WEBVTT
00:00:09.720 --> 00:00:12.130
OK, this is quiz review day.
00:00:12.130 --> 00:00:17.750
The quiz coming up on
Wednesday will before this
00:00:17.750 --> 00:00:22.520
lecture the quiz will be this
hour one o'clock Wednesday
00:00:22.520 --> 00:00:27.740
in Walker, top floor of Walker,
closed book, all normal.
00:00:27.740 --> 00:00:33.760
I wrote down what we've
covered in this second part
00:00:33.760 --> 00:00:38.830
of the course, and actually
I'm impressed as I write it.
00:00:38.830 --> 00:00:42.250
so that's chapter
four on orthogonality
00:00:42.250 --> 00:00:45.790
and you're remembering these --
00:00:45.790 --> 00:00:50.530
what this is suggesting,
these are those columns are
00:00:50.530 --> 00:00:56.810
orthonormal vectors, and then
we call that matrix Q and the --
00:00:56.810 --> 00:01:01.270
what's the key -- how do we
state the fact that those v-
00:01:01.270 --> 00:01:04.989
those columns are
orthonormal in terms of Q,
00:01:04.989 --> 00:01:11.920
it means that Q transpose
Q is the identity.
00:01:11.920 --> 00:01:15.480
So that's the matrix
statement of the --
00:01:15.480 --> 00:01:18.930
of the property that the
columns are orthonormal,
00:01:18.930 --> 00:01:21.710
the dot products are
either one or zero,
00:01:21.710 --> 00:01:30.160
and then we computed the
projections onto lines and onto
00:01:30.160 --> 00:01:35.920
subspaces, and we used that
to solve problems Ax=b in --
00:01:35.920 --> 00:01:38.800
in the least square sense,
when there was no solution,
00:01:38.800 --> 00:01:40.640
we found the best solution.
00:01:40.640 --> 00:01:43.530
And then finally this
Graham-Schmidt idea,
00:01:43.530 --> 00:01:50.790
which takes independent vectors
and lines them up, takes --
00:01:50.790 --> 00:01:54.000
subtracts off the
projections of the part
00:01:54.000 --> 00:01:58.390
you've already done, so that
the new part is orthogonal
00:01:58.390 --> 00:02:03.850
and so it takes a basis
to an orthonormal basis.
00:02:03.850 --> 00:02:07.440
And you -- those calculations
involve square roots a lot
00:02:07.440 --> 00:02:11.300
because you're making
things unit vectors,
00:02:11.300 --> 00:02:13.440
but you should know that step.
00:02:13.440 --> 00:02:16.540
OK, for determinants,
the three big --
00:02:16.540 --> 00:02:19.720
the big picture is the
properties of the determinant,
00:02:19.720 --> 00:02:24.020
one to three d- properties
one, two and three, d-
00:02:24.020 --> 00:02:27.590
that define the determinant,
and then four, five,
00:02:27.590 --> 00:02:31.110
six through ten
were consequences.
00:02:31.110 --> 00:02:35.330
Then the big formula that has
n factorial terms, half of them
00:02:35.330 --> 00:02:38.350
have plus signs and half
minus signs, and then
00:02:38.350 --> 00:02:40.420
the cofactor formula.
00:02:40.420 --> 00:02:46.240
So and which led us to a
formula for the inverse.
00:02:46.240 --> 00:02:48.780
And finally, just
so you know what's
00:02:48.780 --> 00:02:53.550
covered in from chapter
three, it's section six point
00:02:53.550 --> 00:02:58.920
one and two, so that's the
basic idea of eigenvalues
00:02:58.920 --> 00:03:02.660
and eigenvectors, the
equation for the eigenvalues,
00:03:02.660 --> 00:03:07.800
the mechanical step, this
is really Ax equal lambda
00:03:07.800 --> 00:03:11.450
x for all n
eigenvectors at once,
00:03:11.450 --> 00:03:14.480
if we have n independent
eigenvectors,
00:03:14.480 --> 00:03:18.520
and then using that to
compute powers of a matrix.
00:03:18.520 --> 00:03:22.430
So you notice the differential
equations not on this list,
00:03:22.430 --> 00:03:28.390
because that's six point three,
that that's for the third quiz.
00:03:28.390 --> 00:03:30.330
OK.
00:03:30.330 --> 00:03:36.140
Shall I what I usually do for
review is to take an old exam
00:03:36.140 --> 00:03:40.720
and just try to pick out
questions that are significant
00:03:40.720 --> 00:03:44.070
and write them quickly
on the board, shall I --
00:03:44.070 --> 00:03:46.210
shall I proceed that way again?
00:03:46.210 --> 00:03:51.650
This -- this exam is really old.
00:03:51.650 --> 00:03:54.770
November nineteen --
nineteen eighty-four,
00:03:54.770 --> 00:03:59.850
so that was before
the Web existed.
00:03:59.850 --> 00:04:04.760
So not only were the
lectures not on the Web,
00:04:04.760 --> 00:04:07.260
nobody even had a
Web page, my God.
00:04:07.260 --> 00:04:10.760
OK, so can I nevertheless
linear algebra
00:04:10.760 --> 00:04:14.530
was still as great as ever.
00:04:14.530 --> 00:04:22.290
So may I and that wasn't meant
to be a joke, OK, all right,
00:04:22.290 --> 00:04:25.575
so let me just take these
questions as they come.
00:04:28.420 --> 00:04:28.930
All right.
00:04:28.930 --> 00:04:30.260
OK.
00:04:30.260 --> 00:04:32.460
So the first question's
about projections.
00:04:32.460 --> 00:04:35.670
It says we're given
the line, the --
00:04:35.670 --> 00:04:39.930
the vector a is the
vector two, one, two,
00:04:39.930 --> 00:04:44.320
and I want to find the
projection matrix P that
00:04:44.320 --> 00:04:46.900
projects onto the
line through a.
00:04:46.900 --> 00:04:51.510
So my picture is, well I'm in
three dimensions, of course,
00:04:51.510 --> 00:04:55.710
so there's a vector, two --
there's the vector a, two, one,
00:04:55.710 --> 00:04:59.210
two, let me draw the
whole line through it,
00:04:59.210 --> 00:05:05.690
and I want to project any
vector b onto that line,
00:05:05.690 --> 00:05:08.490
and I'm looking for
the projection matrix.
00:05:08.490 --> 00:05:12.360
So -- so this -- the projection
matrix is the matrix that I
00:05:12.360 --> 00:05:15.320
multiply b by to get here.
00:05:15.320 --> 00:05:19.260
And I guess this first
part, this is just part one
00:05:19.260 --> 00:05:23.860
a, I'm really asking you the
-- the quick way to answer,
00:05:23.860 --> 00:05:28.600
to find P, is just to
remember what the formula is.
00:05:28.600 --> 00:05:31.740
And -- and we're in -- we're
projecting onto a line,
00:05:31.740 --> 00:05:37.800
so our formula, our -- our
usual formula is AA transpose A
00:05:37.800 --> 00:05:43.800
inverse A transpose, but
now A is just a column,
00:05:43.800 --> 00:05:49.920
one-column matrix, so it'll be
just a, so I'll just call it
00:05:49.920 --> 00:05:53.850
little a, little a transpose,
and this is just a number now,
00:05:53.850 --> 00:05:56.970
one by one, so I can put
it in the denominator,
00:05:56.970 --> 00:05:58.950
so there's our --
00:05:58.950 --> 00:06:01.560
that's really what we
want to remember, and --
00:06:01.560 --> 00:06:05.490
and using that two, one,
two, what will I get?
00:06:05.490 --> 00:06:08.280
I'm dividing by --
00:06:08.280 --> 00:06:12.020
what's the length
squared of that vector?
00:06:12.020 --> 00:06:13.970
So what's a transpose a?
00:06:13.970 --> 00:06:18.370
Looks like nine, and what's
the matrix, well, I'm --
00:06:18.370 --> 00:06:21.580
I'm doing two, one,
two against two,
00:06:21.580 --> 00:06:28.520
one, two, so it's one-ninth of
this matrix four, two, four,
00:06:28.520 --> 00:06:30.790
two, one, two, four, two, four.
00:06:34.250 --> 00:06:37.410
Now the next part asked
about eigenvalues.
