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- [Voiceover] Hey guys.
There's one more thing I need to talk about
before I can describe the vectorized form
for the quadratic approximation of multivariable functions
which is a mouthful to say
so let's say you have some kind of expression
that looks like a times x squared
and I'm thinking x is a variable
times b times xy,
y is another variable,
plus c times y squared
and I'm thinking of a, b and c as being constants
and x and y as being variables.
Now, this kind of expression has a fancy name.
It's called a quadratic form.
And that always threw me off.
I always kind of was like, what,
what does form mean?
I know what a quadratic expression is
and quadratic typically means something is squared
or you have two variables
but why do they call it a form?
And basically it just means that the only things in here
are quadratic.
It's not the case that you have an x term
sitting on its own or a constant out here
like two when you're adding all of those together
instead it's just you have purely quadratic terms
but of course, mathematicians don't want to call it
just a purely quadratic expression
instead they have to give a fancy name to things
so that it seems more intimidating than it needs to be
but anyways, so we have a quadratic form
and the question is
how can we express this in a vectorized sense?
And for analogy, let's think about linear terms
where let's say you have a times x
plus b times y
and I'll throw another variable in there,
another constant times another variable z.
If you see something like this
where every variable is just being multiplied by a constant
and then you add terms like that to each other,
we can express this nicely with vectors
where you pile all of the constants into their own vector,
a vector containing a, b and c
and you imagine the dot product
between that and a vector that contains
all of the variable components,
x, y and z
and the convenience here is then you can have just a symbol
like a v let's say
which represents this whole constant vector
and then you can write down,
take the dot product between that
and then have another symbol,
maybe a bold faced x
which represents a vector that contains all of the variables
and this way, your notation just kind of looks like
a constant times a variable
just like in the single variable world
when you have a constant number times a variable number,
it's kind of like taking a constant vector
times a variable vector.
And the importance of writing things down like this
is that v could be a vector
that contains not just three numbers
but a hundred numbers
and then x would have a hundred corresponding variables
and the notation doesn't become any more complicated.
It's generalizable at the higher dimensions.
So the question is
can be we do something similar like that
with our quadratic form?
Because you can imagine
let's say we started introducing the variable z
then you would have to have some other term,
some other constant times the xz quadratic term
and then some other constant
times the z squared quadratic term
and another one for the yz quadratic term
and it would get out of hand
and as soon as you start introducing things
like a hundred variables,
it would get seriously out of hand
because there's a lot of different quadratic terms
so we want a nice way to express this.
And I'm just going to kind of show you how we do it
and then we'll work it through to see why it makes sense.
So usually, instead of thinking of b times xy,
we actually think of this as two times some constant
times xy
and this of course doesn't make a difference.
You would just change what b represents
but you'll see why it's more convenient to write it this way
in just a moment.
So the vectorized way to describe a quadratic form like this
is to take a matrix,
a two by two matrix since this is two dimensions
where a and c are in the diagonal
and then b is on the other diagonal
and we always think of these as being symmetric matrices
so if you imagine kind of reflecting the whole matrix
about this line,
you'll get the same number
so it's important that we have that kind of symmetry.
And now what you do is you multiply the vector,
the variable vector that's got x, y
on the right side of this matrix
and then you multiply it again
but you turn it on its side
so instead of being a vertical vector,
you transpose it to being a horizontal vector
on the other side.
And this is a little bit analogous too
having two variables multiplied in.
You have two vectors multiplied in but on either side.
And this is a good point by the way
if you are uncomfortable with matrix multiplication
to maybe pause the video,
go find the videos about matrix multiplication
and kind of refresh or learn about that
because moving forward,
I'm just going to assume
that it's something you're familiar with.
So going about computing this,
first, let's tackle this right multiplication here.
We have a matrix multiplied by a vector.
Well, the first component that we get,
we're going to multiply the top row
by each corresponding term in the vector
so it'll be a times x.
a times x
plus b times y.
Plus b times that second term y
and then similarly for the bottom term,
we'll take the bottom row
and multiply the corresponding terms
so b times x.
b times x
plus c times y.
c times y.
So that's what it looks like
when we do that right multiplication
and of course we've got to keep our transposed vector
over there on the right,
on the left side.
So now, we have,
this is just a two by one vector now
and this is a one by two.
You could think of it as a horizontal vector
or a one by two matrix
but now when we multiply these guys,
you just kind of line up the corresponding terms.
You'll have x multiplied by that entire top expression
so x multiplied by ax plus by.
ax plus by
and then we add that to the second term y
multiplied by the second term of this guy
which is bx plus cy
so y multiplied by
bx plus cy
and all of these are numbers so we can simplify it
once we start distributing the first term
is x times a times x
so that's ax squared
and then the next term is x times b times y
so that's b times xy.
Over here, we have y times b times x
so that's the same thing as b times xy
so that's kind of why we have,
why it's convenient to write a two there
because that naturally comes out of our expansion.
And then the last term is y times c times y
so that's cy squared.
So we get back the original quadratic form
that we were shooting for.
ax squared plus two bxy plus cy squared
That's how this entire term expands.
As you kind of work it through,
you end up with the same quadratic expression.
Now, the convenience of this quadratic form
being written with a matrix like this
is that we can write this more abstractally
and instead of writing the whole matrix in,
you could just let a letter like m
represent that whole matrix
and then take the vector that represents the variable,
maybe a bold faced x
and you would multiply it on the right
and then you transpose it and multiply it on the left
so typically you denote that
by putting a little t as a superscript
so x transposed multiplied by
the matrix from the left
and this expression,
this is what a quadratic form looks like in vectorized form
and the convenience is the same
as it was in the linear case.
Just like v could represent
something that had a hundred different numbers in it
and x would have a hundred different constants,
you could do something similar here
where you can write that same expression
even if the matrix m is super huge.
Let's just see what this would look like
in a three dimensional circumstance so,
actually, I'll need more room
so I'll go down even further.
So we have x transpose multiplied by the matrix
multiplied by x, bold faced x
and let's say instead this represented,
you have x then y then z,
our transposed vector
and then our matrix,
our matrix let's say was a, b, c,
d, e, f and because it needs to be symmetric,
whatever term is in this spot here
needs to be the same as over here
kind of when you reflect it about that diagonal.
Similarly, c, that's going to be the same term here
and e would be over here.
So there's only really six free terms that you have
but if fills up this entire matrix
and then on the right side,
we would multiply that by
x, y, z.
Now, I won't work it out in this video
but you can imagine actually multiplying this matrix
by this vector
and then multiplying the corresponding vector that you get
by this transposed vector
and you'll get some kind of quadratic form
with three variables
and the point is you'll get a very complicated one
but it's very simple to express things like this.
So with that tool in hand,
in the next video,
I will talk about how we can use this notation
to express the quadratic approximations
for multivariable functions.
See you then.