
22:36
Those vectors are perpendicular.
22:38
That'll happen here too.
22:40
Well, let's find the eigenvalues.
22:42
Actually, oh, my example's too easy.
22:45
My example is too easy.
22:48
Let me tell you in advance what's going to happen.
22:53
May I?
22:56
Or shall I do the determinant of A minus lambda,
22:58
and then point out at the end?
23:00
Will you remind me at the -- after I've found
23:03
the eigenvalues to say why they were -- why they were easy from
23:10
That -- it had to be eight, because we factored into lambda
23:17
the, from the example we did?
23:20
OK, let's do the job here.
23:23
Let's compute determinant of A minus lambda I.
23:27
So that's a determinant.
23:29
And what's, what is this thing?
23:32
It's the matrix A with lambda removed from the diagonal.
23:36
for this matrix?
23:38
So the diagonal matrix is shifted,
23:40
and then I'm taking the determinant.
23:43
OK.
23:44
So I multiply this out.
23:47
So what is that determinant?
23:49
Do you notice, I didn't take lambda away
23:52
from all the entries.
23:55
It's lambda I, so it's lambda along the
23:56
Lambda plus three x. diagonal.
23:58
So I get three minus lambda squared and then minus one,
24:04
right?
24:06
And I want that to be zero.
24:08
And what is A minus lambda I x?
24:09
Well, I'm going to simplify it.
24:11
And what will I get?
24:11
So if I multiply this out, I get lambda squared minus six
24:11
What's -- how is this matrix related to that matrix?
24:11
lambda plus what?
24:11
Plus eight.
24:11
But it's out there.
24:11
And that I'm going to set to zero.
24:11
And I'm going to solve it.
24:11
So and it's, it's a quadratic equation.
24:11
I can use factorization, I can use the quadratic formula.
24:11
I'll get two lambdas.
24:12
Before I do it, tell me what's that number six that's
24:30
showing up in this equation?
24:41
It's the trace.
24:48
That number six is three plus three.
25:03
And while we're at it, what's the number eight that's
25:05
showing up in this equation?
25:08
It's the determinant.
25:10
That our matrix has determinant eight.
25:13
So in a two by two case, it's really nice.
25:16
It's lambda squared minus the trace times lambda --
25:21
the trace is the linear coefficient --
25:24
and plus the determinant, the constant term.
25:27
OK.
25:28
So let's -- can, can we find the roots?
25:32
I guess the easy way is to factor that as something times
25:36
something.
25:37
If we couldn't factor it, then we'd have to use the old
25:41
b²-4ac formula, but I, I think we can factor that into lambda
25:46
minus what times lambda minus what?
25:50
Can you do that factorization?
25:54
Four and two?
25:57
Lambda minus four times lambda minus two.
25:59
So the, the eigenvalues are four and two.
26:00
So the eigenvalues are -- one eigenvalue, lambda one,
26:01
Now I'm looking for x, the eigenvector. let's say,
26:01
is four.
26:01
Lambda two, the other eigenvalue, is two.
26:01
The eigenvalues are four and two.
26:03
And then I can go for the eigenvectors.
26:08
Suppose I have a matrix A, and Ax equal lambda x.
26:15
equals zero.
26:16
You see I got the eigenvalues first.
26:20
So if they, if this had eigenvalue lambda,
26:25
Four and two.
26:26
Now for the eigenvectors.
26:29
So what are the eigenvectors?
26:31
They're these guys in the null space when
26:34
I take away, when I make the matrix singular by taking
26:40
four I or two I away.
26:42
So we're -- we got to do those separately.
26:46
I'll -- let me find the eigenvector for four first.
26:50
So I'll subtract four, so A minus four I is --
26:57
so taking four away will put minus ones there.
27:04
And what's the point about that matrix?
27:07
If four is an eigenvalue, then A minus four
27:10
I had better be a what kind of matrix?
27:13
Singular.
