1
00:00:00,000 --> 00:00:11,240
-- two, one and -- okay.
2
00:00:11,240 --> 00:00:18,910
Here is a lecture on the
applications of eigenvalues
3
00:00:18,910 --> 00:00:24,070
and, if I can -- so that
will be Markov matrices.
4
00:00:24,070 --> 00:00:27,260
I'll tell you what
a Markov matrix is,
5
00:00:27,260 --> 00:00:32,070
so this matrix A will
be a Markov matrix
6
00:00:32,070 --> 00:00:38,890
and I'll explain how they
come in applications.
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00:00:38,890 --> 00:00:42,060
And -- and then if I have time,
I would like to say a little
8
00:00:42,060 --> 00:00:46,060
bit about Fourier series, which
is a fantastic application
9
00:00:46,060 --> 00:00:48,540
of the projection chapter.
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00:00:48,540 --> 00:00:49,100
Okay.
11
00:00:49,100 --> 00:00:50,930
What's a Markov matrix?
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00:00:50,930 --> 00:01:01,940
Can I just write down a typical
Markov matrix, say .1, .2, .7,
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00:01:01,940 --> 00:01:08,820
.01, .99 0, let's
say, .3, .3, .4.
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00:01:08,820 --> 00:01:09,970
Okay.
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00:01:09,970 --> 00:01:13,922
There's a -- a totally just
invented Markov matrix.
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00:01:16,640 --> 00:01:19,540
What makes it a Markov matrix?
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00:01:19,540 --> 00:01:22,510
Two properties that
this -- this matrix has.
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00:01:22,510 --> 00:01:24,970
So two properties are --
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00:01:24,970 --> 00:01:30,430
one, every entry is
greater equal zero.
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00:01:30,430 --> 00:01:38,720
All entries greater
than or equal to zero.
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00:01:38,720 --> 00:01:43,100
And, of course, when
I square the matrix,
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00:01:43,100 --> 00:01:45,660
the entries will still
be greater/equal zero.
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00:01:45,660 --> 00:01:50,150
I'm going to be interested
in the powers of this matrix.
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00:01:50,150 --> 00:01:54,250
And this property, of course,
is going to -- stay there.
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00:01:54,250 --> 00:01:58,870
It -- really Markov matrices
you'll see are connected
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00:01:58,870 --> 00:02:04,230
to probability ideas and
probabilities are never
27
00:02:04,230 --> 00:02:05,120
negative.
28
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The other property -- do you
see the other property in there?
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00:02:09,669 --> 00:02:13,590
If I add down the columns,
what answer do I get?
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00:02:13,590 --> 00:02:14,490
One.
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00:02:14,490 --> 00:02:17,640
So all columns add to one.
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00:02:17,640 --> 00:02:25,380
All columns add to one.
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00:02:28,900 --> 00:02:31,580
And actually when I
square the matrix,
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00:02:31,580 --> 00:02:33,820
that will be true again.
35
00:02:33,820 --> 00:02:38,920
So that the powers
of my matrix are all
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00:02:38,920 --> 00:02:42,330
Markov matrices,
and I'm interested
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00:02:42,330 --> 00:02:46,520
in, always, the eigenvalues
and the eigenvectors.
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00:02:46,520 --> 00:02:50,780
And this question of
steady state will come up.
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00:02:50,780 --> 00:02:53,860
You remember we had steady
state for differential equations
40
00:02:53,860 --> 00:02:54,940
last time?
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00:02:54,940 --> 00:02:57,080
When -- what was
the steady state --
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00:02:57,080 --> 00:02:59,100
what was the eigenvalue?
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00:02:59,100 --> 00:03:03,240
What was the eigenvalue in
the differential equation case
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00:03:03,240 --> 00:03:05,620
that led to a steady state?
45
00:03:05,620 --> 00:03:08,360
It was lambda equals zero.
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00:03:08,360 --> 00:03:11,210
When -- you remember that
we did an example and one
47
00:03:11,210 --> 00:03:15,180
of the eigenvalues was lambda
equals zero, and that --
48
00:03:15,180 --> 00:03:20,140
so then we had an E to the
zero T, a constant one --
49
00:03:20,140 --> 00:03:24,430
as time went on, there
that thing stayed steady.
50
00:03:24,430 --> 00:03:30,760
Now what -- in the powers case,
it's not a zero eigenvalue.
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00:03:30,760 --> 00:03:33,740
Actually with powers of a
matrix, a zero eigenvalue,
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00:03:33,740 --> 00:03:36,570
that part is going
to die right away.
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00:03:36,570 --> 00:03:41,710
It's an eigenvalue of
one that's all important.
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00:03:41,710 --> 00:03:44,540
So this steady state
will correspond --
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00:03:44,540 --> 00:03:49,000
will be totally connected with
an eigenvalue of one and its
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00:03:49,000 --> 00:03:49,910
eigenvector.
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00:03:49,910 --> 00:03:53,510
In fact, the steady state
will be the eigenvector
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00:03:53,510 --> 00:03:55,260
for that eigenvalue.
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00:03:55,260 --> 00:03:56,670
Okay.
60
00:03:56,670 --> 00:03:58,580
So that's what's coming.
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00:03:58,580 --> 00:04:04,390
Now, for some reason
then that we have to see,
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00:04:04,390 --> 00:04:08,210
this matrix has an
eigenvalue of one.
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00:04:08,210 --> 00:04:12,040
This property, that the
columns all add to one --
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00:04:12,040 --> 00:04:16,680
turns out -- guarantees
that one is an eigenvalue,
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00:04:16,680 --> 00:04:19,600
so that you can actually
find the eigenvalue --
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00:04:19,600 --> 00:04:24,330
find that eigenvalue of a Markov
matrix without computing any
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00:04:24,330 --> 00:04:27,640
determinants of A
minus lambda I --
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00:04:27,640 --> 00:04:30,310
that matrix will have
an eigenvalue of one,
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00:04:30,310 --> 00:04:32,270
and we want to see why.
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00:04:32,270 --> 00:04:35,850
And then the other thing is --
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00:04:35,850 --> 00:04:39,190
so the key points -- let me --
let me write these underneath.
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00:04:39,190 --> 00:04:45,837
The key points are -- the key
points are lambda equal one is
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00:04:45,837 --> 00:04:46,420
an eigenvalue.
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00:04:53,760 --> 00:04:56,410
I'll add in a little --
an additional -- well,
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00:04:56,410 --> 00:04:59,030
a thing about eigenvalues --
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00:04:59,030 --> 00:05:02,400
key point two, the other
eigenval- values --
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00:05:02,400 --> 00:05:12,340
all other eigenvalues are, in
magnitude, smaller than one --
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00:05:12,340 --> 00:05:14,630
in absolute value,
smaller than one.
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00:05:14,630 --> 00:05:18,390
Well, there could be some
exceptional case when --
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00:05:18,390 --> 00:05:22,380
when an eigen -- another
eigenvalue might have magnitude
81
00:05:22,380 --> 00:05:23,300
equal one.
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00:05:23,300 --> 00:05:26,690
It never has an eigenvalue
larger than one.
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00:05:26,690 --> 00:05:29,230
So these two facts --
somehow we ought to --
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00:05:29,230 --> 00:05:32,600
linear algebra ought to tell us.
85
00:05:32,600 --> 00:05:35,440
And then, of course, linear
algebra is going to tell us
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00:05:35,440 --> 00:05:40,040
what the -- what's -- what
happens if I take -- if --
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00:05:40,040 --> 00:05:42,720
you remember when I solve --
88
00:05:42,720 --> 00:05:47,980
when I multiply by A time
after time the K-th thing is A
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00:05:47,980 --> 00:05:54,940
to the K u0 and I'm asking
what's special about this --
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00:05:54,940 --> 00:06:01,240
these powers of A, and very
likely the quiz will have
91
00:06:01,240 --> 00:06:06,440
a problem to computer s- to
computer some powers of A or --
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00:06:06,440 --> 00:06:08,570
or applied to an initial vector.
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00:06:08,570 --> 00:06:11,040
So, you remember
the general form?
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00:06:11,040 --> 00:06:14,180
The general form is
that there's some amount
95
00:06:14,180 --> 00:06:17,660
of the first eigenvalue
to the K-th power
96
00:06:17,660 --> 00:06:22,790
times the first eigenvector,
and another amount
97
00:06:22,790 --> 00:06:25,340
of the second eigenvalue
to the K-th power
98
00:06:25,340 --> 00:06:27,430
times the second
eigenvector and so on.
99
00:06:27,430 --> 00:06:30,700
A -- just --
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00:06:30,700 --> 00:06:34,380
my conscience always makes me
say at least once per lecture
101
00:06:34,380 --> 00:06:40,060
that this requires a
complete set of eigenvectors,
102
00:06:40,060 --> 00:06:43,080
otherwise we might not
be able to expand u0
103
00:06:43,080 --> 00:06:45,840
in the eigenvectors and
we couldn't get started.
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00:06:45,840 --> 00:06:49,410
But once we're started
with u0 when K is zero,
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00:06:49,410 --> 00:06:53,880
then every A brings
in these lambdas.
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00:06:53,880 --> 00:06:56,830
And now you can see what the
steady state is going to be.
