What exactly is a matrix?

Some say it's just a box of numbers, a

collection of vectors,

mathematical object, linear

transformation,

a function.

Perhaps, no one knows the answer.

But, we do know how to multiply a matrix

and a vector. For example, multiplying

this matrix with vector 1 2, what we get

is a vector 5 2.

And it turns out there's actually a

natural way to visualize this.

Firstly, we represent a vector as a

physical arrow sitting in the

two-dimensional space. And as a vector

gets multiplied by the matrix, the tip

of the arrow moves from the coordinate 1

2 to the coordinate 5 2.

So, that was a visualization for one

vector. How about we try to do this with

more vectors?

Once again, we can represent the vectors

as six physical arrows and observe their

movements as they get multiplied by the

matrix.

But, the original goal was to visualize

a matrix, so we should think what

happens to all vectors as each every one

of them gets multiplied.

Our computer tries its best to add many,

many arrows on the screen to create the

illusion of every vector

and the multiplication.

Well, that was a visualization of matrix

I asked for, but it's just a horrible

visualization. There are too many

vectors moving at the same time, too

much information for my tiny brain to

process, and we need to improve this.

So, to improve the quality, we're going

to apply two tricks. Firstly, let's

squint our eyes and focus on a smaller

region of vectors.

The second trick we do is to represent a

vector as a dot instead. Consider our

old friend vector 1 2. Now it's a dot

sitting over there. The matrix

multiplication would move this dot

to the coordinate 5 2.

And we can replace all the vectors

within our focus region with many, many

dots instead.

And then we observe the matrix

multiplication.

This is much better.

We have essentially built an engine

which creates a visualization for any 2

by 2 matrix.

The very natural thing to do next is to

just plug in a bunch of different

matrices in there and watch the

transformation of the dots on the

screen.

And very soon interesting patterns start

to emerge.

The goal for the next part of the video

is to illustrate the famous matrices

whose transformation are so visually

geometrical that we can even use English

to describe them.

And welcome to the matrix got talent

show. The first matrix on stage is the

identity matrix, which is a square

matrix that has a ones on the diagonal

and zeros everywhere else.

So, tell me about your transformation on

vectors.

Wait.

There There was no transformation?

Nothing at all? Huh.

That's kind of boring start.

But this actually makes sense. If you

multiply any vector with the identity

matrix, you get the same vector back.

So, there will be no movements of the

dots at all.

If we try to label every matrix with a

tag. What would be the tag for identity

matrix? Well, I say it would be no

transformation or does nothing, boring.

The next one is a scalar matrix. The

numbers on the diagonal line are the

same and zeros everywhere else.

So, it's not saying that there is only

one matrix that is the scalar matrix,

but any matrix that satisfies this form

is regarded as a scalar matrix.

Let's try this matrix here.

Wow. That was pretty cool.

Our rectangle got bigger.

And the notable thing is there wasn't

any sense of distortion, but rather like

every single dot moved uniformly.

Let's try one more.

Oh.

This one seems to be moving our dots

even more aggressively outwards.

So, zooming now.

And we still see the contour of the

rectangle and the ratio of width versus

height still remains the same. How about

when the numbers on the diagonal line is

less than one?

We see our dots moving inwards and the

border of the dots still forms a

rectangle.

It seems this kind of matrix has ability

to scale things uniformly.

For K greater than one, the matrix is

stretching all the vectors outwards. For

K less than one, the vectors are

stretched inwards. For K equal to one,

nothing moves because this is the

identity matrix.

What interesting thing can we do about

this?

If you really ponder upon idea of a

shape on 2D space,

then it's really just an infinite number

of dots sitting closely together.

And guess what? If matrix can move dots,

then matrix can transform shape.

And here, we apply the scalar matrix to

the triangle.

To take this idea one step further, a

picture is basically a bunch of pixels

or colored dots.

Then, we can transform pictures well.

Let's apply a scalar matrix to the Mario

here.

Oh, question.

Three dimension?

That is a good question.

A vector in R3 can be represented as

three-dimensional arrow or likewise

three-dimensional dot.

And if you ponder on this again, a

three-dimensional object is just like an

infinite number of dots sitting very

close together, which means we can use a

3 by 3 scalar matrix to transform

three-dimensional objects.

