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MIT courses, visit MITopencourseware@ocw.MIT.edu. PROFESSOR: So
we're really moving along this review of the
highlights of linear algebra. And today it's matrices Q.
They get that name there. They have orthonormal columns. So that's what one looks like. And then the key fact,
orthonormal columns translates directly into that
simple fact that you just keep remembering every
time you see Q transpose Q, you've got the identity matrix. Let's just see why. So Q transpose would be-- I'll take those columns
and make them into rows. And then I multiply
by Q with the columns. And what do I get? Well hopefully, I get
the identity matrix. Why? Because-- oh yeah, the normal
part tells me that the length of each vector-- that's the length
squared Q transpose Q-- the length squared is one. So that gives me the one
in the identity matrix all along the diagonal. And then Q transpose times
a different Q is zero. That's the ortho part. So that gives me the zeros. So that's a simple
identity, but it translates from a lot of words
into a simple expression. Now does that mean that in the
other order, Q, Q transpose, is that the identity? So that's a question
to think about. Is Q, Q transpose
equal the identity? Question, sometimes
yes, sometimes no-- easy to tell which. If the answer is yes, yes when
Q is square, the answer is yes. If Q is a square m
a square matrix-- this is saying that
a square matrix Q has that inverse on its left. But for square matrices,
a left inverse, Q transpose is also
a right inverse. So for a square
matrix, if you have an inverse that
works on one side, it will work on the other side. So the answer is
yes in that case. And then in that case, that's
the case when we call Q is-- well, we really-- I don't know what
the right name would be, but here is the
name everybody uses, an orthogonal matrix. And that's only in their
square case, square. Q is an orthogonal matrix. Do you want to just see an
example of how that works? So if Q is rectangular-- let me do a rectangular
Q and a square Q. So I think there must be a
board up there somewhere. Here. OK, square. All right. Good to see some
orthogonal matrices, because my message
is that they are really important in all
kinds of applications. Let's start two by two. I can think of
two different ways to get an orthogonal matrix. That's a two by two matrix. And one of them you
will know immediately, cos theta sine theta. So that's a unit vector. It's normalized. Cos squared plus
sine squared is one. And this guy has to
be orthogonal to it. So I'll make that minus
sine theta and cos theta. Those are both length
one, they're orthogonal, then this is my Q.
And the inverse of Q will be the transpose. The transpose would put
the minus sign down here and would produce
the inverse matrix. And what is that particular
matrix represent? Geometrically, where
do we see that matrix? It's a rotation, thank you. It's a rotation of the
whole plane by theta. Yeah. So if I apply that to
one, zero for example, I get the first column, which
is cos theta sine theta. And that's just-- let
me draw a picture. That vector one, zero has
gotten rotated up to-- so there's the one, zero. And there, it got rotated
through an angle theta. And similarly,
zero, one will get rotated through an
angle theta to there. So the whole plane rotates. Oh, that makes me remember
a highly important, very important property of
Q. It doesn't change length. The length of any vector is
the same after you rotate it. The length of any vector is the
same after you multiply by Q. Can I just do that? I claim any x, any
vector x, I want to look at the length of Qx. And I claim it has
the same length as x. Actually, that's the
reason in computations that orthogonal matrix
are so much loved, because no overflow can happen
with orthogonal matrices. The lengths don't change. I can multiply by any number
of orthogonal matrices and the length don't change. Can we just see why that's true? So what do we have to go on? What we have to go on is
Q transpose Q equal I. Whatever we're going to prove,
it's got to come out of that, because that's all we know. So how do I use that
to get that one? Well, we haven't said a
whole lot about length, but you'll see it all now. It'll be easier to prove that
the squares are the same. So what's the what's
the matrix expression for the length squared? What's the right hand
side of that equation? X transpose x, right? X transpose x gives me
the sum of the squares. Pythagoras says that's
the length squared. So that right hand
side is x transpose x. What's the left side? It's the length
squared of this, of Qx. So it must be the same
as Qx transpose Qx. And the claim is that
that equation holds. And do you see it? So any property from Q has
to just come out directly from that. Where is it here? Do I just like, push away a