00:06:37.410 --> 00:06:39.450
So you see we're --
since we're learning,
00:06:39.450 --> 00:06:43.070
we know a lot more now, we can
make connections between these
00:06:43.070 --> 00:06:47.880
chapters, so what's
the eigenvector,
00:06:47.880 --> 00:06:50.200
what are the eigenvalues of P?
00:06:50.200 --> 00:06:53.140
I could ask what's
the rank of P. What's
00:06:53.140 --> 00:06:55.860
the rank of that matrix?
00:06:55.860 --> 00:06:58.120
Uh -- one.
00:06:58.120 --> 00:06:59.410
Rank is one.
00:06:59.410 --> 00:07:01.780
What's the column space?
00:07:01.780 --> 00:07:06.780
If I apply P to all vectors
then I fill up the column space,
00:07:06.780 --> 00:07:09.650
it's combinations of the
columns, so what's the column
00:07:09.650 --> 00:07:10.880
space?
00:07:10.880 --> 00:07:13.370
Well, it's just this line.
00:07:13.370 --> 00:07:17.310
The column space is the
line through two, one, two.
00:07:17.310 --> 00:07:20.770
And now what's the eigenvalue?
00:07:20.770 --> 00:07:24.090
So, or since that
matrix has rank one,
00:07:24.090 --> 00:07:26.750
so tell me the eigenvalues
of this matrix.
00:07:29.330 --> 00:07:31.330
It's a singular
matrix, so it certainly
00:07:31.330 --> 00:07:34.970
has an eigenvalue zero.
00:07:34.970 --> 00:07:39.470
Actually, the rank
is only one, so that
00:07:39.470 --> 00:07:42.720
means that there
like going to be
00:07:42.720 --> 00:07:48.120
t- a two-dimensional null
space, there'll be at least two,
00:07:48.120 --> 00:07:50.990
this lambda equals zero
will be repeated twice
00:07:50.990 --> 00:07:55.630
because I can find two
independent eigenvectors
00:07:55.630 --> 00:07:58.740
with lambda equals zero, and
then of course since it's
00:07:58.740 --> 00:08:02.370
got three eigenvalues,
what's the third one?
00:08:02.370 --> 00:08:03.300
It's one.
00:08:03.300 --> 00:08:05.080
How do I know it's one?
00:08:05.080 --> 00:08:09.640
Either from the trace, which is
nine over nine, which is one,
00:08:09.640 --> 00:08:13.810
or by remembering what --
00:08:13.810 --> 00:08:16.350
what is the eigenvector,
and actually now
00:08:16.350 --> 00:08:18.350
it's going to ask
for the eigenvector,
00:08:18.350 --> 00:08:21.545
so what's the eigenvector
for that eigenvalue?
00:08:24.150 --> 00:08:28.310
What's the eigenvector
for that eigenvalue?
00:08:28.310 --> 00:08:35.020
It's the -- it's the vector that
doesn't move, eigenvalue one,
00:08:35.020 --> 00:08:37.460
so the vector that doesn't move
00:08:37.460 --> 00:08:38.630
is a.
00:08:38.630 --> 00:08:41.390
This a is the --
00:08:41.390 --> 00:08:47.040
is -- is also the eigenvector
with lambda equal one,
00:08:47.040 --> 00:08:51.250
because if I apply the
projection matrix to a, I get
00:08:51.250 --> 00:08:52.290
a again.
00:08:52.290 --> 00:08:55.580
Everybody sees that if I apply
that matrix to a, can I do it
00:08:55.580 --> 00:08:59.380
in little letters, if I
apply that matrix to a,
00:08:59.380 --> 00:09:02.320
then I have a transpose
a canceling a transpose a
00:09:02.320 --> 00:09:04.190
and I get a again.
00:09:04.190 --> 00:09:06.720
So sure enough, Pa equals a.
00:09:09.620 --> 00:09:12.220
And the eigenvalue is one.
00:09:12.220 --> 00:09:13.350
OK.
00:09:13.350 --> 00:09:16.140
Good.
00:09:16.140 --> 00:09:22.840
now, actually it asks
you further to solve this
00:09:22.840 --> 00:09:27.200
difference equation, so
this will be -- this is --
00:09:27.200 --> 00:09:36.030
this is solve u(k+1)=Puk,
starting from u0 equal nine,
00:09:36.030 --> 00:09:36.655
nine, zero.
00:09:39.590 --> 00:09:41.740
And find uk.
00:09:44.380 --> 00:09:49.950
So -- so what's up?
00:09:49.950 --> 00:09:53.140
Shall we find u1 first of all?
00:09:53.140 --> 00:09:55.870
So just to get started.
00:09:55.870 --> 00:09:57.780
So what is u1?
00:09:57.780 --> 00:10:01.570
It's Pu0 of course.
00:10:01.570 --> 00:10:04.480
So if I do the projection of --
00:10:04.480 --> 00:10:10.020
of this vector onto the line,
so this is like my vector b now
00:10:10.020 --> 00:10:12.070
that I'm projecting
onto the line,
00:10:12.070 --> 00:10:19.410
I get a times a transpose
u0 over a transpose a.
00:10:19.410 --> 00:10:23.080
Well, one way or another I
just do this multiplication.
00:10:23.080 --> 00:10:26.100
but maybe this is the
easiest way to do it.
00:10:26.100 --> 00:10:29.530
a transpose, can I
remember what a is on
00:10:29.530 --> 00:10:30.340
this board?
00:10:30.340 --> 00:10:37.110
Two, one, two, so I'm projecting
onto the line through there.
00:10:37.110 --> 00:10:41.450
This is the projection,
it's P times the vector u0,
00:10:41.450 --> 00:10:44.590
so what do I have for a
transpose u0 looks like
00:10:44.590 --> 00:10:47.000
eighteen, looks
like twenty-seven,
00:10:47.000 --> 00:10:50.140
and a transpose a
we figured was nine,
00:10:50.140 --> 00:10:57.790
so it's three a, so that this
is the -- this is the x hat,
00:10:57.790 --> 00:11:00.870
the -- the multiple of
a, in -- in our formulas,
00:11:00.870 --> 00:11:03.162
and of course that's
six, three, six.
00:11:06.650 --> 00:11:08.320
So that's u1.
00:11:08.320 --> 00:11:11.500
Computed out directly.
00:11:11.500 --> 00:11:17.530
That's on the line through a and
it's the closest point to u0,
00:11:17.530 --> 00:11:19.600
and it's just Pu0.
00:11:19.600 --> 00:11:23.070
You just straightforward
multiplication produces that.
00:11:23.070 --> 00:11:23.620
OK.
00:11:23.620 --> 00:11:25.300
Now, what's u2?
00:11:28.420 --> 00:11:31.310
Well, u2 is Pu1, I agree.
00:11:31.310 --> 00:11:32.750
Do I need to compute that again?
00:11:35.540 --> 00:11:41.730
No, because once I'm already on
the line through A, uk will be,
00:11:41.730 --> 00:11:45.220
I could do the
projection k times,
00:11:45.220 --> 00:11:47.635
but it's enough
just to do it once.
00:11:50.380 --> 00:11:53.570
It's the same, it's the
same, six, three, six.
00:11:53.570 --> 00:11:59.780
So this is a case
where I could and --
00:11:59.780 --> 00:12:03.620
and actually on the quiz
if you see one of these,
00:12:03.620 --> 00:12:09.360
which could very well be there,
and it could very well be not
00:12:09.360 --> 00:12:16.120
a projection matrix, then we
would use all the eigenvalues
00:12:16.120 --> 00:12:16.940
and eigenvectors.
00:12:16.940 --> 00:12:19.790
Let's think for a moment,
how do you do those?
00:12:19.790 --> 00:12:24.400
M - the point of this small part
of a question was that when P
00:12:24.400 --> 00:12:28.900
is a projection matrix, so that
P squared equals P and P cubed
00:12:28.900 --> 00:12:33.600
equals P, then -- then we don't
need to get into the mechanics
00:12:33.600 --> 00:12:38.520
of all knowing all the other
eigenvalues and eigenvectors.
00:12:38.520 --> 00:12:40.670
We just can go directly.