27:14
If that matrix isn't singular, the four wasn't correct.
27:17
But we're OK, that matrix is singular.
27:21
And what's the x now?
27:23
The x is in the null space.
27:25
So what's the x1 that goes with, with the lambda one?
27:28
eigenvalue, eigenvector, eigenvalue for this,
27:32
So that A -- so this is -- now I'm doing A x1 is lambda one
27:38
x1.
27:41
So I took A minus lambda one I, that's this matrix,
27:45
and now I'm looking for the x1 in its null space,
27:48
and who is he?
27:49
What's the vector x in the null space?
27:51
Of course it's one one.
27:53
So that's the eigenvector that goes with that eigenvalue.
27:56
So, so now --
27:57
Let's just spend one more minute on this bad
28:00
Now how about the eigenvector that
28:02
goes with the other eigenvalue?
28:04
Can I do that with, with erasing?
28:06
I take A minus two I.
28:08
So now I take two away from the diagonal,
28:11
and that leaves me with a one and a one.
28:15
So A minus two I has again produced a singular matrix,
28:19
as it had to.
28:21
I'm looking for the null space of that guy.
28:24
What vector is in its null space?
28:27
Well, of course, a whole line of vectors.
28:32
So when I say the eigenvector, I'm not speaking correctly.
28:35
There's a whole line of eigenvectors, and you just --
28:38
I just want a basis.
28:40
And for a line I just want one vector.
28:43
But -- You could, you're -- there's some freedom
28:48
in choosing that one, but choose a reasonable one.
28:50
What's a vector in the null space of that?
28:54
Well, the natural vector to pick as the eigenvector
28:58
with, with lambda two is minus one one.
29:05
If I did elimination on that vector
29:07
and set that, the free variable to be one,
29:10
I would get minus one and get that eigenvector.
29:15
So you see then that I've got eigenvector,
29:22
Now the other neat fact is that the determinant,
29:32
How are those two matrices related?
29:40
Well, one is just three I more than the other one, right? two.
29:46
I just took that matrix and I --
29:52
I took this matrix and I added three I.
30:29
So my question is, what happened to the minus four times lambda
30:31
minus two. eigenvalues and what happened to the eigenvectors?
30:34
That's the, that's like the question we keep asking now
30:37
in this chapter.
30:39
If I, if I do something to the matrix, what happens if I --
30:42
or I know something about the matrix,
30:44
what's the what's the conclusion for its eigenvectors
30:48
and eigenvalues?
30:49
Because -- those eigenvalues and eigenvectors are going to tell
30:54
us important information about the matrix.
30:57
And here what are we seeing?
31:00
What's happening to these eigenvalues, one and minus
31:04
one, when I add three I?
31:10
It just added three to the eigenvalues.
31:13
I got four and two, three more than one and minus
31:16
one.
31:18
What happened to the eigenvectors?
31:19
Nothing at all.
31:21
One one is -- and minus -- and one -- and minus one one are --
31:25
is still the eigenvectors.
31:27
In other words, simple but useful observation.
31:34
If I add three I to a matrix, its eigenvectors don't change
31:40
and its eigenvalues are three bigger.
31:42
Let's, let's just see why.
31:44
Let me keep all this on the same board. but just so you see --
32:02
so I'll try to do that.
32:08
this has eigenvalue lambda plus three.
32:39
And x, the eigenvector, is the same x for both matrices.
32:46
OK.
32:48
So that's, great.
32:53
Of course, it's special.
32:54
We got the new matrix by adding three I.
32:57
Suppose I had added another matrix.
32:59
Suppose I know the eigenvalues and eigenvectors of A.
33:02
So I took A minus lambda I x, and what kind of a matrix I
33:07
So this is, this, this little board
33:09
here is going to be not so great.
33:17
Suppose I have a matrix A and it has an eigenvector x with
33:21
an eigenvalue lambda.
33:23
You remember, I solve A minus lambda I x
33:28
And now I add on some other matrix.