The next one is not famous enough to

have its own name.

But, the idea is you take the identity

matrix and modify exactly one number on

the diagonal line.

And let's call this the off one matrix.

Dots were moving along the Y axis, while

the X coordinate is unchanged. How about

a different one?

Dots are moving along the X axis, while

the Y coordinate is unchanged.

If we keep playing with the visual

engine we have, the pattern starts to

become clear.

For whichever entry on the diagonal line

that is not one, but K, it scales

everything along the axis by a factor of

K. While the other axis is not changed.

And this really has to do with our way

of representing vector as dot.

We let the first entry be X and the

second entry be Y.

of one matrix behaves very predictably

in three dimension. It's a scaling of

just the X axis,

just the Y axis,

or

just the Z axis.

Hey, you're back. Do you want to get

scale along the Y axis?

I've got a question.

Negative entries.

We did forget to talk about that.

Let's take a look at the simplest case

first. When you change one or more

number on the diagonal to be negative

one for the identity matrix,

seems there's a reflection by the Y axis

and nothing is scale out of proportion.

This one?

A reflection by the X axis.

So, why is this matrix producing some

form of reflection behavior? What's the

intuition here? Let's use the Y

reflection matrix as an example.

If we consider the matrix vector

multiplication, we first know that the

magnitude of vector did not change. So,

nothing is scale,

but the sign of the first entry is

flipped.

Visually, this means arrows from the

first quadrant move to the second

quadrant.

And dots from the third quadrant move to

the fourth quadrant.

And the exact logic of reasoning also

holds for the X reflection matrix, in

which the sign of the Y coordinate is

being flipped.

But what about when both entries are

negative? Is it a reflection by the X

and Y axis?

And you will be exactly right. The sign

of both the X and the Y coordinates are

flipped. Dots from every quadrant would

be going to the opposite quadrant.

And this translates to a reflection

around the origin.

And as always, it's cool to let matrix

transform images.

In three dimension, there is one more

entry that can be flipped, the Z

coordinate.

We will not go into the details of

different cases here, but the overall

spirit of reflections still holds.

Next, enters the diagonal matrix, which

is another square matrix that can have

any numbers on the diagonal line, but

zeros everywhere else.

And the understanding of this matrix

actually ties closely to the reflection

matrix and off one matrix we visualized

earlier.

Before me presenting you with a

visualization, we can even try to guess

what it might look like. But firstly,

allow me to say something very

important.

Matrix to matrix multiplication is

really not multiplication at all. When

someone say, "I want to multiply matrix

B and A to get matrix C."

Okay. What that person really saying is

I just want one matrix, which is matrix

C, which can compose the overall

transformation if first apply A and then

B, but in just one transformation.

For example, we can multiply or compose

these two off-one matrices on the left

to get a diagonal matrix on the right.

And we already know exactly what kind of

visual transformation those two matrices

would have since we have labeled them

earlier.

So, what do you think the diagonal

matrix would do?

Exactly like what you guessed, the

vectors all move along the Y axis by a

factor of two and also along the X axis

by a factor of three.

How about this diagonal matrix here? You

can pause the video and show it's just a

composition of the three matrices on the

right, which we know a lot about

already.

And therefore, we can conclude that the

diagonal matrix must be the overall

transformation of sequentially applying

the three matrices on the right.

Let's use this image as an example.

Firstly, we scale the X axis by 2.5.

Secondly, the Y axis by 0.6.

And thirdly, a reflection.

And this diagonal matrix, which is a

composition of the three matrices on the

right, will just encapsulate the three

transformations but in one go.

Whenever there's a diagonal matrix, you

can always decompose it into a sequence

of the off-one matrix. In case you have

negative numbers on the diagonal, you

can also explicitly factor it out to

include a reflection matrix at the end.

For this 3 by 3 diagonal matrix, we know

it would just be a composition of three

different offline matrices

and we can directly read the numbers

from its diagonal line. So, it will

scale the x-axis by three, y-axis by

two, and z by 0.5.

The next one is a zero matrix, which is

square and everything is zero.

So, regardless of which vector you

multiply with a zero matrix, it becomes

the zero vector.

And visually, it would mean no matter

where the vector starts,

it would move towards and end up in the

origin.