00:12:40.670 --> 00:12:48.200
But if P was now some other
matrix, can you just --
00:12:48.200 --> 00:12:51.140
let's just remember from these
very recent lectures how you
00:12:51.140 --> 00:12:52.560
would proceed.
00:12:52.560 --> 00:12:57.590
from these very recent lectures
we know that uk we would --
00:12:57.590 --> 00:13:02.780
we would expand u0 as a
combination of eigenvectors.
00:13:02.780 --> 00:13:07.020
Let me leave -- yeah, as a
combination of eigenvectors,
00:13:07.020 --> 00:13:12.650
c1x1, some multiple of
the second eigenvector,
00:13:12.650 --> 00:13:15.470
some multiple of the
third eigenvector,
00:13:15.470 --> 00:13:22.320
and then A to the ku0
would be c1, so this --
00:13:22.320 --> 00:13:27.270
we have to find these numbers
here, that's the work actually.
00:13:27.270 --> 00:13:30.280
The work is find
the eigenvalues,
00:13:30.280 --> 00:13:33.500
find the eigenvectors and
find the c-s because they all
00:13:33.500 --> 00:13:35.320
come into the formula.
00:13:35.320 --> 00:13:50.230
We have -- so -- so to do
this, you can see what you have
00:13:50.230 --> 00:13:51.280
to compute.
00:13:51.280 --> 00:13:53.320
You have to compute
the eigenvalues,
00:13:53.320 --> 00:13:55.430
you have to compute
the eigenvectors,
00:13:55.430 --> 00:13:59.230
and then to match u0
you compute the c-s,
00:13:59.230 --> 00:14:01.210
and then you've got it.
00:14:01.210 --> 00:14:06.610
So it's -- it's just that's a
formula that shows what pieces
00:14:06.610 --> 00:14:07.810
we need.
00:14:07.810 --> 00:14:12.550
And what would actually happen
in the case of this projection
00:14:12.550 --> 00:14:13.070
matrix?
00:14:13.070 --> 00:14:15.310
If this A is a
projection matrix,
00:14:15.310 --> 00:14:19.050
then a couple of
eigenvalues are zero.
00:14:19.050 --> 00:14:21.200
That's why we just
throw those away.
00:14:21.200 --> 00:14:27.780
The other eigenvalue was a one,
so that we got the same thing
00:14:27.780 --> 00:14:30.890
every time, c3x3.
00:14:30.890 --> 00:14:33.050
From the first
time, second time,
00:14:33.050 --> 00:14:37.160
third, all iterations pro-
left us with this constant,
00:14:37.160 --> 00:14:39.620
left us right here
at six, three, six.
00:14:39.620 --> 00:14:41.220
But maybe I take --
00:14:41.220 --> 00:14:43.840
I'm taking this chance
to remind you of what
00:14:43.840 --> 00:14:47.900
to do for other matrices.
00:14:47.900 --> 00:14:48.600
OK.
00:14:48.600 --> 00:14:52.910
So that's part way through.
00:14:52.910 --> 00:14:53.460
OK.
00:14:53.460 --> 00:14:55.980
The next question in
nineteen eighty-four
00:14:55.980 --> 00:15:01.030
is fitting a straight
line to points.
00:15:01.030 --> 00:15:04.712
And actually a straight
line through the origin.
00:15:04.712 --> 00:15:06.170
A straight line
through the origin.
00:15:06.170 --> 00:15:10.120
So can I go to question two?
00:15:10.120 --> 00:15:15.360
So this is fitting a straight
line to these points, can I --
00:15:15.360 --> 00:15:26.580
I'll just give you the points
at t=1 the y is four, at t=2,
00:15:26.580 --> 00:15:30.710
y is five, at t=3, y is eight.
00:15:34.190 --> 00:15:42.630
So we've got points one, two,
three, four, five, and eight.
00:15:42.630 --> 00:15:45.720
And I'm trying to fit a
straight line through the origin
00:15:45.720 --> 00:15:48.540
to these three values.
00:15:48.540 --> 00:15:57.170
OK, so my equation that
I'm allowing myself
00:15:57.170 --> 00:16:00.950
is just y equal Dt.
00:16:00.950 --> 00:16:03.470
So I have only one unknown.
00:16:03.470 --> 00:16:04.950
One degree of freedom.
00:16:04.950 --> 00:16:07.730
One parameter D.
00:16:07.730 --> 00:16:12.510
So I'm expecting to end up
my matrix so my -- my --
00:16:12.510 --> 00:16:14.510
when I try to --
00:16:14.510 --> 00:16:17.440
when I try to fit a
straight line, that
00:16:17.440 --> 00:16:19.660
goes through the origin,
that's because it goes
00:16:19.660 --> 00:16:22.570
through the origin, I've
lost the constant c here,
00:16:22.570 --> 00:16:27.330
so I have just this should
be a quick calculation.
00:16:27.330 --> 00:16:32.590
and I can write down the three
equations that -- that --
00:16:32.590 --> 00:16:33.200
that would --
00:16:33.200 --> 00:16:36.990
I'd like to solve if the line
went through the points, that's
00:16:36.990 --> 00:16:38.080
a good start.
00:16:38.080 --> 00:16:40.015
Because that
displays the matrix.
00:16:43.410 --> 00:16:45.270
So can I continue that problem?
00:16:45.270 --> 00:16:48.760
We would like to solve--
00:16:48.760 --> 00:16:54.200
so y is Dt, so I'd like
to solve D times one times
00:16:54.200 --> 00:17:00.030
D equals four, two times
D equals five and three
00:17:00.030 --> 00:17:02.690
times D equals eight.
00:17:02.690 --> 00:17:04.500
That would be perfection.
00:17:04.500 --> 00:17:09.150
If I could find such
a D, then the line y
00:17:09.150 --> 00:17:13.349
equal Dt would satisfy
all three equations,
00:17:13.349 --> 00:17:16.500
would go through all three
points, but it doesn't exist.
00:17:16.500 --> 00:17:19.140
So -- so I have to
solve this -- so the --
00:17:19.140 --> 00:17:23.680
my matrix is now you can see my
matrix, it just has one column.
00:17:23.680 --> 00:17:27.589
Multiplying a scalar D.
00:17:27.589 --> 00:17:30.260
And you can see the
right-hand side.
00:17:30.260 --> 00:17:31.500
This is my Ax=b.
00:17:34.280 --> 00:17:38.860
I don't need three equals signs
now because I've got vectors.
00:17:38.860 --> 00:17:39.620
OK.
00:17:39.620 --> 00:17:43.610
There's Ax=b and you
take it from there.
00:17:43.610 --> 00:17:47.570
You the -- the best x will
be -- will come from --
00:17:47.570 --> 00:17:49.510
so what's the --
the key equation?
00:17:49.510 --> 00:17:54.960
So this is the A, this is
the Ax hat equal b equation.
00:17:54.960 --> 00:17:57.900
Well, Ax=b.
00:17:57.900 --> 00:18:00.590
And what's the
equation for x hat?
00:18:00.590 --> 00:18:08.490
The best D, so to find
the best D, the best x,
00:18:08.490 --> 00:18:14.660
the equation is A
transpose A, the best D,
00:18:14.660 --> 00:18:20.900
is A transpose times
the right-hand side.
00:18:20.900 --> 00:18:24.340
This is all coming from
projection on a line, our --
00:18:24.340 --> 00:18:26.620
our matrix only has one column.
00:18:26.620 --> 00:18:30.070
So A transpose A would
be maybe fourteen,
00:18:30.070 --> 00:18:35.410
D hat, and A transpose
b I'm getting four, ten,
00:18:35.410 --> 00:18:37.520
and twenty-four.
00:18:37.520 --> 00:18:38.590
Is that right?
00:18:38.590 --> 00:18:40.950
Four, ten and twenty-four.
00:18:40.950 --> 00:18:43.000
So thirty-eight.
00:18:43.000 --> 00:18:47.530
So that tells me the
best D hat is D hat
00:18:47.530 --> 00:18:52.150
is thirty-eight over fourteen.
00:18:52.150 --> 00:18:53.310
OK.
00:18:53.310 --> 00:18:54.930
Fine.
00:18:54.930 --> 00:18:57.620
All right.
00:18:57.620 --> 00:18:59.240
so we found the best line.