33:31
So, so what I'm asking you is, if you know the eigenvalues
33:34
of A and you know the eigenvalues of B,
33:38
let me say suppose B -- so this is if -- let me put an if here.
33:43
If Ax equals lambda x, fine, and B has, eigenvalues,
33:49
has eigenvalues --
33:52
what shall we call them?
33:56
Alpha, alpha one and alpha --
34:02
let's say --
34:05
I'll use alpha for the eigenvalues of B
34:07
for no good reason.
34:11
What a- you see what I'm going to ask is, how about A plus B?
34:20
Let me, let me give you the, let me give you,
34:24
what you might think first.
34:27
OK.
34:28
If Ax equals lambda x and if B has an eigenvalue alpha,
34:35
then I allowed to say -- what's the matter with this argument?
34:40
That gave us the constant term eight.
34:43
It's wrong.
34:44
What I'm going to write up is wrong.
34:47
I'm going to say Bx is alpha x.
34:50
Add those up, and you get A plus B x equals lambda plus alpha x.
34:55
So you would think that if you know the eigenvalues of A
35:00
and you knew the eigenvalues of B,
35:03
then if you added you would know the eigenvalues of A plus B.
35:07
But that's false.
35:11
A plus B -- well, when B was three I, that worked great.
35:17
But this is not so great.
35:21
And what's the matter with that argument there?
35:25
We have no reason to believe that x is also
35:32
an eigenvector of
35:33
B has some eigenvalues, B. but it's
35:38
got some different eigenvectors normally.
35:43
It's a different matrix.
35:45
I don't know anything special.
35:47
If I don't know anything special, then as far as I know,
35:50
it's got some different eigenvector y,
35:52
and when I add I get just rubbish.
35:55
I mean, I get --
35:57
I can add, but I don't learn anything.
35:59
So not so great is A plus B.
36:05
Or A times B.
36:09
Normally the eigenvalues of A plus B
36:12
or A times B are not eigenvalues of A plus eigenvalues of B.
36:18
Ei- eigenvalues are not, like, linear.
36:22
Or -- and they don't multiply.
36:24
Because, eigenvectors are usually different
36:27
and, and there's just no way to find out
36:31
what A plus B does to affect
36:33
What do I do now? it.
36:34
OK.
36:35
So that's, like, a caution.
36:41
Don't, if B is a multiple of the identity, great,
36:44
but if B is some general matrix, then for A plus B you've got
36:50
to find -- you've got to solve the eigenvalue problem.
36:54
Now I want to do another example that brings out a,
36:59
OK. another point about eigenvalues.
37:02
Let me make this example a rotation matrix.
37:06
possibility of complex numbers.
37:09
OK.
37:10
So here's another example.
37:13
So a rotate --
37:16
oh, I'd better call it Q.
37:21
I often use Q for, for, rotations
37:26
because those are the, like, very important examples
37:32
of orthogonal matrices.
37:34
Let me make it a ninety degree rotation.
37:38
So -- my matrix is going to be the one that rotates every
37:41
And that's the sum, that's lambda one plus lambda
37:43
vector by ninety degrees.
37:46
So do you remember that matrix?
37:49
It's the cosine of ninety degrees, which
37:51
is zero, the sine of ninety degrees,
37:54
which is one, minus the sine of ninety, the cosine of ninety.
38:03
So that matrix deserves the letter Q.
38:09
It's an orthogonal matrix, very, very orthogonal matrix.
38:15
Now I'm interested in its eigenvalues and eigenvectors.
38:21
Two by two, it can't be that tough.
38:24
We know that the eigenvalues add to zero.
38:30
Actually, we know something already here.
38:33
The eigen- what's the sum of the two eigenvalues?
38:35
Just tell me what I just said.
38:38
Zero, right.
38:40
From that trace business.
38:42
The sum of the eigenvalues is, is going to come out zero.