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00:06:56,830 --> 00:07:01,630
If lambda one is one --
so lambda one equals one
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00:07:01,630 --> 00:07:07,900
to the K-th power and these
other eigenvalues are smaller
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00:07:07,900 --> 00:07:10,280
than one --
110
00:07:10,280 --> 00:07:14,560
so I've sort of scratched
over the equation there to --
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00:07:14,560 --> 00:07:18,260
we had this term, but what
happens to this term --
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00:07:18,260 --> 00:07:21,180
if the lambda's smaller than
one, then the -- when --
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00:07:21,180 --> 00:07:25,160
as we take powers, as
we iterate as we --
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00:07:25,160 --> 00:07:30,662
as we go forward in time, this
goes to zero, Can I just --
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00:07:30,662 --> 00:07:32,370
having scratched over
it, I might as well
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00:07:32,370 --> 00:07:34,050
scratch right? further.
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00:07:34,050 --> 00:07:36,880
That term and all the other
terms are going to zero
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00:07:36,880 --> 00:07:39,750
because all the other
eigenvalues are smaller than
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00:07:39,750 --> 00:07:45,440
one and the steady state that
we're approaching is just --
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00:07:45,440 --> 00:07:48,780
whatever there was -- this
was -- this was the --
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00:07:48,780 --> 00:07:56,990
this is the x1 part of un- of
the initial condition u0 --
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00:07:56,990 --> 00:07:59,150
is the steady state.
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00:08:05,120 --> 00:08:08,350
This much we know from
general -- from -- you know,
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00:08:08,350 --> 00:08:11,390
what we've already done.
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00:08:11,390 --> 00:08:15,740
So I want to see why -- let's
at least see number one,
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00:08:15,740 --> 00:08:18,160
why one is an eigenvalue.
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00:08:18,160 --> 00:08:20,580
And then there's actually --
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00:08:20,580 --> 00:08:23,370
in this chapter we're interested
not only in eigenvalues,
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00:08:23,370 --> 00:08:25,390
but also eigenvectors.
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00:08:25,390 --> 00:08:28,760
And there's something special
about the eigenvector.
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00:08:28,760 --> 00:08:30,800
Let me write down what that is.
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The eigenvector x1 --
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00:08:36,700 --> 00:08:41,909
x1 is the eigenvector and all
its components are positive,
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00:08:41,909 --> 00:08:49,100
so the steady state is
positive, if the start was.
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00:08:49,100 --> 00:08:51,480
If the start was -- so --
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00:08:51,480 --> 00:08:54,780
well, actually, in general, I --
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00:08:54,780 --> 00:09:00,710
this might have a -- might have
some component zero always,
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00:09:00,710 --> 00:09:03,880
but no negative components
in that eigenvector.
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00:09:03,880 --> 00:09:04,410
Okay.
140
00:09:04,410 --> 00:09:08,470
Can I come to that point?
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00:09:08,470 --> 00:09:11,370
How can I look at that matrix --
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00:09:11,370 --> 00:09:15,280
so that was just an example.
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00:09:15,280 --> 00:09:21,000
How could I be sure -- how
can I see that a matrix --
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00:09:21,000 --> 00:09:24,750
if the columns add to zero
-- add to one, sorry --
145
00:09:24,750 --> 00:09:30,930
if the columns add to one,
this property means that lambda
146
00:09:30,930 --> 00:09:32,780
equal one is an eigenvalue.
147
00:09:32,780 --> 00:09:33,540
Okay.
148
00:09:33,540 --> 00:09:37,080
So let's just
think that through.
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00:09:37,080 --> 00:09:41,760
What I saying about -- let
me ca- let me look at A,
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00:09:41,760 --> 00:09:44,940
and if I believe that
one is an eigenvalue,
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00:09:44,940 --> 00:09:47,810
then I should be able to
subtract off one times
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00:09:47,810 --> 00:09:56,150
the identity and then I would
get a matrix that's, what, -.9,
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00:09:56,150 --> 00:09:59,160
-.01 and -.6 --
154
00:09:59,160 --> 00:10:02,700
wh- I took the ones away and
the other parts, of course,
155
00:10:02,700 --> 00:10:11,430
are still what they were, and
this is still .2 and .7 and --
156
00:10:11,430 --> 00:10:14,200
okay, what's --
157
00:10:14,200 --> 00:10:15,700
what's up with this matrix now?
158
00:10:15,700 --> 00:10:19,380
I've shifted the matrix,
this Markov matrix by one,
159
00:10:19,380 --> 00:10:24,350
by the identity, and
what do I want to prove?
160
00:10:24,350 --> 00:10:28,940
I -- what is it that I
believe this matrix --
161
00:10:28,940 --> 00:10:30,760
about this matrix?
162
00:10:30,760 --> 00:10:33,190
I believe it's singular.
163
00:10:33,190 --> 00:10:36,580
Singular will -- if A
minus I is singular,
164
00:10:36,580 --> 00:10:40,020
that tells me that one
is an eigenvalue, right?
165
00:10:40,020 --> 00:10:43,420
The eigenvalues are the
numbers that I subtract off --
166
00:10:43,420 --> 00:10:45,010
the shifts --
167
00:10:45,010 --> 00:10:47,880
the numbers that I subtract
from the diagonal --
168
00:10:47,880 --> 00:10:48,810
to make it singular.
169
00:10:48,810 --> 00:10:50,610
Now why is that matrix singular?
170
00:10:50,610 --> 00:10:55,570
I -- we could compute
its determinant,
171
00:10:55,570 --> 00:11:00,000
but we want to see a reason
that would work for every Markov
172
00:11:00,000 --> 00:11:05,550
matrix not just this
particular random example.
173
00:11:05,550 --> 00:11:07,780
So what is it about that matrix?
174
00:11:07,780 --> 00:11:11,040
Well, I guess you could
look at its columns now --
175
00:11:11,040 --> 00:11:12,890
what do they add up to?
176
00:11:12,890 --> 00:11:13,390
Zero.
177
00:11:13,390 --> 00:11:21,660
The columns add to
zero, so all columns --
178
00:11:21,660 --> 00:11:25,240
let me put all columns
now of -- of --
179
00:11:25,240 --> 00:11:32,450
of A minus I add
to zero, and then I
180
00:11:32,450 --> 00:11:40,305
want to realize that this
means A minus I is singular.
181
00:11:45,400 --> 00:11:47,790
Okay.
182
00:11:47,790 --> 00:11:49,560
Why?
183
00:11:49,560 --> 00:11:52,210
So I could I -- you know,
that could be a quiz question,
184
00:11:52,210 --> 00:11:54,280
a sort of theoretical
quiz question.
185
00:11:54,280 --> 00:11:56,070
If I give you a
matrix and I tell you
186
00:11:56,070 --> 00:12:00,740
all its columns add to
zero, give me a reason,
187
00:12:00,740 --> 00:12:06,270
because it is true, that
the matrix is singular.
188
00:12:06,270 --> 00:12:06,940
Okay.
189
00:12:06,940 --> 00:12:09,290
I guess actually -- now what --
190
00:12:09,290 --> 00:12:11,680
I think of -- you know, I'm
thinking of two or three ways
191
00:12:11,680 --> 00:12:12,480
to see that.
192
00:12:17,370 --> 00:12:18,410
How would you do it?
193
00:12:18,410 --> 00:12:21,880
We don't want to take
its determinant somehow.
194
00:12:21,880 --> 00:12:24,520
For the matrix to
be singular, well,
195
00:12:24,520 --> 00:12:31,180
it means that these three
columns are dependent, right?
196
00:12:31,180 --> 00:12:33,530
The determinant will be zero
when those three columns
197
00:12:33,530 --> 00:12:34,260
are dependent.
198
00:12:34,260 --> 00:12:36,650
You see, we're -- we're at
a point in this course, now,
199
00:12:36,650 --> 00:12:40,100
where we have several
ways to look at an idea.
200
00:12:40,100 --> 00:12:42,390
We can take the determinant
-- here we don't want to.
201
00:12:42,390 --> 00:12:45,810
B- but we met singular
before that --
202
00:12:45,810 --> 00:12:47,830
those columns are dependent.
203
00:12:47,830 --> 00:12:50,670
So how do I see that those
columns are dependent?
204
00:12:50,670 --> 00:12:51,925
They all add to zero.
205
00:12:57,020 --> 00:13:00,440
Let's see, whew --
206
00:13:00,440 --> 00:13:05,070
well, oh, actually, what --
207
00:13:05,070 --> 00:13:07,010
another thing I
know is that the --
208
00:13:07,010 --> 00:13:11,410
I would like to be able to show
is that the rows are dependent.
209
00:13:11,410 --> 00:13:14,250
Maybe that's easier.
210
00:13:14,250 --> 00:13:15,940
If I know that all
the columns add
211
00:13:15,940 --> 00:13:18,480
to zero, that's my
information, how
212
00:13:18,480 --> 00:13:25,120
do I see that those three
rows are linearly dependent?
213
00:13:25,120 --> 00:13:30,210
What -- what combination of
those rows gives the zero row?
214
00:13:30,210 --> 00:13:32,920
How -- how could I combine
those three rows --
215
00:13:32,920 --> 00:13:38,930
those three row vectors to
produce the zero row vector?
216
00:13:38,930 --> 00:13:42,360
And that would tell me
those rows are dependent,
217
00:13:42,360 --> 00:13:43,910
therefore the columns
are dependent,
218
00:13:43,910 --> 00:13:46,940
the matrix is singular, the
determinant is zero -- well,
219
00:13:46,940 --> 00:13:47,890
you see it.