little bit at that left hand side and see it? Qx transpose is the same
as x transpose Q transpose. And Qx is Qx. And now I'm seeing-- well, you might
say, wait a minute. The parentheses were
there and there. But I say the most important
law for matrix multiplication is you can move parentheses
or throw them away. Let's throw them away. So in here I'm
seeing Q transpose Q, which is the identity. So it's true, yeah. So that means that you're
never under flow or overflow when you're multiplying by
Q. Every numerical algorithm is written to use orthogonal
matrices wherever it can. And here's the first example. I think it may be good for
me to think of other examples or for us to think
of other examples of orthogonal matrices. So I'm using that word
orthogonal matrix. We should really be
saying orthonormal. And I'm really thinking
mostly of square ones. So in this square case when
Q transpose is Q inverse. Of course, that fact
makes it easy to solve all equations that have Q
as a coefficient matrix, because you want the inverse
and you just use the transpose. Let's just take some minutes
to think of examples of Q's. If they're so
important, there have to be interesting examples. And that was a first one. Now there's one more two by two
example that you should know. Do you know what that would be? This will be an example two, and
it's also going to be only two by two and real. And what possibility
have I got left here? I'll use this same first
column, cos theta sine theta, because that's more or less any
unit vector in two dimensions, this has got that form. So what do you propose
for the second column? Yes? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, put the
minus sign down here. So you think does that
make any difference? So sine theta and
minus cos theta. I don't know if you've
ever looked at that matrix. We're trying to collect
together a few matrices that are worth knowing,
are worth looking at. Now what's happened here? You may say that was a trivial
change, which it kind of was. But it's a different matrix now. It's not a rotation anymore. That's not a rotation. And yeah, somehow
now it's symmetric. And yeah, it's eigenvectors
must be something or other. We'll get to those. But what does that matrix do? I don't know if you've seen it. If you haven't, it doesn't jump
out, but it's a important case. This is a reflection matrix. Notice that it's
determinant is minus one. You have minus cos square
theta, minus sine squared theta. It's determinant is minus one. There's some eigenvalue
coming up that's got a minus. So what do I mean by
a reflection matrix? Let me draw the plane. So one, zero, let's
follow that, follow again. One, zero, where does that go? That gives me the first column. So as before, it goes
to cos theta sine theta. And when I say
reflection, let me put the mirror into
the picture so you see what reflection it is. The mirror is along here
at angle theta over two, theta over two line. So sure enough, one,
zero at angle zero got reflected into a unit
vector at angle theta, and halfway between was
theta over two line. That's OK. Now what about the
other guy, zero, one? Here's zero, one. I multiply that. Can I put the zero,
one up here, so your I does the multiplication? Where is the result
of zero, one? What's the output from
Q applied to zero, one? Sine theta minus
cos theta, right? It's the second column. And so where is that? Well, it's perpendicular
to that guy. That's what I know. That was the point, that the
two columns are perpendicular. So it must go down
this way, right? Sine theta cos theta. And it doesn't
change the length. All these facts that we
just learned are key. So there's zero, one. And it goes to this guy, which
is whatever that second column is sine theta minus cos theta. And if you check
that, actually, gosh! This is like plain geometry. I believe-- it never occurred
to me before-- but I believe it, that this angle
going down to there, that that goes straight through,
and that the halfway one is that line. Yeah, I think that
picture has got it. And I think it's in the note. So that's a reflection matrix. Well, that's a two by
two reflection matrix. Would you like to see some
other matrices like this one, but larger? They're named after a
guy named Householder. So these are
Householder reflections. What am I doing here? I'm collecting together
some orthogonal matrices that are useful and important. And Householder found
a whole bunch of them. And his algorithm
is a much used part of numerical linear algebra. So he started with
a unit vector. Start with a unit vector
u transpose u equal one. So the length of
the vector is one. And then he created-- let's name it after him-- H. He created this matrix,
the identity minus two u, u transpose. And I believe that that's
a really useful matrix. I think this review is