38:46
And the product of the eigenvalues,
38:48
did I tell you about the determinant being
38:50
the product of the eigenvalues?
38:50
No.
38:50
But that's a good thing to know.
38:50
We pointed out how that eight appeared in
38:51
the, in the quadratic equation. eigenvalues,
38:52
we can postpone that evil day,
39:07
So let me just say this.
39:23
The trace is zero plus zero, obviously.
39:49
And that was the determinant.
39:52
OK.
39:56
What I'm leading up to with this example
39:58
is that something's going to go wrong.
40:03
Something goes wrong for rotation
40:09
because what vector can come out parallel to itself
40:16
after a rotation?
40:18
If this matrix rotates every vector by ninety degrees,
40:23
what could be an eigenvector?
40:26
Do you see we're, we're, we're going to have trouble.
40:31
eigenvectors are --
40:36
Well.
40:36
Our, our picture of eigenvectors,
40:39
of, of coming out in the same direction that they went in,
40:42
there won't be it.
40:45
And with, and with eigenvalues we're going to have trouble.
40:48
From these equations.
40:50
Let's see.
40:51
Why I expecting trouble?
40:53
The, the first equation says that the eigenvalues
40:56
add to zero.
40:59
So there's a plus and a minus.
41:01
So I take the eigenvalue.
41:04
But then the second equation says
41:06
that the product is plus one.
41:09
We're in trouble.
41:10
But there's a way out.
41:14
So how -- let's do the usual stuff.
41:17
Look at determinant of Q minus lambda I.
41:21
So I'll just follow the rules, take the determinant,
41:27
subtract lambda from the diagonal, where I had zeros,
41:31
the rest is the same.
41:34
Rest of Q is just copied.
41:37
Compute that determinant.
41:38
OK, so what does that determinant equal?
41:42
Lambda squared minus minus one plus what?
41:51
What's up?
41:54
There's my equation.
41:56
My equation for the eigenvalues is lambda
41:59
squared plus one equals zero.
42:00
What are the eigenvalues lambda one and lambda two?
42:04
They're I, whatever that is, and minus it, right.
42:27
Those are the right numbers.
42:30
To be real numbers even though the matrix was perfectly real.
42:36
So this can happen.
42:41
Complex numbers are going to -- have to enter eighteen oh six
42:44
at this moment.
42:51
Boo, right.
42:54
All right.
42:56
If I just choose good matrices that have real
43:02
supposed to have here?
43:24
We do know a little information about the,
43:35
the two complex numbers.
43:38
They're complex conjugates of each other.
43:45
If, if lambda is an eigenvalue, then when I change,
43:51
when I go -- you remember what complex conjugates are?
43:54
You switch the sign of the imaginary part.
43:57
Well, this was only imaginary, had no real part,
43:59
so we just switched its sign.
44:02
So that eigenvalues come in pairs like that,
44:06
but they're complex.
44:08
A complex conjugate pair.
44:11
And that can happen with a perfectly real matrix.
44:14
And as a matter of fact --
44:17
so that was my, my point earlier,
44:18
that if a matrix was symmetric, it wouldn't happen.
44:23
So if we stick to matrices that are symmetric or, like, close
44:27
to symmetric, then the eigenvalues will stay real.
44:32
But if we move far away from symmetric --
44:35
and that's as far as you can move, because that matrix is --
44:39
how is Q transpose related to Q for that matrix?
44:44
That matrix is anti-symmetric.
44:46
Q transpose is minus Q.
44:49
That's the very opposite of symmetry.
44:52
When I flip across the diagonal I get --
44:54
I reverse all the signs.
44:56
Those are the guys that have pure imaginary eigenvalues.
45:00
So they're the extreme case.
45:03
And in between are, are matrices that
45:07
are not symmetric or anti-symmetric but,
45:10
but they have partly a symmetric part
45:13
and an anti-symmetric part.
45:15
OK.
45:16
So I'm doing a bunch of examples here to show the possibilities.