220
00:13:47,890 --> 00:13:49,010
I just add the rows.
221
00:13:49,010 --> 00:13:52,610
One times that row plus one
times that row plus one times
222
00:13:52,610 --> 00:13:53,770
that row --
223
00:13:53,770 --> 00:13:56,990
it's the zero row.
224
00:13:56,990 --> 00:13:58,960
The rows are dependent.
225
00:13:58,960 --> 00:14:03,470
In a way, that one one one,
because it's multiplying
226
00:14:03,470 --> 00:14:07,810
the rows, is like an
eigenvector in the --
227
00:14:07,810 --> 00:14:10,700
it's in the left
null space, right?
228
00:14:10,700 --> 00:14:12,870
One one one is in
the left null space.
229
00:14:12,870 --> 00:14:23,190
It's singular because
the rows are dependent --
230
00:14:23,190 --> 00:14:25,380
and can I just keep
the reasoning going?
231
00:14:25,380 --> 00:14:32,790
Because this vector
one one one is --
232
00:14:32,790 --> 00:14:34,940
it's not in the null
space of the matrix,
233
00:14:34,940 --> 00:14:37,410
but it's in the null
space of the transpose --
234
00:14:37,410 --> 00:14:43,160
is in the null space
of the transpose.
235
00:14:43,160 --> 00:14:45,600
And that's good enough.
236
00:14:45,600 --> 00:14:48,890
If we have a square matrix
-- if we have a square matrix
237
00:14:48,890 --> 00:14:52,930
and the rows are dependent,
that matrix is singular.
238
00:14:52,930 --> 00:14:58,630
So it turned out that the
immediate guy we could identify
239
00:14:58,630 --> 00:15:00,470
was one one one.
240
00:15:00,470 --> 00:15:04,650
Of course, the --
241
00:15:04,650 --> 00:15:06,480
there will be somebody
in the null space,
242
00:15:06,480 --> 00:15:07,300
too.
243
00:15:07,300 --> 00:15:10,280
And actually, who will it be?
244
00:15:10,280 --> 00:15:13,110
So what's -- so --
245
00:15:13,110 --> 00:15:15,830
so now I want to ask
about the null space of --
246
00:15:15,830 --> 00:15:17,400
of the matrix itself.
247
00:15:17,400 --> 00:15:20,890
What combination of
the columns gives zero?
248
00:15:20,890 --> 00:15:23,450
I -- I don't want to compute
it because I just made up this
249
00:15:23,450 --> 00:15:24,830
matrix and --
250
00:15:24,830 --> 00:15:28,020
it will -- it would
take me a while --
251
00:15:28,020 --> 00:15:31,390
it looks sort of doable because
it's three by three but wh-
252
00:15:31,390 --> 00:15:34,720
my point is, what --
253
00:15:34,720 --> 00:15:37,350
what vector is it if we
-- once we've found it,
254
00:15:37,350 --> 00:15:41,360
what have we got that's in
the -- in the null space of A?
255
00:15:41,360 --> 00:15:44,060
It's the eigenvector, right?
256
00:15:44,060 --> 00:15:46,060
That's where we find X one.
257
00:15:46,060 --> 00:15:54,350
Then X one, the eigenvector,
is in the null space of A.
258
00:15:54,350 --> 00:15:57,690
That's the eigenvector
corresponding to the eigenvalue
259
00:15:57,690 --> 00:15:58,470
one.
260
00:15:58,470 --> 00:15:58,980
Right?
261
00:15:58,980 --> 00:16:01,330
That's how we find eigenvectors.
262
00:16:01,330 --> 00:16:05,280
So those three columns
must be dependent --
263
00:16:05,280 --> 00:16:08,090
some combination of columns
-- of those three columns is
264
00:16:08,090 --> 00:16:11,930
the zero column and that --
the three components in that
265
00:16:11,930 --> 00:16:14,020
combination are the eigenvector.
266
00:16:14,020 --> 00:16:17,250
And that guy is
the steady state.
267
00:16:17,250 --> 00:16:18,280
Okay.
268
00:16:18,280 --> 00:16:21,110
So I'm happy about the --
269
00:16:21,110 --> 00:16:26,010
the thinking here,
but I haven't given --
270
00:16:26,010 --> 00:16:29,620
I haven't completed it
because I haven't found x1.
271
00:16:29,620 --> 00:16:32,560
But it's there.
272
00:16:32,560 --> 00:16:36,330
Can I -- another thought came
to me as I was doing this,
273
00:16:36,330 --> 00:16:39,280
another little
comment that -- you --
274
00:16:39,280 --> 00:16:41,500
about eigenvalues
and eigenvectors,
275
00:16:41,500 --> 00:16:45,330
because of A and A transpose.
276
00:16:45,330 --> 00:16:50,800
What can you tell me
about eigenvalues of A --
277
00:16:50,800 --> 00:16:56,275
of A and eigenvalues
of A transpose?
278
00:17:01,160 --> 00:17:01,660
Whoops.
279
00:17:04,952 --> 00:17:05,660
They're the same.
280
00:17:05,660 --> 00:17:11,079
They're -- so this is a little
comment -- we -- it's useful,
281
00:17:11,079 --> 00:17:14,550
since eigenvalues are
generally not easy to find --
282
00:17:14,550 --> 00:17:19,180
it's always useful to know some
cases where you've got them,
283
00:17:19,180 --> 00:17:20,210
where --
284
00:17:20,210 --> 00:17:23,089
and this is -- if you
know the eigenvalues of A,
285
00:17:23,089 --> 00:17:25,260
then you know the
eigenvalues of A transpose.
286
00:17:25,260 --> 00:17:27,500
eigenvalues of A
transpose are the same.
287
00:17:32,560 --> 00:17:37,890
And can I just, like,
review why that is?
288
00:17:37,890 --> 00:17:44,280
So to find the eigenvalues of A,
this would be determinate of A
289
00:17:44,280 --> 00:17:53,490
minus lambda I equals zero, that
gives me an eigenvalue of A --
290
00:17:53,490 --> 00:17:58,520
now how can I get A transpose
into the picture here?
291
00:17:58,520 --> 00:18:00,690
I'll use the fact
that the determinant
292
00:18:00,690 --> 00:18:06,690
of a matrix and the determinant
of its transpose are the same.
293
00:18:06,690 --> 00:18:09,550
The determinant of a matrix
equals the determinant of a --
294
00:18:09,550 --> 00:18:10,520
of the transpose.
295
00:18:10,520 --> 00:18:13,950
That was property
ten, the very last guy
296
00:18:13,950 --> 00:18:15,860
in our determinant list.
297
00:18:15,860 --> 00:18:18,280
So I'll transpose that matrix.
298
00:18:18,280 --> 00:18:21,680
This leads to --
299
00:18:21,680 --> 00:18:24,690
I just take the matrix
and transpose it,
300
00:18:24,690 --> 00:18:29,520
but now what do I get
when I transpose lambda I?
301
00:18:29,520 --> 00:18:33,920
I just get lambda I.
302
00:18:33,920 --> 00:18:37,800
So that's -- that's all
there was to the reasoning.
303
00:18:37,800 --> 00:18:40,000
The reasoning is that
the eigenvalues of A
304
00:18:40,000 --> 00:18:42,370
solved that equation.
305
00:18:42,370 --> 00:18:44,530
The determinant of a
matrix is the determinant
306
00:18:44,530 --> 00:18:47,480
of its transpose, so that
gives me this equation
307
00:18:47,480 --> 00:18:50,160
and that tells me
that the same lambdas
308
00:18:50,160 --> 00:18:53,240
are eigenvalues of A transpose.
309
00:18:53,240 --> 00:18:56,660
So that, backing up
to the Markov case,
310
00:18:56,660 --> 00:19:00,880
one is an eigenvalue of A
transpose and we actually found
311
00:19:00,880 --> 00:19:05,480
its eigenvector, one one one,
and that tell us that one is
312
00:19:05,480 --> 00:19:08,090
also an eigenvalue of
A -- but, of course,
313
00:19:08,090 --> 00:19:10,360
it has a different
eigenvector, the --
314
00:19:10,360 --> 00:19:13,530
the left null space isn't the
same as the null space and we
315
00:19:13,530 --> 00:19:14,750
would have to find it.
316
00:19:14,750 --> 00:19:20,960
So there's some vector here
which is x1 that produces zero
317
00:19:20,960 --> 00:19:22,460
zero zero.
318
00:19:22,460 --> 00:19:24,900
Actually, it wouldn't be that
hard to find, you know, I --
319
00:19:24,900 --> 00:19:26,520
as I'm talking I'm
thinking, okay,
320
00:19:26,520 --> 00:19:29,440
I going to follow through
and actually find it?
321
00:19:29,440 --> 00:19:32,750
Well, I can tell from
this one -- look,
322
00:19:32,750 --> 00:19:37,650
if I put a point six there
and a point seven there,
323
00:19:37,650 --> 00:19:43,510
that's what -- then I'll be
okay in the last row, right?
324
00:19:43,510 --> 00:19:47,840
Now I only -- remains
to find one guy.
325
00:19:47,840 --> 00:19:49,860
And let me take the
first row, then.
326
00:19:49,860 --> 00:19:53,500
Minus point 54 plus point 21 --
327
00:19:53,500 --> 00:19:56,790
there's some big number
going in there, right?