like going beyond 18.06, into what ones are
really worth knowing, worth knowing individually. Could we just check what are
the properties of Householders reflection of that I minus two? You recognize here a
column times a row. So that's a matrix. And what could you tell me about
that matrix u, u transpose? It's yeah? What can we say about H? So I guess I'm believing that
H is a orthogonal matrix, otherwise it wouldn't
be here today. So I believe that-- and that
not only is it orthogonal, it is also-- have a look at it-- symmetric. It's also symmetric. The identity is symmetric. u, u transpose is symmetric. So this is a family of
symmetric orthogonal matrices. And that was one of them. That's a symmetric
orthogonal matrix. These matrices are
really great to have. In using linear
algebra, you just get a collection of useful
matrices that you can call on. And these are
definitely one family. Well, it's obviously symmetric. Shall we check that
it's orthogonal? So to check that
it's orthogonal, so I'm going to check that H
transpose H is the identity. Can I just check that? Well, H transpose is the same
as H because it was symmetric. So I'm going to square this guy. This is really H times H. I'm squaring it. And what do I get if I square-- So I get-- I hope I get the identity,
but let's see it. What do I get when
I square this? I get little-- multiply it out. So I times I is I. And then I get some
number of u, u transposes. How many do I get from that? So I'm squaring this
thing because H transpose J is the same as H times
H. So I'm squaring it. So what do I put here? Four, thanks. And now I've got this
guy squared with a plus. So that's four, u, you,
transpose u, u transpose. Yeah, I'm totally
realizing I've practiced for a lifetime doing these
dinky little calculations. But they are dinky. And you'll get the
hang of it quickly. Now what am I hoping
out of that bottom line? That it is I. We're hoping we're
going to get I. Do we get I? Yes. Who sees how to get
I out of that thing? Yeah, u transpose u
in here is a number. That was u-- that was column
times row times column times row. And I look at in the middle
here is row times column. And that's the number one
right because it's there. So that's one and
then I have minus 4 of that plus 4 that that. They cancel each
other, and I get I. So those are good matrices. We'll use them. We'll use them. Actually, they're better
than Gram-Schmidt. So we'll use them in
making things orthogonal. So what other
orthogonal matrices? Let's create some. Creating good
orthogonal matrices is-- you know, it pays off. Let's think. So there a family named
after this French guy who lived to 100. He was a real old timer. Well, MIT had a faculty
member in math, when I came, Professor Struik,
who lived to 106. And I heard him give a lecture
at Brown University at age 100. And it was perfect. You could not have
done it better. So he's my inspiration. I'm keep going. I only have I like, n
more years to get there. And then it's-- well,
it's too many anyway. So Hadamard, he created-- well, that's the
simplest, the smallest. Now the next guy is
going to be four by four. I'm going to put that-- so where I see a
one, I'm going to put Hadamard one, one, one minus
one, one, one, one minus one. And then when I say a minus, I'm
going to put an n with a minus. You saw that picture? It was a picture of H2,
H2, H2, and minus H2. That's what I've got there. And I believe those
columns are orthogonal. Right? Now what could I do? Well it's not quite
an orthogonal matrix. What do I have to do to
make-- this isn't quite an orthogonal matrix either. What do I do to make that
an orthogonal matrix? Divide by square root of two? I need unit vectors there. And here, those links are one
squared, one squared, got four, square root of four is two. So I better divide
by the two there. And now here I'm up to-- yeah. So that was that one. Tell me the next one up. What's that going to be? Eight by eight? What's that? So this is H4 here. Oops, four. So tell me, what I should
do for eight by eight. You know, it's simple. But that's a good
thing to say about it. You know, they're
in coding theory, all sorts of places you want
matrices of ones and minus ones. What's H8? I'm going to build it out of H4. So what's it going to be? I'm going to put an H4 there. What am I going to put here? Another H4. And up here? Another H4. And finally, here? Minus H4. And I think I've got
orthogonal columns again. Because the columns within these
dot products with themselves give zero and zero. The dot products from these
columns and these columns obviously, have the minus. And the dot products in here
are zero from that and zero from that. Yeah, it works. And we could keep going
to 16 and 32 and 64. But then up comes a question. What about H12? Is there ones and minus ones
matrix of size 12, 12 by 12? It doesn't come directly from