00:18:59.240 --> 00:19:01.830
And now here's a --
here's the next question.
00:19:01.830 --> 00:19:07.050
What vector did I just
project onto what line?
00:19:07.050 --> 00:19:12.210
See in this section on least
squares here's the key point,
00:19:12.210 --> 00:19:12.740
I'm --
00:19:12.740 --> 00:19:14.700
I'm asking you to think
of the least squares
00:19:14.700 --> 00:19:17.240
problem in two ways.
00:19:17.240 --> 00:19:18.890
Two different pictures.
00:19:18.890 --> 00:19:20.080
Two different graphs.
00:19:20.080 --> 00:19:22.320
One graph is this.
00:19:22.320 --> 00:19:28.060
This is a graph in the -- in the
b -- in the tb plane, ty plane.
00:19:28.060 --> 00:19:30.900
The -- the -- the line itself.
00:19:30.900 --> 00:19:32.940
The other picture I'm
asking you to think of
00:19:32.940 --> 00:19:35.050
is like my projection picture.
00:19:35.050 --> 00:19:38.610
What -- what projection --
what -- what vector I --
00:19:38.610 --> 00:19:42.940
I projecting onto what line or
what subspace when I -- when I
00:19:42.940 --> 00:19:44.050
do this?
00:19:44.050 --> 00:19:48.700
So the -- my second picture is
a projection picture that --
00:19:48.700 --> 00:19:51.280
that sees the whole
thing with vectors.
00:19:51.280 --> 00:19:53.970
Here's my vector of course
that I'm projecting.
00:19:53.970 --> 00:20:06.800
I'm projecting that vector b
onto the column space of A.
00:20:06.800 --> 00:20:15.800
Of if you like -- it's just
a line onto that's the line
00:20:15.800 --> 00:20:18.620
it's just a line, of course.
00:20:18.620 --> 00:20:21.070
That's what this
calculation is doing.
00:20:21.070 --> 00:20:25.595
This is computing the best D,
which is -- this is the x hat.
00:20:30.230 --> 00:20:34.090
So -- so seeing it as a
projection means I don't see
00:20:34.090 --> 00:20:36.570
the projection in
this figure, right?
00:20:36.570 --> 00:20:39.060
In this figure I'm not
projecting those points
00:20:39.060 --> 00:20:41.760
onto that line or
anything of the sort.
00:20:41.760 --> 00:20:48.430
The projection s-picture for --
for least squares is in the --
00:20:48.430 --> 00:20:51.900
in the space where b
lies, the whole vector b,
00:20:51.900 --> 00:20:54.870
and the columns of A.
00:20:54.870 --> 00:21:02.570
And then the x is
the best combination
00:21:02.570 --> 00:21:04.600
that gives the projection.
00:21:04.600 --> 00:21:05.100
OK.
00:21:05.100 --> 00:21:07.980
So that's a chance
to tell me that.
00:21:07.980 --> 00:21:08.760
OK.
00:21:08.760 --> 00:21:12.310
I'll go -- OK now
finally in orthogonality
00:21:12.310 --> 00:21:14.610
there's the Graham-Schmidt idea.
00:21:14.610 --> 00:21:20.230
So that's problem two D here.
00:21:20.230 --> 00:21:24.840
It asks me if I have two
vectors, a1 equal one, two,
00:21:24.840 --> 00:21:32.650
three, and a2 equal
one, one, one, find
00:21:32.650 --> 00:21:37.650
two orthogonal
vectors in that plane.
00:21:37.650 --> 00:21:43.480
So those two vectors give
a plane, they give a plane.
00:21:43.480 --> 00:21:47.860
Which is of course the --
the column space of the --
00:21:47.860 --> 00:21:50.710
of the matrix.
00:21:50.710 --> 00:21:55.050
And I'm looking for
an orthogonal basis
00:21:55.050 --> 00:21:55.710
for that plane.
00:21:55.710 --> 00:21:58.300
So I'm looking for two
orthogonal vectors.
00:21:58.300 --> 00:22:02.100
And of course there
are lots of --
00:22:02.100 --> 00:22:05.330
I mean, I've got a plane there.
00:22:05.330 --> 00:22:08.700
If I get one orthogonal
pair, I can rotate it.
00:22:08.700 --> 00:22:10.910
There's not just
one answer here.
00:22:10.910 --> 00:22:16.140
But Graham-Schmidt says OK,
start with the first vector,
00:22:16.140 --> 00:22:19.840
and let that be --
and keep that one.
00:22:19.840 --> 00:22:23.300
And then take the second
one orthogonal to this.
00:22:23.300 --> 00:22:27.400
So -- so Graham-Schmidt says
start with this one and then
00:22:27.400 --> 00:22:31.220
make a second vector B, can
I call that second vector B,
00:22:31.220 --> 00:22:35.970
this is going to be orthogonal
to, so perpendicular to a1.
00:22:35.970 --> 00:22:40.650
If I can with my chalk
create the key equation.
00:22:40.650 --> 00:22:46.740
This vector B is going
to be this one, one, one,
00:22:46.740 --> 00:22:50.720
but that one, one -- one, one,
one is not perpendicular to a1,
00:22:50.720 --> 00:22:54.340
so I have to subtract
off its projection,
00:22:54.340 --> 00:23:00.820
I have to subtract off the B,
the -- the B trans- ye the B --
00:23:00.820 --> 00:23:07.780
the -- the I should say the a1
transpose b over a1 transpose
00:23:07.780 --> 00:23:11.400
a1, that multiple of
a1, I've got to remove.
00:23:14.990 --> 00:23:17.090
So I just have to
compute what that is,
00:23:17.090 --> 00:23:21.440
and I get ano- I get a vector
B that's orthogonal to a1.
00:23:21.440 --> 00:23:24.440
It's the -- it's --
00:23:24.440 --> 00:23:29.630
it's the original vector
minus its projection.
00:23:29.630 --> 00:23:32.110
Oh, so what is --
00:23:32.110 --> 00:23:33.490
I mean this to be a2.
00:23:36.130 --> 00:23:39.430
So I'm projecting a2
onto the line through a1.
00:23:39.430 --> 00:23:40.190
Yeah.
00:23:40.190 --> 00:23:42.190
That's the part that I
don't want because that's
00:23:42.190 --> 00:23:45.250
in the direction I already
have, so I subtract off
00:23:45.250 --> 00:23:48.170
that projection and I
get the part I want,
00:23:48.170 --> 00:23:50.240
the orthogonal part.
00:23:50.240 --> 00:23:52.540
So that's the
Graham-Schmidt thing
00:23:52.540 --> 00:23:54.640
and we can put numbers in.
00:23:54.640 --> 00:23:55.370
OK.
00:23:55.370 --> 00:24:00.710
one, one, one take away
a1 transpose a2 is six,
00:24:00.710 --> 00:24:06.900
a1 transpose a1 is
fourteen,multiplying a1.
00:24:06.900 --> 00:24:13.400
And that gives us the
new orthogonal vector B.
00:24:13.400 --> 00:24:15.340
Because I only ask for
orthogonal right now,
00:24:15.340 --> 00:24:19.200
I don't have to divide
by the length which
00:24:19.200 --> 00:24:20.590
will involve a square root.
00:24:20.590 --> 00:24:23.210
OK.
00:24:23.210 --> 00:24:23.940
Third question.
00:24:27.540 --> 00:24:28.677
Third question.
00:24:28.677 --> 00:24:29.510
All right, let me --
00:24:29.510 --> 00:24:33.270
I'll move this board up.
00:24:33.270 --> 00:24:37.930
third question will probably
be about eigenvalues.
00:24:37.930 --> 00:24:38.610
OK.
00:24:38.610 --> 00:24:41.200
Three.
00:24:41.200 --> 00:24:44.020
This is a four-by-four matrix.
00:24:44.020 --> 00:24:47.320
Its eigenvalues are lambda
one, lambda two, lambda three,
00:24:47.320 --> 00:24:50.310
lambda four.
00:24:50.310 --> 00:24:52.930
Question one.
00:24:52.930 --> 00:24:55.870
What's the condition
on the lambdas so
00:24:55.870 --> 00:24:59.380
that the matrix is invertible?
00:24:59.380 --> 00:24:59.990
OK.