328
00:19:56,790 --> 00:20:00,320
So I have -- just to make
the first row come out zero,
329
00:20:00,320 --> 00:20:03,980
I'm getting minus
point 54 plus point 21,
330
00:20:03,980 --> 00:20:09,420
so that was minus point 33
and what -- what do I want?
331
00:20:09,420 --> 00:20:11,350
Like thirty three hundred?
332
00:20:11,350 --> 00:20:13,980
This is the first time in
the history of linear algebra
333
00:20:13,980 --> 00:20:16,980
that an eigenvector has
every had a component
334
00:20:16,980 --> 00:20:18,980
thirty three hundred.
335
00:20:18,980 --> 00:20:21,030
But I guess it's true.
336
00:20:21,030 --> 00:20:24,760
Because then I multiply by minus
one over a hundred -- oh no,
337
00:20:24,760 --> 00:20:26,830
it was point 33.
338
00:20:26,830 --> 00:20:29,180
So is this just -- oh, shoot.
339
00:20:29,180 --> 00:20:30,800
Only 33.
340
00:20:30,800 --> 00:20:31,880
Okay.
341
00:20:31,880 --> 00:20:33,030
Only 33.
342
00:20:33,030 --> 00:20:35,610
Okay, so there's
the eigenvector.
343
00:20:35,610 --> 00:20:39,150
Oh, and notice that it -- that
it turned -- did turn out,
344
00:20:39,150 --> 00:20:42,940
at least, to be all positive.
345
00:20:42,940 --> 00:20:45,990
So that was, like, the theory
-- predicts that part, too.
346
00:20:45,990 --> 00:20:48,500
I won't give the
proof of that part.
347
00:20:48,500 --> 00:20:50,600
So 30 -- 33 --
348
00:20:50,600 --> 00:20:52,770
point six 33 point seven.
349
00:20:52,770 --> 00:20:53,420
Okay.
350
00:20:53,420 --> 00:20:58,750
Now those are the ma- that's
the linear algebra part.
351
00:20:58,750 --> 00:21:00,980
Can I get to the applications?
352
00:21:00,980 --> 00:21:03,120
Where do these Markov
matrices come from?
353
00:21:03,120 --> 00:21:06,360
Because that's -- that's part of
this course and absolutely part
354
00:21:06,360 --> 00:21:07,570
of this lecture.
355
00:21:07,570 --> 00:21:08,080
Okay.
356
00:21:08,080 --> 00:21:12,270
So where's -- what's an
application of Markov matrices?
357
00:21:12,270 --> 00:21:12,770
Okay.
358
00:21:17,600 --> 00:21:21,180
Markov matrices --
so, my equation, then,
359
00:21:21,180 --> 00:21:24,842
that I'm solving and studying
is this equation u(k+1)=Auk.
360
00:21:28,290 --> 00:21:31,280
And now A is a Markov matrix.
361
00:21:31,280 --> 00:21:31,930
A is Markov.
362
00:21:36,060 --> 00:21:39,110
And I want to give an example.
363
00:21:39,110 --> 00:21:41,190
Can I just create an example?
364
00:21:41,190 --> 00:21:44,120
It'll be two by two.
365
00:21:44,120 --> 00:21:48,870
And it's one I've used
before because it seems
366
00:21:48,870 --> 00:21:50,990
to me to bring out the idea.
367
00:21:50,990 --> 00:21:55,740
It's -- because we have two
by two, we have two states,
368
00:21:55,740 --> 00:22:01,510
let's say California
and Massachusetts.
369
00:22:01,510 --> 00:22:05,310
And I'm looking at the
populations in those two
370
00:22:05,310 --> 00:22:08,720
states, the people in those
two states, California
371
00:22:08,720 --> 00:22:10,900
and Massachusetts.
372
00:22:10,900 --> 00:22:17,040
And my matrix A is going to
tell me in a -- in a year,
373
00:22:17,040 --> 00:22:19,230
some movement has happened.
374
00:22:19,230 --> 00:22:21,180
Some people stayed
in Massachusetts,
375
00:22:21,180 --> 00:22:24,050
some people moved to
California, some smart people
376
00:22:24,050 --> 00:22:26,300
moved from California
to Massachusetts,
377
00:22:26,300 --> 00:22:29,240
some people stayed in
California and made a billion.
378
00:22:29,240 --> 00:22:29,740
Okay.
379
00:22:29,740 --> 00:22:36,150
So that -- there's a matrix
there with four entries
380
00:22:36,150 --> 00:22:41,044
and those tell me the
fractions of my population --
381
00:22:41,044 --> 00:22:41,710
so I'm making --
382
00:22:41,710 --> 00:22:45,140
I'm going to use fractions,
so they won't be negative,
383
00:22:45,140 --> 00:22:48,500
of course, because -- because
only positive people are
384
00:22:48,500 --> 00:22:51,550
in- involved here -- and
they'll add up to one,
385
00:22:51,550 --> 00:22:54,620
because I'm accounting
for all people.
386
00:22:54,620 --> 00:22:57,830
So that's why I have
these two key properties.
387
00:22:57,830 --> 00:22:59,650
The entries are
greater equal zero
388
00:22:59,650 --> 00:23:02,890
because I'm looking
at probabilities.
389
00:23:02,890 --> 00:23:06,050
Do they move, do they stay?
390
00:23:06,050 --> 00:23:09,130
Those probabilities are
all between zero and one.
391
00:23:09,130 --> 00:23:12,290
And the probabilities add
to one because everybody's
392
00:23:12,290 --> 00:23:12,990
accounted for.
393
00:23:12,990 --> 00:23:17,970
I'm not losing anybody, gaining
anybody in this Markov chain.
394
00:23:17,970 --> 00:23:22,490
It's -- it conserves
the total population.
395
00:23:22,490 --> 00:23:22,990
Okay.
396
00:23:22,990 --> 00:23:25,450
So what would be a
typical matrix, then?
397
00:23:25,450 --> 00:23:34,820
So this would be u, California
and u Massachusetts at time t
398
00:23:34,820 --> 00:23:36,740
equal k+1.
399
00:23:36,740 --> 00:23:40,680
And it's some
matrix, which we'll
400
00:23:40,680 --> 00:23:48,730
think of, times u California
and u Massachusetts at time k.
401
00:23:51,280 --> 00:23:54,460
And notice this matrix is
going to stay the same,
402
00:23:54,460 --> 00:23:57,480
you know, forever.
403
00:23:57,480 --> 00:24:01,940
So that's a severe
limitation on the example.
404
00:24:01,940 --> 00:24:05,090
The example has a --
the same Markov matrix,
405
00:24:05,090 --> 00:24:08,510
the same probabilities
act at every time.
406
00:24:08,510 --> 00:24:09,010
Okay.
407
00:24:09,010 --> 00:24:11,030
So what's a reasonable, say --
408
00:24:11,030 --> 00:24:16,750
say point nine of the people
in California at time k
409
00:24:16,750 --> 00:24:19,410
stay there.
410
00:24:19,410 --> 00:24:24,440
And point one of the
people in California
411
00:24:24,440 --> 00:24:26,930
move to Massachusetts.
412
00:24:26,930 --> 00:24:28,900
Notice why that
column added to one,
413
00:24:28,900 --> 00:24:32,170
because we've now accounted for
all the people in California
414
00:24:32,170 --> 00:24:33,890
at time k.
415
00:24:33,890 --> 00:24:36,250
Nine tenths of them are
still in California,
416
00:24:36,250 --> 00:24:41,220
one tenth are here at time k+1.
417
00:24:41,220 --> 00:24:42,130
Okay.
418
00:24:42,130 --> 00:24:45,120
What about the people
who are in Massachusetts?
419
00:24:45,120 --> 00:24:47,530
This is going to multiply
column two, right,
420
00:24:47,530 --> 00:24:52,370
by our fundamental rule of
multiplying matrix by vector,
421
00:24:52,370 --> 00:24:57,050
it's the -- it's the
population in Massachusetts.
422
00:24:57,050 --> 00:25:06,350
Shall we say that -- that
after the Red Sox, fail again,
423
00:25:06,350 --> 00:25:11,630
eight -- only 80 percent of the
people in Massachusetts stay
424
00:25:11,630 --> 00:25:15,120
and 20 percent
move to California.
425
00:25:15,120 --> 00:25:15,870
Okay.
426
00:25:15,870 --> 00:25:18,960
So again, this
adds to one, which
427
00:25:18,960 --> 00:25:23,070
accounts for all people in
Massachusetts where they are.
428
00:25:23,070 --> 00:25:26,770
So there is a Markov matrix.
429
00:25:26,770 --> 00:25:28,400
Non-negative entries
adding to one.
430
00:25:28,400 --> 00:25:29,980
What's the steady state?
431
00:25:29,980 --> 00:25:33,314
If everybody started in
Massachusetts, say, at --
432
00:25:33,314 --> 00:25:34,980
you know, when the
Pilgrims showed up or
433
00:25:34,980 --> 00:25:35,600
something.
434
00:25:35,600 --> 00:25:40,280
Then where are they now?
435
00:25:40,280 --> 00:25:45,211
Where are they at time
100, let's say, or maybe --
436
00:25:45,211 --> 00:25:47,210
I don't know, how many
years since the Pilgrims?
437
00:25:47,210 --> 00:25:49,090
300 and something.