our little pattern, which is doubling size every time. But you could still hope. And it works. I don't know what
it is, but there's a there's a matrix of
ones and minus ones 12 orthogonal columns. So the answer is yes. So we make an 18.065 conjecture. Every matrix size-- well,
not every matrix size, because three by three
is not going to work. One by one is not
going to work either. But H12 works, H8, H16-- What's our conjecture? We won't be the first
to conjecture it, that there is a
ones and minus ones orthogonal matrix with
orthogonal columns of every size n. So I'll start the
conjecture, always possible if n, which is the size
of the matrix, let's say. What would you guess? Just take a shot. You won't be asked for a proof,
because nobody has a proof. Even? Well, you could hope for even. You could look for try six, I
guess, would be the first guy there. And I don't think it's possible. I think six is not possible. So every even size
is, I think, not going to work, but a
natural idea to try. What's the next thought? Every multiple of four. If n over a four
is a whole number. But nobody has a systematic
way to create these things. So like some of
them, at this point, we're down to doing
it one at a time. And we're up to 668. But we haven't got that one yet. Isn't that crazy? So all this is coming
from Wikipedia. My source of all that's
good in mathematics is there on Wikipedia. Anyway, this is the conjecture. Conjecture means you
don't have any damn idea of whether it's true or not. Is that divisible by four? Yeah, I guess it would be. 600 certainly is,
and 68 certainly is. Yeah, OK. So I think that's the first one. If you find one of size
668, just skip the homework and tell us about that one. But I don't think
I'll assign that. Yeah, I must have
searched for it online. But yeah. Anyway, so those are
the Hadamard matrices. Now where else do I remember
orthogonal matrices coming from? Well, yeah really,
the biggest source. So when I'm looking for
orthogonal matrices, I'm looking for a basis
of orthogonal vectors. And where in math am I
going to find vectors that come out to be orthogonal? We haven't seen-- that's the
next section of the notes, but maybe you are remembering. Where will we sort of
like, automatically show up with orthogonal vectors? They could be the
eye eigenvectors of symmetric matrix. And that's where the most
important ones come from. Oh, I could tell you
about wavelets though. Wavelets are more
like this picture. They're ones and minus ones,
or the simplest wavelets are ones and minus ones. Before I go on to the
eigenvector business, can I mention the
wavelets matrices? Yeah, these are really
important simple and important constructions. So wavelets- let me
draw a picture of-- I'm going to come up with four-- I'll do the four by four case. And these are the
orthogonal guys. And then the next one
is and down and zero. And the last one is
zero and up and down. So that's four things. But let me show you the matrix. So I'll call it W for wavelets. So that guy, I'm thinking
of was one, one, one, one. This guy I'm thinking of as
one, one, minus one, minus one. It's looking sort
of Hadamard's way. But there's a difference here. This guy is one minus
one, zero, zero. So the wavelengths rescale. That's the difference between
Hadamard and wavelets. Wavelets are self-scaling. And what's the last guy here? What's the fourth column? From that fourth wavelet? Zero, zero, one minus one, yeah. So Haar came up with that. This is the Haar wavelet, which
was many years before the word wavelets was invented. He came up with this
construction, the Haar matrix, the Haar functions. So they're very simple
functions, but you know, that makes them usable that the
fact that they're so simple. Now I don't know if you want
to see the pattern in eight by eight but let me start
the eight by eight so you'll know what wavelets are about. You'll know what these Haar
wavelets are about, anyway. They're the ones that were
kind of easy to visualize. So if that's W4, let's
just take a minute. It won't take long for W8. So the first column is
going to be eight, one. And what's the next
column going to be? Four ones and four
minus ones, like so. So four ones and
four minus ones. And now the next column, two
ones two minus ones and zeros. One, one, minus one,
minus one and zero. And the fourth will be zeros
and two ones and two minus ones. We got half a matrix now. Now if we just tell me the
fifth, what do you think? What do I put in the fifth one? So again, it's going to
squeeze down and rescale. And what's fifth
column up here that's going to be ones and
minus ones and zeros now? So it's not Hadamard, it's Haar. And what shall I put? One, one. Shall I start with one, one? AUDIENCE: 1 minus 1. PROFESSOR: Oh! And then all zeros? Oh yeah, thanks! Perfect! One minus and then all zeros. And then the next