00:24:59.990 --> 00:25:02.510
So under what conditions
on the lambdas
00:25:02.510 --> 00:25:05.900
will the matrix be invertible?
00:25:05.900 --> 00:25:08.850
So that's easy.
00:25:08.850 --> 00:25:17.780
Invertible if what's the
condition on the lambdas?
00:25:17.780 --> 00:25:19.370
None of them are zero.
00:25:19.370 --> 00:25:23.900
A zero eigenvalue would mean
something in the null space
00:25:23.900 --> 00:25:28.850
would mean a solution to
Ax=0x, but we're invertible,
00:25:28.850 --> 00:25:32.170
so none of them is
zero, the product --
00:25:32.170 --> 00:25:34.120
however you want to say, no --
00:25:34.120 --> 00:25:38.510
no zero eigenvalues.
00:25:38.510 --> 00:25:39.170
Good.
00:25:39.170 --> 00:25:43.200
OK, what's the
determinant of A inverse?
00:25:43.200 --> 00:25:45.555
The determinant of A inverse?
00:25:49.280 --> 00:25:52.120
So where is that
going to come from?
00:25:52.120 --> 00:25:55.470
Well, if we knew the
eigenvalues of A inverse,
00:25:55.470 --> 00:25:59.780
we could multiply them together
to find the determinant.
00:25:59.780 --> 00:26:02.110
And we do know the
eigenvalues of A inverse.
00:26:02.110 --> 00:26:04.370
What are they?
00:26:04.370 --> 00:26:08.580
They're just one
over lambda one times
00:26:08.580 --> 00:26:11.470
one over lambda two, that's
the second eigenvalue,
00:26:11.470 --> 00:26:13.640
the third eigenvalue and the
00:26:13.640 --> 00:26:14.760
fourth.
00:26:14.760 --> 00:26:18.770
So the product of the four
eigenvalues of the inverse
00:26:18.770 --> 00:26:21.450
will give us the
determinant of the inverse.
00:26:21.450 --> 00:26:21.950
Fine.
00:26:21.950 --> 00:26:23.110
OK.
00:26:23.110 --> 00:26:39.600
And what's the
trace of A plus I?
00:26:39.600 --> 00:26:41.170
So what do we know about trace?
00:26:44.160 --> 00:26:46.510
It's the sum down
the diagonal, but we
00:26:46.510 --> 00:26:48.630
don't know what our matrix is.
00:26:48.630 --> 00:26:51.960
The trace is also the
sum of the eigenvalues,
00:26:51.960 --> 00:26:55.710
and we do know the
eigenvalues of A plus I.
00:26:55.710 --> 00:26:57.410
So we just add them up.
00:26:57.410 --> 00:27:02.520
So what -- what's the first
eigenvalue of A plus I?
00:27:02.520 --> 00:27:05.610
When the matrix A has
eigenvalues lambda one, two,
00:27:05.610 --> 00:27:08.030
three and four,
then the eigenvalues
00:27:08.030 --> 00:27:11.610
if I add the identity, that
moves all the eigenvalues
00:27:11.610 --> 00:27:17.450
by one, so I just add
up lambda one plus one,
00:27:17.450 --> 00:27:22.520
lambda two plus one, and so on,
lambda three plus one, lambda
00:27:22.520 --> 00:27:27.180
four plus one, so it's lambda
one plus lambda two plus lambda
00:27:27.180 --> 00:27:32.180
three plus lambda
four plus four.
00:27:32.180 --> 00:27:34.850
Right.
00:27:34.850 --> 00:27:38.110
That movement by the identity
moved all the eigenvalues
00:27:38.110 --> 00:27:42.290
by one, so it moved the
whole trace by four.
00:27:42.290 --> 00:27:45.690
So it was the trace
of A plus four more.
00:27:45.690 --> 00:27:46.880
OK.
00:27:46.880 --> 00:27:48.240
Let's see.
00:27:48.240 --> 00:27:51.600
We may be finished this
quiz twenty minutes early.
00:27:51.600 --> 00:27:52.420
No.
00:27:52.420 --> 00:27:54.700
There's another question.
00:27:54.700 --> 00:27:57.590
Oh, God, OK.
00:27:57.590 --> 00:27:59.270
How did this class ever do it?
00:27:59.270 --> 00:28:02.710
Well, you'll see.
you'll be able to do it.
00:28:02.710 --> 00:28:03.510
OK.
00:28:03.510 --> 00:28:06.950
this has got to be a
determinant question.
00:28:06.950 --> 00:28:09.290
All right.
00:28:09.290 --> 00:28:13.810
More determinants and cofactors
and big formula question.
00:28:13.810 --> 00:28:14.430
OK.
00:28:14.430 --> 00:28:18.760
Let me do that.
00:28:18.760 --> 00:28:22.820
So it's about a matrix, a --
a whole family of matrices.
00:28:22.820 --> 00:28:26.200
Here's the four-by-four one.
00:28:26.200 --> 00:28:29.990
The four-by-four one is, and
-- and all the matrices in this
00:28:29.990 --> 00:28:35.240
family are tridiagonal with --
00:28:35.240 --> 00:28:38.230
with ones.
00:28:38.230 --> 00:28:40.110
Otherwise zeroes.
00:28:40.110 --> 00:28:43.480
So that's the pattern, and
we've seen this matrix.
00:28:43.480 --> 00:28:44.840
OK.
00:28:44.840 --> 00:28:47.850
So the -- it's tridiagonal
with ones on the diagonal,
00:28:47.850 --> 00:28:52.790
ones above and ones below, and
you see the general formula An,
00:28:52.790 --> 00:28:58.780
so I'll use Dn for
the determinant of An.
00:28:58.780 --> 00:28:59.950
OK.
00:28:59.950 --> 00:29:00.990
All right.
00:29:00.990 --> 00:29:02.950
So I'm going to do a --
00:29:02.950 --> 00:29:14.740
the first question is use
cofactors to show that Dn is
00:29:14.740 --> 00:29:20.680
something times D(n-1) plus
something times D(n-2).
00:29:20.680 --> 00:29:23.150
And find those somethings.
00:29:23.150 --> 00:29:23.650
OK.
00:29:26.310 --> 00:29:31.020
So this -- the fact that it's
tridiagonal with these constant
00:29:31.020 --> 00:29:37.290
diagonals means that there
is such a recurrence formula.
00:29:37.290 --> 00:29:39.660
And so the first
question is find it.
00:29:39.660 --> 00:29:41.810
Well, what's the
recurrence formula?
00:29:41.810 --> 00:29:43.550
OK, how does it go?
00:29:43.550 --> 00:29:47.390
So I'll use cofactors
along the first row.
00:29:47.390 --> 00:29:51.570
So I take that number
times its cofactor.
00:29:51.570 --> 00:29:56.730
So it's one times its cofactor
and what is its cofactor?
00:29:56.730 --> 00:30:00.120
D(n-1), right, exactly,
the cofactor is this --
00:30:00.120 --> 00:30:03.510
is this guy uses up
row one and column one,
00:30:03.510 --> 00:30:07.670
so the cofactor is down
here, so it's one of those.
00:30:10.230 --> 00:30:12.550
OK, that's the
first cofactor term.
00:30:12.550 --> 00:30:17.090
Now the other cofactor
term is this guy.
00:30:17.090 --> 00:30:20.740
Which uses up row
one and column two
00:30:20.740 --> 00:30:24.870
and what's surprising
about that?
00:30:24.870 --> 00:30:30.660
When you use row one and column
two that brings in a minus.
00:30:30.660 --> 00:30:32.940
There'll be a minus
because the --
00:30:32.940 --> 00:30:38.490
the cofactor is this
determinant times minus one.
00:30:38.490 --> 00:30:44.440
The the one-two cofactor is
that determinant with its sign
00:30:44.440 --> 00:30:45.460
changed.
00:30:45.460 --> 00:30:45.960
OK.
00:30:45.960 --> 00:30:47.270
So I have to look
at that determinant
00:30:47.270 --> 00:30:48.645
and I have to
remember in my head
00:30:48.645 --> 00:30:50.620
a sign is going to get changed.
00:30:50.620 --> 00:30:51.160
OK.
00:30:51.160 --> 00:30:54.970
Now how do I do
that determinant?