438
00:25:49,090 --> 00:25:51,520
Or -- and actually where
will they be, like,
439
00:25:51,520 --> 00:25:54,820
way out a million
years from now?
440
00:25:54,820 --> 00:26:00,980
I -- I could multiply --
441
00:26:00,980 --> 00:26:03,360
take the powers of this matrix.
442
00:26:03,360 --> 00:26:07,040
In fact, you'll -- you would --
ought to be able to figure out
443
00:26:07,040 --> 00:26:10,490
what is the hundredth
power of that matrix?
444
00:26:10,490 --> 00:26:12,520
Why don't we do that?
445
00:26:12,520 --> 00:26:15,050
But let me follow
the steady state.
446
00:26:15,050 --> 00:26:17,760
So what -- what's my starting --
447
00:26:17,760 --> 00:26:28,560
my starting u Cal, u Mass at
time zero is, shall we say --
448
00:26:28,560 --> 00:26:30,400
shall we put anybody
in California?
449
00:26:30,400 --> 00:26:32,920
Let's make -- let's
make zero there,
450
00:26:32,920 --> 00:26:36,400
and say the population
of Massachusetts is --
451
00:26:36,400 --> 00:26:38,780
let's say a thousand
just to -- okay.
452
00:26:43,330 --> 00:26:46,030
So the population is --
453
00:26:46,030 --> 00:26:49,400
so the populations are zero
and a thousand at the start.
454
00:26:49,400 --> 00:26:52,790
What can you tell me about
this population after --
455
00:26:52,790 --> 00:26:55,990
after k steps?
456
00:26:55,990 --> 00:27:01,420
What will u Cal
plus u Mass add to?
457
00:27:01,420 --> 00:27:02,790
A thousand.
458
00:27:02,790 --> 00:27:06,260
Those thousand people
are always accounted for.
459
00:27:06,260 --> 00:27:10,590
But -- so u Mass will start
dropping from a thousand and u
460
00:27:10,590 --> 00:27:11,930
Cal will start growing.
461
00:27:11,930 --> 00:27:15,060
Actually, we could see -- why
don't we figure out what it is
462
00:27:15,060 --> 00:27:16,420
after one?
463
00:27:16,420 --> 00:27:22,100
After one time step, what are
the populations at time one?
464
00:27:24,890 --> 00:27:26,530
So what happens in one step?
465
00:27:26,530 --> 00:27:30,230
You multiply once by that
matrix and, let's see,
466
00:27:30,230 --> 00:27:33,460
zero times this column -- so
it's just a thousand times this
467
00:27:33,460 --> 00:27:39,940
column, so I think we're
getting 200 and 800.
468
00:27:39,940 --> 00:27:42,860
So after the first
step, 200 people have --
469
00:27:42,860 --> 00:27:44,760
are in California.
470
00:27:44,760 --> 00:27:49,620
Now at the following step, I'll
multiply again by this matrix
471
00:27:49,620 --> 00:27:52,740
-- more people will
move to California.
472
00:27:52,740 --> 00:27:54,340
Some people will move back.
473
00:27:54,340 --> 00:28:00,390
Twenty people will
come back and, the --
474
00:28:00,390 --> 00:28:03,010
the net result will be that the
California population will be
475
00:28:03,010 --> 00:28:08,310
above 200 and the Massachusetts
below 800 and they'll still add
476
00:28:08,310 --> 00:28:10,191
up to a thousand.
477
00:28:10,191 --> 00:28:10,690
Okay.
478
00:28:10,690 --> 00:28:14,100
I do that a few times.
479
00:28:14,100 --> 00:28:15,500
I do that 100 times.
480
00:28:15,500 --> 00:28:18,220
What's the population?
481
00:28:18,220 --> 00:28:20,770
Well, okay, to answer
any question like that,
482
00:28:20,770 --> 00:28:22,700
I need the eigenvalues
and eigenvectors,
483
00:28:22,700 --> 00:28:23,200
right?
484
00:28:23,200 --> 00:28:24,090
As soon as I've --
485
00:28:24,090 --> 00:28:26,340
I've created an
example, but as soon
486
00:28:26,340 --> 00:28:28,570
as I want to solve
anything, I have
487
00:28:28,570 --> 00:28:31,610
to find eigenvalues and
eigenvectors of that matrix.
488
00:28:31,610 --> 00:28:32,230
Okay.
489
00:28:32,230 --> 00:28:33,890
So let's do it.
490
00:28:33,890 --> 00:28:40,010
So there's the matrix
.9, .2, .1, .8 and tell
491
00:28:40,010 --> 00:28:43,150
me its eigenvalues.
492
00:28:43,150 --> 00:28:46,390
Lambda equals -- so
tell me one eigenvalue?
493
00:28:46,390 --> 00:28:48,680
One, thanks.
494
00:28:48,680 --> 00:28:49,960
And tell me the other one.
495
00:28:53,120 --> 00:28:55,300
What's the other eigenvalue --
496
00:28:55,300 --> 00:28:59,370
from the trace or the
determinant -- from the --
497
00:28:59,370 --> 00:29:02,510
I -- the trace is what
-- is, like, easier.
498
00:29:02,510 --> 00:29:06,110
So the trace of that
matrix is one point seven.
499
00:29:06,110 --> 00:29:10,040
So the other eigenvalue
is point seven.
500
00:29:10,040 --> 00:29:13,540
And it -- notice that
it's less than one.
501
00:29:13,540 --> 00:29:17,780
And notice that that
determinant is point 72-.02,
502
00:29:17,780 --> 00:29:18,960
which is point seven.
503
00:29:18,960 --> 00:29:19,490
Right.
504
00:29:19,490 --> 00:29:20,190
Okay.
505
00:29:20,190 --> 00:29:21,900
Now to find the eigenvectors.
506
00:29:21,900 --> 00:29:26,120
This is -- so that's lambda
one and the eigenvector --
507
00:29:26,120 --> 00:29:29,950
I'll subtract one from
the diagonal, right?
508
00:29:29,950 --> 00:29:33,440
So can I do that in light
let -- in light here?
509
00:29:33,440 --> 00:29:35,300
Subtract one from
the diagonal, I
510
00:29:35,300 --> 00:29:38,410
have minus point one
and minus point two,
511
00:29:38,410 --> 00:29:39,730
and of course these are still
512
00:29:39,730 --> 00:29:40,940
there.
513
00:29:40,940 --> 00:29:44,220
And I'm looking for its --
514
00:29:44,220 --> 00:29:47,360
here's -- here's --
this is going to be x1.
515
00:29:47,360 --> 00:29:52,170
It's the null
space of A minus I.
516
00:29:52,170 --> 00:29:56,820
Okay, everybody sees
that it's two and one.
517
00:29:56,820 --> 00:29:57,940
Okay?
518
00:29:57,940 --> 00:29:59,820
And now how about --
so that -- and it --
519
00:29:59,820 --> 00:30:03,160
notice that that
eigenvector is positive.
520
00:30:03,160 --> 00:30:08,770
And actually, we can jump
to infinity right now.
521
00:30:08,770 --> 00:30:14,100
What's the population
at infinity?
522
00:30:14,100 --> 00:30:17,170
It's a multiple -- this is
-- this eigenvector is giving
523
00:30:17,170 --> 00:30:19,910
the steady state.
524
00:30:19,910 --> 00:30:23,280
It's some multiple of this, and
how is that multiple decided?
525
00:30:23,280 --> 00:30:26,940
By adding up to a
thousand people.
526
00:30:26,940 --> 00:30:31,200
So the steady state, the
c1x1 -- this is the x1,
527
00:30:31,200 --> 00:30:36,740
but that adds up to three,
so I really want two --
528
00:30:36,740 --> 00:30:39,820
it's going to be two thirds
of a thousand and one third
529
00:30:39,820 --> 00:30:42,980
of a thousand, making
a total of the thousand
530
00:30:42,980 --> 00:30:43,550
people.
531
00:30:43,550 --> 00:30:45,480
That'll be the steady state.
532
00:30:45,480 --> 00:30:48,390
That's really all I need
to know at infinity.
533
00:30:48,390 --> 00:30:50,160
But if I want to
know what's happened
534
00:30:50,160 --> 00:30:53,280
after just a finite
number like 100 steps,
535
00:30:53,280 --> 00:30:55,380
I'd better find
this eigenvector.
536
00:30:55,380 --> 00:30:57,840
So can I -- can I look at --
537
00:30:57,840 --> 00:31:00,130
I'll subtract point
seven time -- ti-
538
00:31:00,130 --> 00:31:06,250
from the diagonal and I'll get
that and I'll look at the null
539
00:31:06,250 --> 00:31:11,280
space of that one and I -- and
this is going to give me x2,
540
00:31:11,280 --> 00:31:13,540
now, and what is it?
541
00:31:13,540 --> 00:31:16,380
So what's in the null space of
-- that's certainly singular,
542
00:31:16,380 --> 00:31:20,920
so I know my calculation
is right, and --
543
00:31:20,920 --> 00:31:24,290
one and minus one.
544
00:31:24,290 --> 00:31:26,240
One and minus one.
545
00:31:26,240 --> 00:31:30,580
So I'm prepared
now to write down
546
00:31:30,580 --> 00:31:32,470
the solution after 100 time
547
00:31:32,470 --> 00:31:33,180
steps.
548
00:31:33,180 --> 00:31:35,670
The -- the populations
after 100 time steps,
549
00:31:35,670 --> 00:31:36,420
right?