three columns, we'll have the one minus one here
and the one minus one here, and the one minus one here. And otherwise all zeros. Yeah. So you see the pattern. It's scaling at every step. So that matrix has
the advantage of being quite sparse, short of-- This, in my mind is a-- or four ones, that get involved
with like, taking the average. Then this guy is like,
taking the differences between those and those. And then this is like taking the
difference at a smaller scale. And that also at
a smaller scale. So that's what we keep doing. Yeah. Yeah. So that wavelets. It looks so simple
right, but just a one minute
history of wavelets, so Haar invented
this in like, 1910. I mean, a long-- forever. But then you wanted
wavelets said we're a little
better, and not just ones and minus ones and zeros. And that took a lot of thinking. A lot of people were
searching for it. And Ingrid Daubechies-- so
I'll just put her name-- became famous for finding them. So in about 1988 she
found a whole lot of families of wavelets. And when I say
wavelets, she found a whole lot of
orthogonal matrices that had good properties. Yeah. So that's the wavelet picture. OK. Now to close today
and to connect with the next lecture on
eigenvalues, eigenvectors, positive definite matrices--
we're really getting to the heart of things here. Let me follow
through on that idea. So the eigenvectors
of a symmetric matrix, but also of an orthogonal
matrix are orthogonal. And that is really
where, people-- where you can invent-- because you don't
have to work hard. You just find a
symmetric matrix. Its eigenvectors are
automatically orthogonal. That doesn't mean they're
great for use, but some of them are really important. And that maybe the
most important of all is the Fourier. So you probably have
seen Fourier series sines and cosines. Those guys are
orthogonal functions. But the discrete Fourier series
is what everybody computes. And those are
orthogonal vectors. So the are orthogonal
vectors that go into the discrete
Fourier transform. And then they're
done at high speed by the fast Fourier transform. Those are eigenvectors of Q,
eigenvectors of the right Q. So let me just tell you
in the right Q that get-- who's eigenvectors-- So here we go. Eigenvectors of
Q-- you will just be amazed by how
simple this matrix is. It's just that matrix. It's called a
permutation matrix. It just puts the-- those are the four, eight,
four Fourier discrete-- let me put that word
discrete up here-- transform. So I really meant to put
discrete Fourier transform. Yeah. The eigenvectors of that
matrix, first of all, they are orthogonal, and then
second, and more important, they're tremendously useful. They're the heart of
signal processing. In signal processing, they
just take a discrete Fourier transform of a vector
before they even look at it. I mean, like that's
the way to see it, is split it into
its frequencies. And that's what the
eigenvectors of this will do. So we're going to see the
discrete Fourier transform. But my point here is to know
that they're orthogonal just comes out of this fact
that the eigenvectors are orthogonal for any Q.
And that is certainly a Q. Everybody can see that
those columns are orthogonal. That's a permutation matrix. You've taken the columns
of the identity, which are totally orthogonal,
and you just put them in a different order. So a permutation
matrix is a reordering of the identity matrix. It's got to be a Q, and
therefore, its eigenvectors are orthogonal. And they're just the winners. I mean, the matrix
of the Fourier matrix with those four eigenvectors
in it, I'll show you now. We're finishing today by leading
into Wednesday, eigenvectors and eigenvalues. And we happen to be doing
eigenvectors and eigenvalues of a Q, not today, of
an S. Most of the time, it's a symmetric matrix
whose eigenvectors we take, but here that happens
to be a Q. Can I show you the eigenvectors,
the four eigenvectors of that? Now, oh! The complex number I
is going to come in. You have to let it in here. Yeah, sorry! If S is a real symmetric matrix,
its eigenvectors are real. But if Q is a orthogonal
matrix, its eigenvectors-- even though you couldn't ask for
a more real matrix than that-- but its eigenvectors
are at least, the good way are
complex numbers. So can I just show you
the eigenvectors of-- So again, overall
the point today is to see orthogonal matrices. So I'll just repeat
now while I can-- rotations here, reflections,
wavelets, the Householder idea of reflections of
big, large matrices that have this form I minus 2
u, u transpose are orthogonal. And now we're going
to see the big guys, the eigenvectors of Q. Yes? AUDIENCE: [INAUDIBLE]? PROFESSOR: Ah yeah, we
don't have orthonormal. That's right. We don't have orthonormal. I better divide by the