00:30:54.970 --> 00:30:58.180
How do I make that one clear?
00:30:58.180 --> 00:31:01.810
I -- the -- the neat way to
do is -- is here I see I --
00:31:01.810 --> 00:31:05.570
I'll use cofactors
down the first column.
00:31:05.570 --> 00:31:08.880
Because the first column is
all zeroes except for that one,
00:31:08.880 --> 00:31:13.220
so this one is now --
and what's its cofactor?
00:31:13.220 --> 00:31:17.490
Within this three-by-three its
cofactor will be two-by-two,
00:31:17.490 --> 00:31:19.080
and what is it?
00:31:19.080 --> 00:31:20.970
It's this, right?
00:31:20.970 --> 00:31:25.730
So -- so that part is all gone,
so I'm taking that times its
00:31:25.730 --> 00:31:29.550
cofactor, then zero times
whatever its cofactor is,
00:31:29.550 --> 00:31:34.040
so it's really just one times
and what's this in the general
00:31:34.040 --> 00:31:35.510
n-by-n case?
00:31:35.510 --> 00:31:39.420
It's Dn minus two.
00:31:39.420 --> 00:31:43.660
But now so is this a plus or
sign or a minus sign, it's --
00:31:43.660 --> 00:31:48.360
it's just a one, because there's
a one from there and a one from
00:31:48.360 --> 00:31:50.050
there.
00:31:50.050 --> 00:31:52.930
And is it a plus or a minus?
00:31:52.930 --> 00:31:56.010
It's minus I guess because
there was a minus the first time
00:31:56.010 --> 00:31:57.970
and then the second
time it's a plus,
00:31:57.970 --> 00:32:00.240
so it's overall it's a minus.
00:32:00.240 --> 00:32:04.864
So there's my a and b
were one and minus one.
00:32:04.864 --> 00:32:05.530
Those constants.
00:32:05.530 --> 00:32:09.140
Th- that's the --
that's the recurrence.
00:32:09.140 --> 00:32:10.220
OK.
00:32:10.220 --> 00:32:19.180
And oh, then it asks you to then
it asks you to solve this thing
00:32:19.180 --> 00:32:22.070
first by writing it as a --
00:32:22.070 --> 00:32:25.120
as a system.
00:32:25.120 --> 00:32:28.200
So now I'd like to
know the solution.
00:32:28.200 --> 00:32:30.600
I -- I better know
how it starts, right?
00:32:30.600 --> 00:32:34.020
It starts with D1,
what was D1, that's
00:32:34.020 --> 00:32:39.550
just the one-by-one case, so
D1 is one, and what is D2?
00:32:39.550 --> 00:32:41.880
Just to get us started and
then this would give us
00:32:41.880 --> 00:32:45.520
D3, D4, and forever.
00:32:45.520 --> 00:32:48.390
D2 is this two-by-two
that I'm seeing here
00:32:48.390 --> 00:32:51.620
and that determinant
is obviously zero.
00:32:51.620 --> 00:32:57.410
So those little ones will start
the recurrence and then we take
00:32:57.410 --> 00:32:58.030
off.
00:32:58.030 --> 00:33:02.070
And then the idea is to
write this recurrence as --
00:33:02.070 --> 00:33:11.990
as a Dn, D(n-1) is some
matrix times the one before,
00:33:11.990 --> 00:33:16.613
the D(n-1), D(n-2).
00:33:20.250 --> 00:33:22.270
What's the matrix?
00:33:22.270 --> 00:33:26.570
You see, you remember this step
of taking a single second order
00:33:26.570 --> 00:33:31.090
equation and by introducing
a vector unknown to make it
00:33:31.090 --> 00:33:32.600
into a --
00:33:32.600 --> 00:33:35.600
to a first order system.
00:33:35.600 --> 00:33:36.200
OK.
00:33:36.200 --> 00:33:41.860
So Dn is one of Dn minus one
minus one, I think that --
00:33:41.860 --> 00:33:43.530
that goes in the
first row, right?
00:33:43.530 --> 00:33:45.640
From the equation above?
00:33:45.640 --> 00:33:48.200
And the second one is
this is the same as this,
00:33:48.200 --> 00:33:50.010
so one and zero are fine.
00:33:53.500 --> 00:33:55.330
So there's the matrix.
00:33:55.330 --> 00:33:55.920
OK.
00:33:55.920 --> 00:33:58.820
So now how do I proceed?
00:33:58.820 --> 00:34:01.590
We can guess what
this examiner's
00:34:01.590 --> 00:34:02.845
got in his little mind.
00:34:07.740 --> 00:34:09.120
well, find the eigenvalues.
00:34:12.760 --> 00:34:19.469
And actually it tells
us that the sixth power
00:34:19.469 --> 00:34:23.679
of these eigenvalues
turns out to be one.
00:34:23.679 --> 00:34:30.065
Uh, well, can -- can we get the
equation for the eigenvalues?
00:34:30.065 --> 00:34:31.940
Let's do it and let's
get a formula for them.
00:34:31.940 --> 00:34:32.880
OK.
00:34:32.880 --> 00:34:35.260
So what are the eigenvalues?
00:34:35.260 --> 00:34:39.590
I look at the -- the matrix,
this determinant one minus
00:34:39.590 --> 00:34:44.420
lambda and zero minus lambda,
and these guys are still there,
00:34:44.420 --> 00:34:49.400
I compute that determinant, I
get lambda squared minus lambda
00:34:49.400 --> 00:34:52.310
and then plus one.
00:34:52.310 --> 00:34:54.820
And I set that to zero.
00:34:54.820 --> 00:34:55.820
OK.
00:34:55.820 --> 00:34:59.540
So we're not Fibonacci here.
00:34:59.540 --> 00:35:02.930
We're -- we're not
seeing Fibonacci numbers.
00:35:02.930 --> 00:35:07.500
Because the sign -- we
had a sign change there.
00:35:07.500 --> 00:35:11.040
And it's not clear right
away whether these --
00:35:11.040 --> 00:35:13.380
whether this -- is it clear?
00:35:13.380 --> 00:35:18.400
Is this matrix
stable or unstable?
00:35:18.400 --> 00:35:21.000
When we take -- when we go
further and further out?
00:35:21.000 --> 00:35:23.520
Are these Ds increasing?
00:35:23.520 --> 00:35:25.050
Are they going to zero?
00:35:25.050 --> 00:35:27.860
Are they bouncing
around periodically?
00:35:27.860 --> 00:35:30.600
the answers have to be here.
00:35:30.600 --> 00:35:34.840
I would like to know how big
these lambdas are, right?
00:35:34.840 --> 00:35:37.880
And the point is probably
these -- let's -- let's see,
00:35:37.880 --> 00:35:38.930
what's lambda?
00:35:38.930 --> 00:35:42.780
From the quadratic
formula lambda is one,
00:35:42.780 --> 00:35:45.390
I switch the sign of
that, plus or minus
00:35:45.390 --> 00:35:50.570
the square root of one minus
4ac, I getting a minus three
00:35:50.570 --> 00:35:51.970
there?
00:35:51.970 --> 00:35:52.710
Over two.
00:35:58.860 --> 00:36:00.960
What's up?
00:36:00.960 --> 00:36:03.340
They're complex.
00:36:03.340 --> 00:36:08.240
The -- the eigenvalues are one
plus square root of three I
00:36:08.240 --> 00:36:15.600
over two and one minus square
root of three I over two.
00:36:15.600 --> 00:36:18.240
What's the magnitude of lambda?
00:36:18.240 --> 00:36:19.765
That's the key
point for stability.
00:36:22.560 --> 00:36:26.480
These are two numbers
in the complex plane.
00:36:26.480 --> 00:36:30.650
One plus some -- somewhere
here, and its complex conjugate
00:36:30.650 --> 00:36:32.820
there.
00:36:32.820 --> 00:36:37.580
I want to know how far from
the origin are those numbers.
00:36:37.580 --> 00:36:40.840
What's the magnitude of lambda?
00:36:40.840 --> 00:36:43.230
And do you see what it is?
00:36:43.230 --> 00:36:46.690
Do you recognize this
-- a number like that?
00:36:46.690 --> 00:36:50.220
Take the real part squared
and the imaginary part squared
00:36:50.220 --> 00:36:51.510
and add.