550
00:31:36,420 --> 00:31:39,290
Can -- can we remember
the point one -- the --
551
00:31:39,290 --> 00:31:43,100
the one with this two one
eigenvector and the point seven
552
00:31:43,100 --> 00:31:44,930
with the minus one
one eigenvector.
553
00:31:44,930 --> 00:31:46,160
So I'll -- let me --
554
00:31:46,160 --> 00:31:48,960
I'll just write it above here.
555
00:31:48,960 --> 00:31:53,810
u after k steps is
some multiple of one
556
00:31:53,810 --> 00:31:57,610
to the k times the
two one eigenvector
557
00:31:57,610 --> 00:32:04,230
and some multiple of point seven
to the k times the minus one
558
00:32:04,230 --> 00:32:07,430
one eigenvector.
559
00:32:07,430 --> 00:32:09,910
Right?
560
00:32:09,910 --> 00:32:11,290
That's -- I --
561
00:32:11,290 --> 00:32:15,010
this is how I take -- how
powers of a matrix work.
562
00:32:15,010 --> 00:32:21,360
When I apply those powers to
a u0, what I -- so it's u0,
563
00:32:21,360 --> 00:32:26,390
which was zero a thousand --
564
00:32:26,390 --> 00:32:29,220
that has to be corrected k=0.
565
00:32:29,220 --> 00:32:35,720
So I'm plugging in k=0 and I get
c1 times two one and c2 times
566
00:32:35,720 --> 00:32:38,640
minus one one.
567
00:32:38,640 --> 00:32:44,990
Two equations, two
constants, certainly
568
00:32:44,990 --> 00:32:49,710
independent eigenvectors,
so there's a solution
569
00:32:49,710 --> 00:32:51,770
and you see what it is?
570
00:32:51,770 --> 00:32:56,450
Let's see, I guess we already
figured that c1 was a thousand
571
00:32:56,450 --> 00:32:59,390
over three, I think -- did we
think that had to be a thousand
572
00:32:59,390 --> 00:33:02,200
over three?
573
00:33:02,200 --> 00:33:05,840
And maybe c2 would be --
574
00:33:05,840 --> 00:33:07,920
excuse me, let --
get an eraser --
575
00:33:07,920 --> 00:33:08,930
we can --
576
00:33:08,930 --> 00:33:11,330
I just -- I think
we've -- get it here.
577
00:33:11,330 --> 00:33:15,590
c2, we want to get a
zero here, so maybe we
578
00:33:15,590 --> 00:33:21,400
need plus two
thousand over three.
579
00:33:21,400 --> 00:33:22,840
I think that has to work.
580
00:33:22,840 --> 00:33:26,010
Two times a thousand over
three minus two thousand
581
00:33:26,010 --> 00:33:29,160
over three, that'll give
us the zero and a thousand
582
00:33:29,160 --> 00:33:32,020
over three and the two thousand
over three will give us
583
00:33:32,020 --> 00:33:33,290
three thousand over three,
584
00:33:33,290 --> 00:33:34,160
the thousand.
585
00:33:34,160 --> 00:33:37,820
So this is what we approach --
586
00:33:37,820 --> 00:33:42,000
this part, with the point
seven to the k-th power
587
00:33:42,000 --> 00:33:45,050
is the part that's disappearing.
588
00:33:45,050 --> 00:33:48,270
That's -- that's
Markov matrices.
589
00:33:48,270 --> 00:33:48,770
Okay.
590
00:33:48,770 --> 00:33:52,610
That's an example of
where they come from,
591
00:33:52,610 --> 00:34:00,220
from modeling movement of
people with no gain or loss,
592
00:34:00,220 --> 00:34:03,690
with total -- total
count conserved.
593
00:34:03,690 --> 00:34:04,440
Okay.
594
00:34:04,440 --> 00:34:07,260
I -- just if I can
add one more comment,
595
00:34:07,260 --> 00:34:11,550
because you'll see Markov
matrices in electrical
596
00:34:11,550 --> 00:34:16,949
engineering courses and often
you'll see them -- sorry,
597
00:34:16,949 --> 00:34:19,679
here's my little comment.
598
00:34:19,679 --> 00:34:22,280
Sometimes -- in a lot of
applications they prefer
599
00:34:22,280 --> 00:34:25,590
to work with row vectors.
600
00:34:25,590 --> 00:34:29,480
So they -- instead of -- this
was natural for us, right?
601
00:34:29,480 --> 00:34:32,429
For all the eigenvectors
to be column vectors.
602
00:34:32,429 --> 00:34:37,139
So our columns added to
one in the Markov matrix.
603
00:34:37,139 --> 00:34:39,889
Just so you don't
think, well, what --
604
00:34:39,889 --> 00:34:41,820
what's going on?
605
00:34:41,820 --> 00:34:48,170
If we work with row vectors and
we multiply vector times matrix
606
00:34:48,170 --> 00:34:51,179
-- so we're multiplying
from the left --
607
00:34:51,179 --> 00:34:55,679
then it'll be the then we'll
be using the transpose of --
608
00:34:55,679 --> 00:35:00,100
of this matrix and it'll be
the rows that add to one.
609
00:35:00,100 --> 00:35:05,390
So in other textbooks,
you'll see -- instead of col-
610
00:35:05,390 --> 00:35:08,200
columns adding to one,
you'll see rows add to one.
611
00:35:08,200 --> 00:35:09,320
Okay.
612
00:35:09,320 --> 00:35:10,410
Fine.
613
00:35:10,410 --> 00:35:13,030
Okay, that's what I wanted
to say about Markov,
614
00:35:13,030 --> 00:35:17,560
now I want to say something
about projections and even
615
00:35:17,560 --> 00:35:22,190
leading in -- a little
into Fourier series.
616
00:35:22,190 --> 00:35:24,960
Because -- but before
any Fourier stuff,
617
00:35:24,960 --> 00:35:28,150
let me make a comment
about projections.
618
00:35:28,150 --> 00:35:33,310
This -- so this is a comment
about projections onto --
619
00:35:33,310 --> 00:35:36,526
with an orthonormal basis.
620
00:35:42,420 --> 00:35:49,770
So, of course, the basis
vectors are q1 up to qn.
621
00:35:49,770 --> 00:35:50,860
Okay.
622
00:35:50,860 --> 00:35:52,330
I have a vector b.
623
00:35:52,330 --> 00:35:58,700
Let -- let me imagine -- let
me imagine this is a basis.
624
00:35:58,700 --> 00:36:00,640
Let -- let's say I'm in n by n.
625
00:36:00,640 --> 00:36:06,660
I'm -- I've got, eh,
n orthonormal vectors,
626
00:36:06,660 --> 00:36:09,350
I'm in n dimensional space
so they're a complete --
627
00:36:09,350 --> 00:36:11,110
they're a basis --
628
00:36:11,110 --> 00:36:16,170
any vector v could be
expanded in this basis.
629
00:36:16,170 --> 00:36:22,050
So any vector v is some
combination, some amount of q1
630
00:36:22,050 --> 00:36:28,060
plus some amount of q2
plus some amount of qn.
631
00:36:28,060 --> 00:36:35,220
So -- so any v.
632
00:36:35,220 --> 00:36:41,740
I just want you to tell
me what those amounts are.
633
00:36:41,740 --> 00:36:46,510
What are x1 -- what's
x1, for example?
634
00:36:46,510 --> 00:36:49,980
So I'm looking
for the expansion.
635
00:36:49,980 --> 00:36:51,900
This is -- this is
really our projection.
636
00:36:51,900 --> 00:36:56,330
I could -- I could really
use the word expansion.
637
00:36:56,330 --> 00:37:01,450
I'm expanding the
vector in the basis.
638
00:37:01,450 --> 00:37:05,770
And the special thing about the
basis is that it's orthonormal.
639
00:37:05,770 --> 00:37:10,470
So that should give me a
special formula for the answer,
640
00:37:10,470 --> 00:37:12,220
for the coefficients.
641
00:37:12,220 --> 00:37:14,010
So how do I get x1?
642
00:37:14,010 --> 00:37:15,510
What -- what's a formula for x1?
643
00:37:18,250 --> 00:37:23,040
I could -- I can go
through the projection --
644
00:37:23,040 --> 00:37:26,940
the Q transpose Q, all that --
645
00:37:26,940 --> 00:37:31,450
normal equations, but --
646
00:37:31,450 --> 00:37:32,140
and I'll get --
647
00:37:32,140 --> 00:37:33,960
I'll come out with
this nice answer
648
00:37:33,960 --> 00:37:36,060
that I think I can
see right away.
649
00:37:36,060 --> 00:37:38,370
How can I pick --
650
00:37:38,370 --> 00:37:42,920
get a hold of x1 and get these
other x-s out of the equation?
651
00:37:42,920 --> 00:37:47,460
So how can I get a nice,
simple formula for x1?
652
00:37:47,460 --> 00:37:50,751
And then we want to see, sure,
we knew that all the time.
653
00:37:50,751 --> 00:37:51,250
Okay.
654
00:37:51,250 --> 00:37:52,610
So what's x1?
655
00:37:52,610 --> 00:37:59,020
The good way is take the inner
product of everything with q1.
656
00:37:59,020 --> 00:38:02,430
Take the inner product of that
whole equation, every term,
657
00:38:02,430 --> 00:38:03,850
with q1.