45:23
The good possibilities being perpendicular eigenvectors,
45:28
real eigenvalues.
45:30
The bad possibilities being complex eigenvalues.
45:33
We could say that's bad.
45:37
There's another even worse.
45:42
I'm getting through the, the bad things here today.
45:45
Then, then the next lecture can, can,
45:51
can be like pure happiness.
45:56
OK.
45:57
Here's one more bad thing that could happen.
46:03
So I, again, I'll do it with an example.
46:05
Suppose my matrix is, suppose I take this three three one
46:10
and I change that guy to zero.
46:18
What are the eigenvalues of that matrix?
46:21
What are the eigenvectors?
46:22
This is always our question.
46:25
Of course, the next section we're
46:26
going to show why are, why do we care.
46:29
But for the moment, this lecture is introducing
46:33
them.
46:33
And let's just find them.
46:35
OK.
46:36
What are the eigenvalues of that matrix?
46:40
Let me tell you -- at a glance we could answer that question.
46:45
Because the matrix is triangular.
46:49
It's really useful to know -- if you've got properties like
46:53
a triangular matrix.
46:55
It's very useful to know you can read the eigenvalues
46:57
off.
46:58
They're right on the diagonal.
47:01
So the eigenvalue is three and also three.
47:05
Three is a repeated eigenvalue.
47:07
But let's see that happen.
47:09
Let me do it right.
47:10
The determinant of A minus lambda I, what I always
47:15
have to do is this determinant.
47:17
I take away lambda from the diagonal.
47:21
I leave the rest.
47:24
I compute the determinant, so I get a three minus lambda
47:28
times a three minus lambda.
47:32
And nothing.
47:35
So that's where the triangular part came in.
47:38
Triangular part, the one thing we know about triangular
47:41
matrices is the determinant is just the product down
47:44
the diagonal.
47:45
And in this case, it's this same, repeated --
47:48
so lambda one is one --
47:51
sorry, lambda one is three and lambda two is three.
47:53
That was easy.
47:58
I mean, no -- why should I be pessimistic about a matrix
48:05
whose eigenvalues can be read off right away?
48:10
The problem with this matrix is in the eigenvectors.
48:14
So let's go to the eigenvectors.
48:16
So how do I find the eigenvectors?
48:18
I'm looking for a couple of eigenvectors.
48:22
Singular, right?
48:22
It's supposed to be singular.
48:31
And then it's got some vectors -- which it is.
48:40
So it's got some vector x in the null space.
49:12
And what, what's the, what's -- give me a basis for the null
49:15
space for that guy.
49:18
Tell me, what's a vector x in the null space, so that'll
49:21
be the, the eigenvector that goes with lambda one
49:25
equals three.
49:27
The eigenvector is -- so what's in the null space?
49:31
One zero, is it?
49:34
Great.
49:38
Now, what's the other eigenvector?
49:40
What's, what's the eigenvector that goes with lambda two?
49:47
Well, lambda two is three again.
49:51
So I get the same thing again.
49:53
Give me another vector --
49:55
I want it to be independent.
49:57
If I'm going to write down an x2,
49:59
I'm never going to let it be dependent on x1.
50:02
I'm looking for independent eigenvectors,
50:05
and what's the conclusion?
50:08
There isn't one.
50:11
This is a degenerate matrix.
50:17
It's only got one line of eigenvectors instead of two.
50:22
It's this possibility of a repeated eigenvalue
50:27
opens this further possibility of a shortage of eigenvectors.
50:34
And so there's no second independent eigenvector x2.
50:43
So it's a matrix, it's a two by two matrix,
50:48
but with only one independent eigenvector.
50:51
So that will be -- the matrices that --
50:56
where eigenvectors are -- don't give the complete story.
51:01
OK.
51:02
My lecture on Monday will give the complete story
51:05
for all the other matrices.
51:11
Thanks.
51:12
Have a good weekend.
51:16
A real New England weekend.</p>