00:36:51.510 --> 00:36:53.840
What do you get?
00:36:53.840 --> 00:36:57.010
So the real part
squared is a quarter.
00:36:57.010 --> 00:36:59.990
The imaginary part
squared is three-quarters.
00:36:59.990 --> 00:37:01.200
They add to one.
00:37:01.200 --> 00:37:06.090
That's a number with --
that's on the unit circle.
00:37:06.090 --> 00:37:08.540
That's an e to the i theta.
00:37:08.540 --> 00:37:10.180
That's a cos(theta)+isin(theta).
00:37:10.180 --> 00:37:14.650
And what's theta?
00:37:14.650 --> 00:37:19.330
This -- this is like a complex
number that's worth knowing,
00:37:19.330 --> 00:37:23.670
it's not totally
obvious but it's nice.
00:37:23.670 --> 00:37:27.480
That's -- I should see that
as cos(theta)+isin(theta),
00:37:27.480 --> 00:37:30.770
and the angle that would
do that is sixty degrees,
00:37:30.770 --> 00:37:32.580
pi over three.
00:37:32.580 --> 00:37:36.520
So that's a -- let me
improve my picture.
00:37:36.520 --> 00:37:39.990
So those -- that's e
to the i pi over six --
00:37:39.990 --> 00:37:40.920
pi over three.
00:37:40.920 --> 00:37:46.840
This is -- this number is e
to the i pi over three and e
00:37:46.840 --> 00:37:49.840
to the minus i pi over three.
00:37:49.840 --> 00:37:54.400
We'll be doing more
complex numbers briefly
00:37:54.400 --> 00:37:56.545
but a little more in
the next two days.
00:37:59.560 --> 00:38:00.940
next two lectures.
00:38:00.940 --> 00:38:05.670
Anyway, the -- so what's
the deal with stability,
00:38:05.670 --> 00:38:08.600
what do the Dn-s do?
00:38:08.600 --> 00:38:13.570
Well, look, if -- if I take the
sixth power I'm around at one,
00:38:13.570 --> 00:38:16.300
the problem actually
told me this.
00:38:16.300 --> 00:38:18.950
The sixth power of those
eigenvalues brings me around to
00:38:18.950 --> 00:38:23.350
What does that tell you about
the matrix, by the way? one.
00:38:23.350 --> 00:38:26.010
Suppose you know -- this
was a great quiz question,
00:38:26.010 --> 00:38:29.510
so I should never have just
said it, but popped out.
00:38:29.510 --> 00:38:33.600
Suppose lambda one to the sixth
and lambda two to the sixth are
00:38:33.600 --> 00:38:36.930
-- are one, which they are.
00:38:36.930 --> 00:38:40.090
What does that tell me
about a m- a matrix?
00:38:40.090 --> 00:38:43.890
About my matrix A here.
00:38:43.890 --> 00:38:47.290
Well, what -- what matrix
is connected with lambda one
00:38:47.290 --> 00:38:49.180
to the sixth and lambda
two to the sixth?
00:38:49.180 --> 00:38:51.890
It's got to be the
matrix A to the sixth.
00:38:51.890 --> 00:38:55.700
So what is A to the
sixth for that matrix?
00:38:55.700 --> 00:39:00.170
It's got eigenvalues
one and one.
00:39:00.170 --> 00:39:04.720
Because when I take the
sixth power, actually, ye,
00:39:04.720 --> 00:39:09.610
if I take the sixth power b- all
the sixth power of that is one
00:39:09.610 --> 00:39:12.990
and the sixth power of that is
one, the sixth power of this
00:39:12.990 --> 00:39:16.660
is e to the two pi i, that's
one, the sixth power of this
00:39:16.660 --> 00:39:19.330
is e to the minus
two pi i, that's one.
00:39:19.330 --> 00:39:24.190
So the sixth powers, the -- the
sixth power of that matrix has
00:39:24.190 --> 00:39:27.850
eigenvalues one and
one, so what is it?
00:39:27.850 --> 00:39:30.400
It's the identity, right.
00:39:30.400 --> 00:39:33.940
So if I operate this -- if
I run this thing six times,
00:39:33.940 --> 00:39:35.930
I'm back where I was.
00:39:35.930 --> 00:39:39.340
The sixth power of that
matrix is the identity.
00:39:39.340 --> 00:39:40.550
Good.
00:39:40.550 --> 00:39:41.670
OK.
00:39:41.670 --> 00:39:44.890
So it'll loop around, it's
-- it doesn't go to zero,
00:39:44.890 --> 00:39:49.080
it doesn't blow up, it just
periodically goes around with
00:39:49.080 --> 00:39:50.360
period six.
00:39:50.360 --> 00:39:51.460
OK.
00:39:51.460 --> 00:39:54.440
let's just see if there's a --
00:39:54.440 --> 00:39:55.095
all right.
00:39:58.520 --> 00:40:00.020
I'll -- let's see.
00:40:00.020 --> 00:40:02.250
Could I also look at a --
00:40:02.250 --> 00:40:05.155
at a final exam from
nineteen ninety-two.
00:40:07.870 --> 00:40:11.570
I think that's yeah, let me
do that on this last board.
00:40:14.740 --> 00:40:18.080
It starts -- a lot of the
questions in this exam are
00:40:18.080 --> 00:40:20.760
about a family of matrices.
00:40:20.760 --> 00:40:25.300
Let me give you the fourth, the
fourth guy in the family is --
00:40:25.300 --> 00:40:29.880
has a one, so it's
zeroes on the diagonal,
00:40:29.880 --> 00:40:34.130
but these are going one,
two, three and so on.
00:40:34.130 --> 00:40:37.420
One, two, three, and so on.
00:40:37.420 --> 00:40:42.190
But, for the four-by-four
case I'm stopping at four.
00:40:42.190 --> 00:40:47.070
You see the pattern?
00:40:47.070 --> 00:40:49.670
It's a family of matrices
which is growing,
00:40:49.670 --> 00:40:52.760
and actually the numbers
-- it's symmetric, right,
00:40:52.760 --> 00:40:56.490
it's equal to A4 transpose.
00:40:56.490 --> 00:41:00.880
And we can ask all sorts of
questions about its null space,
00:41:00.880 --> 00:41:07.030
its range, r- its column
space find the projection
00:41:07.030 --> 00:41:12.330
matrix onto the column space
of A3, for example, is in here.
00:41:12.330 --> 00:41:21.730
So -- so one -- so A3 is zero,
one, zero, one, zero, two,
00:41:21.730 --> 00:41:24.100
zero, two, zero.
00:41:32.400 --> 00:41:36.900
OK, find the projection
matrix onto the column space.
00:41:36.900 --> 00:41:41.540
By the way, is that matrix
singular or invertible?
00:41:41.540 --> 00:41:42.310
Singular.
00:41:42.310 --> 00:41:45.750
Why do we know it's singular?
00:41:45.750 --> 00:41:50.500
I see that column three is
a multiple of column one.
00:41:50.500 --> 00:41:53.090
Or we could take
its determinant.
00:41:53.090 --> 00:41:55.730
So it's certainly singular.
00:41:55.730 --> 00:42:00.200
The projection will be
matrix will be three-by-three
00:42:00.200 --> 00:42:03.190
but it will project
onto the column space,
00:42:03.190 --> 00:42:05.610
it'll project onto this plane.
00:42:05.610 --> 00:42:09.240
The column space of A3, and
I guess I would find it from
00:42:09.240 --> 00:42:11.100
the formula AA --
00:42:11.100 --> 00:42:15.120
AA transpose A inverse,
I would have to --
00:42:15.120 --> 00:42:17.120
I would -- I guess
I would do all this.
00:42:20.970 --> 00:42:23.050
There may be a
better way, perhaps
00:42:23.050 --> 00:42:25.700
I could think there might
be a slightly quicker way,
00:42:25.700 --> 00:42:27.390
but that would come
out pretty fast.
00:42:27.390 --> 00:42:28.450
OK.
00:42:28.450 --> 00:42:31.260
So that's be the
projection matrix.
00:42:31.260 --> 00:42:32.360
Next question.
00:42:32.360 --> 00:42:35.010
Find the eigenvalues and
eigenvectors of that matrix.