658
00:38:03,850 --> 00:38:08,370
What will happen
to that last term?
659
00:38:08,370 --> 00:38:11,560
The inner product -- when -- if
I take the dot product with q1
660
00:38:11,560 --> 00:38:13,970
I get zero, right?
661
00:38:13,970 --> 00:38:17,160
Because this basis
was orthonormal.
662
00:38:17,160 --> 00:38:20,330
If I take the dot product
with q2 I get zero.
663
00:38:20,330 --> 00:38:24,570
If I take the dot product
with q1 I get one.
664
00:38:24,570 --> 00:38:28,740
So that tells me what
x1 is. q1 transpose
665
00:38:28,740 --> 00:38:31,250
v, that's taking
the dot product,
666
00:38:31,250 --> 00:38:41,060
is x1 times q1 transpose
q1 plus a bunch of zeroes.
667
00:38:41,060 --> 00:38:45,330
And this is a one,
so I can forget that.
668
00:38:45,330 --> 00:38:47,820
I get x1 immediately.
669
00:38:47,820 --> 00:38:49,840
So -- do you see
what I'm saying --
670
00:38:49,840 --> 00:38:53,070
is that I have an
orthonormal basis,
671
00:38:53,070 --> 00:38:58,760
then the coefficient that I
need for each basis vector is
672
00:38:58,760 --> 00:38:59,340
a cinch to
673
00:38:59,340 --> 00:39:00,070
find.
674
00:39:00,070 --> 00:39:02,310
Let me -- let me just --
675
00:39:02,310 --> 00:39:05,580
I have to put this into
matrix language, too,
676
00:39:05,580 --> 00:39:07,350
so you'll see it there also.
677
00:39:07,350 --> 00:39:10,690
If I write that first equation
in matrix language, what --
678
00:39:10,690 --> 00:39:12,010
what is it?
679
00:39:12,010 --> 00:39:13,870
I'm writing -- in
matrix language,
680
00:39:13,870 --> 00:39:19,020
this equation says I'm taking
these columns -- are --
681
00:39:19,020 --> 00:39:20,620
are you guys good at this now?
682
00:39:20,620 --> 00:39:29,180
I'm taking those columns times
the Xs and getting V, right?
683
00:39:29,180 --> 00:39:30,440
That's the matrix form.
684
00:39:30,440 --> 00:39:37,464
Okay, that's the
matrix Q. Qx is v.
685
00:39:37,464 --> 00:39:39,005
What's the solution
to that equation?
686
00:39:41,570 --> 00:39:44,920
It's -- of course, it's
x equal Q inverse v.
687
00:39:44,920 --> 00:39:51,000
So x is Q inverse v,
but what's the point?
688
00:39:51,000 --> 00:39:54,060
Q inverse in this case
is going to -- is simple.
689
00:39:54,060 --> 00:39:58,220
I don't have to work to
invert this matrix Q,
690
00:39:58,220 --> 00:40:03,110
because of the fact that the --
these columns are orthonormal,
691
00:40:03,110 --> 00:40:05,110
I know the inverse to that.
692
00:40:05,110 --> 00:40:10,070
And it is Q transpose.
693
00:40:10,070 --> 00:40:15,050
When you see a Q, a square
matrix with that letter Q,
694
00:40:15,050 --> 00:40:17,090
the -- that just triggers --
695
00:40:17,090 --> 00:40:19,670
Q inverse is the
same as Q transpose.
696
00:40:19,670 --> 00:40:22,100
So the first component, then --
697
00:40:22,100 --> 00:40:25,970
the first component of
x is the first row times
698
00:40:25,970 --> 00:40:29,550
v, and what's that?
699
00:40:29,550 --> 00:40:32,650
The first component
of this answer
700
00:40:32,650 --> 00:40:36,570
is the first row of Q transpose.
701
00:40:36,570 --> 00:40:42,110
That's just -- that's
just q1 transpose times v.
702
00:40:42,110 --> 00:40:46,320
So that's what we
concluded here, too.
703
00:40:46,320 --> 00:40:46,820
Okay.
704
00:40:49,590 --> 00:40:55,090
So -- so nothing Fourier here.
705
00:40:55,090 --> 00:40:59,650
The -- the key ingredient
here was that the q-s are
706
00:40:59,650 --> 00:41:01,460
orthonormal.
707
00:41:01,460 --> 00:41:04,900
And now that's what Fourier
series are built on.
708
00:41:04,900 --> 00:41:08,520
So now, in the
remaining time, let
709
00:41:08,520 --> 00:41:11,760
me say something
about Fourier series.
710
00:41:11,760 --> 00:41:12,400
Okay.
711
00:41:12,400 --> 00:41:20,120
So Fourier series is --
712
00:41:20,120 --> 00:41:25,150
well, we've got a
function f of x.
713
00:41:25,150 --> 00:41:28,160
And we want to write it
as a combination of --
714
00:41:28,160 --> 00:41:30,900
maybe it has a constant term.
715
00:41:30,900 --> 00:41:34,910
And then it has
some cos(x) in it.
716
00:41:34,910 --> 00:41:38,120
And it has some sin(x) in it.
717
00:41:38,120 --> 00:41:42,330
And it has some cos(2x) in it.
718
00:41:42,330 --> 00:41:45,230
And a -- and some
sin(2x), and forever.
719
00:41:50,780 --> 00:41:54,600
So what's -- what's the
difference between this type
720
00:41:54,600 --> 00:41:56,900
problem and the one above it?
721
00:41:56,900 --> 00:42:02,090
This one's infinite,
but the key property
722
00:42:02,090 --> 00:42:06,110
of things being
orthogonal is still
723
00:42:06,110 --> 00:42:09,790
true for sines and cosines,
so it's the property that
724
00:42:09,790 --> 00:42:11,170
makes Fourier series work.
725
00:42:11,170 --> 00:42:12,860
So that's called
a Fourier series.
726
00:42:12,860 --> 00:42:14,620
Better write his name up.
727
00:42:14,620 --> 00:42:15,530
Fourier series.
728
00:42:22,170 --> 00:42:25,960
So it was Joseph Fourier
who realized that, hey, I
729
00:42:25,960 --> 00:42:29,870
could work in function space.
730
00:42:29,870 --> 00:42:33,970
Instead of a vector v, I
could have a function f of x.
731
00:42:33,970 --> 00:42:38,590
Instead of orthogonal
vectors, q1, q2 , q3,
732
00:42:38,590 --> 00:42:42,180
I could have orthogonal
functions, the constant,
733
00:42:42,180 --> 00:42:45,370
the cos(x), the
sin(x), the s- cos(2x),
734
00:42:45,370 --> 00:42:47,450
but infinitely many of them.
735
00:42:47,450 --> 00:42:49,930
I need infinitely
many, because my space
736
00:42:49,930 --> 00:42:52,440
is infinite dimensional.
737
00:42:52,440 --> 00:42:56,970
So this is, like, the moment
in which we leave finite
738
00:42:56,970 --> 00:43:00,680
dimensional vector spaces and go
to infinite dimensional vector
739
00:43:00,680 --> 00:43:03,090
spaces and our basis --
740
00:43:03,090 --> 00:43:07,250
so the vectors are
now functions --
741
00:43:07,250 --> 00:43:09,650
and of course, there are so
many functions that it's --
742
00:43:09,650 --> 00:43:13,620
that we've got an infin-
infinite dimensional space --
743
00:43:13,620 --> 00:43:17,320
and the basis vectors
are functions, too.
744
00:43:17,320 --> 00:43:25,400
a0, the constant function one
-- so my basis is one cos(x),
745
00:43:25,400 --> 00:43:29,190
sin(x), cos(2x),
sin(2x) and so on.
746
00:43:29,190 --> 00:43:32,710
And the reason Fourier
series is a success
747
00:43:32,710 --> 00:43:35,170
is that those are orthogonal.
748
00:43:35,170 --> 00:43:35,760
Okay.
749
00:43:35,760 --> 00:43:37,190
Now what do I mean
by orthogonal?
750
00:43:40,110 --> 00:43:44,490
I know what it means for two
vectors to be orthogonal --
751
00:43:44,490 --> 00:43:46,870
y transpose x
equals zero, right?
752
00:43:46,870 --> 00:43:48,600
Dot product equals zero.
753
00:43:48,600 --> 00:43:52,070
But what's the dot
product of functions?
754
00:43:52,070 --> 00:43:55,340
I'm claiming that whatever
it is, the dot product --
755
00:43:55,340 --> 00:43:59,840
or we would more likely use
the word inner product of, say,
756
00:43:59,840 --> 00:44:02,430
cos(x) with sin(x) is zero.
757
00:44:02,430 --> 00:44:06,340
And cos(x) with
cos(2x), also zero.
758
00:44:06,340 --> 00:44:08,780
So I -- let me tell you
what I mean by that,
759
00:44:08,780 --> 00:44:10,410
by that dot product.
760
00:44:10,410 --> 00:44:12,850
Well, how do I
compute a dot product?
761
00:44:12,850 --> 00:44:16,970
So, let's just remember
for vectors v trans-
762
00:44:16,970 --> 00:44:23,050
v transpose w for vectors,
so this was vectors,
763
00:44:23,050 --> 00:44:30,145
v transpose w was
v1w1 +...+vnwn.
764
00:44:33,510 --> 00:44:34,140
Okay.