square root of eight. AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh yes! Oh, you're right! Sorry. I just thought I'd get away
with that, but I didn't. Yeah. So these guys these
words of eight. So are these. But these guys are
square roots of four. These guys are
square roots of two. Thank you. Absolutely right. Absolutely. Yeah. OK, so what are the
eigenvectors of a permutation. This is going to be nice to see. And I'll use the matrix
F for the eigenvector matrix of that Q up there,
and F is for Fourier. And it'd be the four
by four Fourier matrix. OK. What are the eigenvectors of Q? OK. So Q is a permutation. So like, I'm going to ask
you for one eigenvector of every permutation matrix. What vector can you tell me
that actually the eigenvalue will be one? What vector can you tell
me where if I permute it, I don't change it? One, one, one, one. Like, it's everywhere
here, one, one, one, one. So that's the zero
frequency for a vector, the constant vector, all ones. Everybody sees if I'm multiply
by Q, it doesn't change. OK. Now the next one-- I'll show you the four now. The next one will be 1
I, I squared and I cubed. Of course, I squared-- I don't know how many course
6J people are in this audience, but this is a math building. We paid for it. It's 2-190. And it's I in this room. Anyway, I is the
first letter in what? Imaginary. Thank you, the first
letter in imaginary. You can't say jimaginary. So that's it. OK. And then the next one is 1 I
squared, I fourth, I sixth. And the next one is
1 I cubed, I6 and I9. Isn't that just beautiful? And you could show that every
one of those four columns, if you multiply them by Q,
you would get the eigenvalues. And you would see that
it's an eigenvector. And this is just sort of like a
discrete Fourier stuff, instead of e to the I, e to the Ix,
e to the 2Ix, e to the 3Ix and so on. We just have vectors. So those are the
four eigenvectors of that permutation. And those are orthogonal. Could I just check that? How do you know that this first
column and the second-- well, I should really say zeros
column and first column, if I'm talking frequencies. Do you see that that's
orthogonal to that? Well, y is one plus
I plus I squared plus I cubed equal zero. That's the dot product is this
column one dot column two. Yeah, you're right. This happens to come out zero. Is that right? AUDIENCE: Yeah, that
will come out zero. PROFESSOR: That
will come out zero. But somebody mentioned
that this isn't right. It's true that
that came out zero, but when I have imaginary
numbers anywhere around, this isn't a correct dot
product to test orthogonal. If I have imaginary-- complex vectors-- if I
have complex numbers, complex vectors, I should test
column one conjugate dotted with column two-- column I conjugate-- well,
let me take column see, which ones shall I take? Maybe that guy and that guy. Many of them, you luck out here. But really, I should be
taking that conjugate. So these ones-- But the thing is, the complex
conjugate of one is one. So that was OK. But in general, if
I wanted to take column two dotted
with column maybe four would be a little dodgy-- yeah. Look what happens. Take that column
and that column. Take their dot product. Do it the wrong way. So what's the wrong way? Forget about the
complex conjugate and just do it the usual way. So one times one is one. I times I cube is? One. I squared times I6 is? One. I'm getting all ones. I'm not getting
orthogonality there. And that's because I forgot
that I should take the complex conjugate-- well, of
these guys-- one-- I should take minus I,
minus I, minus I squared-- well, that's real. So it's OK. Minus there. So minus I squared
is still minus one. So now if I do it,
it comes out zero. So let me repeat again. Let me just make this statement. If Q transpose Q is
I, and Qx is lambda x, and Qy is different
eigenvalue y-- so I'm setting up the main fact. In the last minute, I'm
just going to write down. So I have an orthogonal matrix. I have an eigenvector
with eigenvalue lambda. I have another eigenvector with
a different eigenvalue, mu. Then the claim is that-- what is the claim
about eigenvectors? So here, this has an eigenvalue. This has a different eigenvalue. I need them to be
different to really know that the x's and the
y's can't be the same. So what is it that
I want to show? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes! That x-- and I have to
remember to do that. x transpose y is zero. That's orthogonality. That's orthogonality
for a complex vectors. I have to remember to change
are every I to a minus I in one of the vectors. And I can prove that fact
by playing with these. By starting from here,
I can get to that. OK. That's it. We've done a lot
today, a lot of stuff about orthogonal matrices. Important ones, and those
sources of important ones, eigenvectors. And so it will be
eigenvectors on Wednesday.