00:42:35.010 --> 00:42:36.020
OK.
00:42:36.020 --> 00:42:38.590
There's a three-by-three
matrix, oh, yeah,
00:42:38.590 --> 00:42:40.560
so what are its eigenvalues
and eigenvectors,
00:42:40.560 --> 00:42:42.840
we haven't done any
three-by-threes.
00:42:42.840 --> 00:42:45.070
Let's do one.
00:42:45.070 --> 00:42:48.770
I want to find, so how
do I find eigenvalues?
00:42:48.770 --> 00:42:53.280
I take the determinant
of A3 minus lambda I.
00:42:53.280 --> 00:42:57.200
So this is you just have to --
so I'm subtracting lambda from
00:42:57.200 --> 00:43:02.650
the diagonal, and I have a
one, one, zero, zero, two,
00:43:02.650 --> 00:43:07.200
two there, and I just have
to find that determinant.
00:43:07.200 --> 00:43:11.030
OK, since it's three-by-three
I'll just go for it.
00:43:11.030 --> 00:43:16.850
This way gives me minus lambda
cubed and a zero and zero.
00:43:16.850 --> 00:43:20.060
Then in this direction which
has the minus sign, that's
00:43:20.060 --> 00:43:25.090
a zero, four lambdas, I
mean minus four lambdas,
00:43:25.090 --> 00:43:29.110
and minus another lambda, so
that's minus five lambdas,
00:43:29.110 --> 00:43:32.550
but that direction
goes with a minus sign,
00:43:32.550 --> 00:43:36.850
so I think it's
plus five lambda.
00:43:36.850 --> 00:43:40.330
That looks like the determinant
of A3 minus lambda I,
00:43:40.330 --> 00:43:43.030
so I set it to zero.
00:43:43.030 --> 00:43:45.520
So what are the eigenvalues?
00:43:45.520 --> 00:43:48.950
Well, lambda equals zero --
lambda factors out of this,
00:43:48.950 --> 00:43:52.790
times minus lambda
squared plus four,
00:43:52.790 --> 00:44:03.610
so the eigenvalues are
five, thanks, thanks,
00:44:03.610 --> 00:44:08.470
so the eigenvalues are
zero, square root of five,
00:44:08.470 --> 00:44:11.410
and minus square root of five.
00:44:11.410 --> 00:44:13.970
And I would never write
down those three eigenvalues
00:44:13.970 --> 00:44:17.120
without checking the
trace to tell the truth.
00:44:17.120 --> 00:44:19.520
Because -- because we did a
bunch of calculations here
00:44:19.520 --> 00:44:22.990
but then I can quickly add up
the eigenvalues to get zero,
00:44:22.990 --> 00:44:27.270
add up the trace to get
zero, and feel that I'm --
00:44:27.270 --> 00:44:30.510
well, I guess that wouldn't have
caught my error if I'd made it
00:44:30.510 --> 00:44:34.280
-- if -- if that had been a
four I wouldn't have noticed,the
00:44:34.280 --> 00:44:37.660
determinant isn't anything
greatly useful here, right,
00:44:37.660 --> 00:44:41.310
because the determinant
is just zero.
00:44:41.310 --> 00:44:44.190
And so I never would
know whether that five
00:44:44.190 --> 00:44:48.990
was right or wrong, but
thanks for making it right.
00:44:48.990 --> 00:44:49.490
OK.
00:44:52.690 --> 00:44:54.380
Ha.
00:44:54.380 --> 00:45:00.760
Question two c, whoever
wrote this, probably me,
00:45:00.760 --> 00:45:03.270
said this is not difficult.
00:45:03.270 --> 00:45:06.570
I don't know why I put that in.
00:45:06.570 --> 00:45:10.640
just -- it asks for the
projection matrix onto
00:45:10.640 --> 00:45:12.050
the column space of A4.
00:45:15.402 --> 00:45:17.360
How could I have thought
that wasn't difficult?
00:45:17.360 --> 00:45:25.580
It looks extremely difficult.
what's the projection matrix
00:45:25.580 --> 00:45:27.505
onto the column space of A4?
00:45:32.250 --> 00:45:34.680
I don't know whether that --
this is not difficult is just
00:45:34.680 --> 00:45:37.650
like helpful or -- or insulting.
00:45:37.650 --> 00:45:41.380
Uh, what do you think?
00:45:41.380 --> 00:45:43.870
The -- what's the
column space of A4 here?
00:45:46.910 --> 00:45:52.800
Well, what's our first question
is is the matrix singular
00:45:52.800 --> 00:45:54.280
or invertible?
00:45:54.280 --> 00:45:58.210
If the answer is invertible,
then what's the column space?
00:46:00.770 --> 00:46:03.820
If -- if this matrix A4 is
invertible, so that's my guess,
00:46:03.820 --> 00:46:07.200
if this problem's easy it has
to be because this matrix is
00:46:07.200 --> 00:46:08.600
probably invertible.
00:46:08.600 --> 00:46:12.940
Then its column
space is R^4, good,
00:46:12.940 --> 00:46:15.260
the column space
is the whole space,
00:46:15.260 --> 00:46:18.320
and the answer to this easy
question is the projection
00:46:18.320 --> 00:46:23.180
matrix is the identity, it's the
four-by-four identity matrix.
00:46:23.180 --> 00:46:26.130
If this matrix is invertible.
00:46:26.130 --> 00:46:27.510
Shall we check invertibility?
00:46:27.510 --> 00:46:29.840
How would you find
its determinant?
00:46:29.840 --> 00:46:33.770
Can we just like take the
determinant of that matrix?
00:46:33.770 --> 00:46:37.010
I could ask you how -- so there
-- there are twenty-four terms,
00:46:37.010 --> 00:46:39.950
do we want to write all
twenty-four terms down?
00:46:39.950 --> 00:46:42.470
not in the remaining
ten seconds.
00:46:42.470 --> 00:46:43.930
Better to use cofactors.
00:46:43.930 --> 00:46:47.890
So I go along row
one, I see one --
00:46:47.890 --> 00:46:52.060
the only nonzero is this guy,
so I should take that one times
00:46:52.060 --> 00:46:53.380
the cofactor.
00:46:53.380 --> 00:46:55.680
Now so I'm down to
this determinant.
00:46:55.680 --> 00:46:56.640
OK.
00:46:56.640 --> 00:47:00.970
So now I'm -- look at this first
column, I see one times this,
00:47:00.970 --> 00:47:05.640
there's the cofactor of the
one, so I'm using up row one --
00:47:05.640 --> 00:47:09.530
row one and column one of
this three-by-three matrix,
00:47:09.530 --> 00:47:12.060
I'm down to this
cofactor, and by the way,
00:47:12.060 --> 00:47:14.370
those were both
plus signs, right?
00:47:14.370 --> 00:47:15.410
No, they weren't.
00:47:15.410 --> 00:47:17.240
That was a minus sign.
00:47:17.240 --> 00:47:18.490
That was a --
00:47:18.490 --> 00:47:22.540
that was a minus, and then that
was a plus, and then this, so
00:47:22.540 --> 00:47:24.670
what's the determinant?
00:47:24.670 --> 00:47:25.400
Nine.
00:47:25.400 --> 00:47:25.900
Nine.
00:47:25.900 --> 00:47:27.960
Determinant is nine.
00:47:27.960 --> 00:47:31.190
Determinant of A4 is nine.
00:47:31.190 --> 00:47:32.580
OK.
00:47:32.580 --> 00:47:38.640
Where A3, so my guess is I'll
put that on the final this
00:47:38.640 --> 00:47:43.320
year, the -- probably the odd-
numbered ones are singular
00:47:43.320 --> 00:47:47.830
and the even-numbered
ones are invertible.
00:47:47.830 --> 00:47:50.710
And I don't know
what the determinants
00:47:50.710 --> 00:47:55.750
are but I'm betting that
they have some nice formula.
00:47:55.750 --> 00:47:56.290
OK.
00:47:56.290 --> 00:48:02.770
So, recitations this week
will also be quiz review
00:48:02.770 --> 00:48:08.740
and then the quiz is
Wednesday at one o'clock.
00:48:08.740 --> 00:48:10.290
Thanks.