765
00:44:34,140 --> 00:44:34,750
Now functions.
766
00:44:40,100 --> 00:44:42,720
Now I have two functions,
let's call them f and g.
767
00:44:45,260 --> 00:44:46,700
What's with them now?
768
00:44:46,700 --> 00:44:49,730
The vectors had n
components, but the functions
769
00:44:49,730 --> 00:44:53,060
have a whole, like, continuum.
770
00:44:53,060 --> 00:44:55,640
To graph the function, I
just don't have n points,
771
00:44:55,640 --> 00:44:57,780
I've got this whole graph.
772
00:44:57,780 --> 00:44:59,780
So I have functions --
773
00:44:59,780 --> 00:45:01,590
I'm really trying
to ask you what's
774
00:45:01,590 --> 00:45:03,840
the inner product
of this function
775
00:45:03,840 --> 00:45:05,170
f with another function
776
00:45:05,170 --> 00:45:06,130
g?
777
00:45:06,130 --> 00:45:11,870
And I want to make it parallel
to this the best I can.
778
00:45:11,870 --> 00:45:20,080
So the best parallel is to
multiply f (x) times g(x)
779
00:45:20,080 --> 00:45:23,060
at every x --
780
00:45:23,060 --> 00:45:25,120
and here I just had
n multiplications,
781
00:45:25,120 --> 00:45:28,330
but here I'm going to
have a whole range of x-s,
782
00:45:28,330 --> 00:45:31,980
and here I added the results.
783
00:45:31,980 --> 00:45:34,600
What do I do here?
784
00:45:34,600 --> 00:45:38,400
So what's the analog of
addition when you have --
785
00:45:38,400 --> 00:45:40,350
when you're in a continuum?
786
00:45:40,350 --> 00:45:41,830
It's integration.
787
00:45:41,830 --> 00:45:47,630
So that the -- the dot product
of two functions will be
788
00:45:47,630 --> 00:45:49,525
the integral of
those functions, dx.
789
00:45:52,240 --> 00:45:55,010
Now I have to say -- say,
well, what are the limits
790
00:45:55,010 --> 00:45:56,800
of integration?
791
00:45:56,800 --> 00:46:03,980
And for this Fourier series,
this function f(x) --
792
00:46:03,980 --> 00:46:08,000
if I'm going to have -- if that
right hand side is going to be
793
00:46:08,000 --> 00:46:11,920
f(x), that function that
I'm seeing on the right,
794
00:46:11,920 --> 00:46:16,130
all those sines and cosines,
they're all periodic, with --
795
00:46:16,130 --> 00:46:18,210
with period two pi.
796
00:46:18,210 --> 00:46:21,650
So -- so that's what
f(x) had better be.
797
00:46:21,650 --> 00:46:24,870
So I'll integrate
from zero to two pi.
798
00:46:24,870 --> 00:46:29,160
My -- all -- everything -- is
on the interval zero two pi now,
799
00:46:29,160 --> 00:46:33,690
because if I'm going to use
these sines and cosines,
800
00:46:33,690 --> 00:46:39,950
then f(x) is equal to f(x+2pi).
801
00:46:39,950 --> 00:46:42,428
This is periodic --
802
00:46:45,180 --> 00:46:48,250
periodic functions.
803
00:46:48,250 --> 00:46:49,800
Okay.
804
00:46:49,800 --> 00:46:52,770
So now I know what --
805
00:46:52,770 --> 00:46:56,120
I've got all the
right words now.
806
00:46:56,120 --> 00:47:00,490
I've got a vector space, but
the vectors are functions.
807
00:47:00,490 --> 00:47:05,450
I've got inner products and
-- and the inner product gives
808
00:47:05,450 --> 00:47:07,760
a number, all right.
809
00:47:07,760 --> 00:47:12,260
It just happens to be an
integral instead of a sum.
810
00:47:12,260 --> 00:47:15,230
I've got -- and that -- then I
have the idea of orthogonality
811
00:47:15,230 --> 00:47:15,860
--
812
00:47:15,860 --> 00:47:18,470
because, actually, just
-- let's just check.
813
00:47:18,470 --> 00:47:21,910
Orthogonality -- if I take
the integral -- s- I --
814
00:47:21,910 --> 00:47:25,890
let me do sin(x) times cos(x) --
815
00:47:25,890 --> 00:47:31,106
sin(x) times cos(x) dx
from zero to two pi --
816
00:47:34,680 --> 00:47:36,700
I think we get zero.
817
00:47:36,700 --> 00:47:40,670
That's the differential of
that, so it would be one half
818
00:47:40,670 --> 00:47:42,890
sine x squared, was that right?
819
00:47:47,220 --> 00:47:50,100
Between zero and two pi --
820
00:47:50,100 --> 00:47:52,910
and, of course, we get zero.
821
00:47:52,910 --> 00:47:58,150
And the same would be
true with a little more --
822
00:47:58,150 --> 00:48:02,430
some trig identities to help
us out -- of every other pair.
823
00:48:02,430 --> 00:48:05,650
So we have now an
orthonormal infinite
824
00:48:05,650 --> 00:48:10,160
basis for function space,
and all we want to do
825
00:48:10,160 --> 00:48:12,980
is express a function in that
826
00:48:12,980 --> 00:48:14,090
basis.
827
00:48:14,090 --> 00:48:20,120
And so I -- the end of my
lecture is, okay, what is a1?
828
00:48:20,120 --> 00:48:24,210
What's the coefficient --
how much cos(x) is there
829
00:48:24,210 --> 00:48:30,230
in a function compared
to the other harmonics?
830
00:48:30,230 --> 00:48:32,940
How much constant
is in that function?
831
00:48:32,940 --> 00:48:35,310
That'll -- that would
be an easy question.
832
00:48:35,310 --> 00:48:39,110
The answer a0 will come out
to be the average value of f.
833
00:48:39,110 --> 00:48:40,620
That's the amount
of the constant
834
00:48:40,620 --> 00:48:42,630
that's in there,
its average value.
835
00:48:42,630 --> 00:48:45,350
But let's take a1
as more typical.
836
00:48:45,350 --> 00:48:48,170
How will I get -- here's the
end of the lecture, then --
837
00:48:48,170 --> 00:48:49,400
how do I get a1?
838
00:48:52,070 --> 00:48:54,720
The first Fourier coefficient.
839
00:48:54,720 --> 00:48:56,350
Okay.
840
00:48:56,350 --> 00:48:59,790
I do just as I did
in the vector case.
841
00:48:59,790 --> 00:49:03,850
I take the inner product
of everything with cos(x)
842
00:49:03,850 --> 00:49:07,010
Take the inner product of
everything with cos(x).
843
00:49:07,010 --> 00:49:08,800
Then on the left --
844
00:49:08,800 --> 00:49:13,990
on the left I have -- the inner
product is the integral of f(x)
845
00:49:13,990 --> 00:49:15,365
times cos(x) cx.
846
00:49:18,740 --> 00:49:22,080
And on the right,
what do I have?
847
00:49:22,080 --> 00:49:24,970
When I -- so what I -- when I
say take the inner product with
848
00:49:24,970 --> 00:49:28,850
cos(x), let me put it in
ordinary calculus words.
849
00:49:28,850 --> 00:49:32,610
Multiply by cos(x)
and integrate.
850
00:49:32,610 --> 00:49:34,540
That's what inner products are.
851
00:49:34,540 --> 00:49:36,940
So if I multiply that
whole thing by cos(x)
852
00:49:36,940 --> 00:49:40,990
and I integrate, I get
a whole lot of zeroes.
853
00:49:40,990 --> 00:49:45,830
The only thing that
survives is that term.
854
00:49:45,830 --> 00:49:47,250
All the others disappear.
855
00:49:47,250 --> 00:49:53,950
So -- and that term is a1 times
the integral of cos(x) squared
856
00:49:53,950 --> 00:50:01,580
dx zero to 2pi equals -- so this
was the left side and this is
857
00:50:01,580 --> 00:50:04,840
all that's left on
the right-hand side.
858
00:50:04,840 --> 00:50:09,700
And this is not zero of
course, because it's the length
859
00:50:09,700 --> 00:50:13,570
of the function squared, it's
the inner product with itself,
860
00:50:13,570 --> 00:50:18,050
and -- and a simple calculation
gives that answer to be pi.
861
00:50:18,050 --> 00:50:23,360
So that's an easy integral
and it turns out to be pi,
862
00:50:23,360 --> 00:50:31,900
so that a1 is one over pi times
there -- times this integral.
863
00:50:31,900 --> 00:50:35,490
So there is, actually --
that's Euler's famous formula
864
00:50:35,490 --> 00:50:39,110
for the -- or maybe
Fourier found it --
865
00:50:39,110 --> 00:50:41,405
for the coefficients
in a Fourier series.
866
00:50:43,980 --> 00:50:47,940
And you see that it's
exactly an expansion
867
00:50:47,940 --> 00:50:50,790
in an orthonormal basis.
868
00:50:50,790 --> 00:50:51,540
Okay, thanks.
869
00:50:51,540 --> 00:50:56,660
So I'll do a quiz review on
Monday and then the quiz itself
870
00:50:56,660 --> 00:50:59,200
in Walker on Wednesday.
871
00:50:59,200 --> 00:51:00,590
Okay, see you Monday.
872
00:51:00,590 --> 00:51:02,140
Thanks.