- [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.

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.