Continued from part one, we have the

shear matrix,

which is a matrix that produces a shear

transformation.

But, what is a shear transformation?

Turns out, it's not the easiest to

describe verbally, so you have to see it

for yourself.

The matrix 1101 is a shear matrix.

And this is what it does.

What we see is all the vectors above the

origin were moving towards the right.

And all the vectors below the origin

were moving towards the left. And the

way they move was actually parallel to

the x-axis.

So, overall, the rectangle was slanted

into parallelogram.

In 2D, there are four directions we can

apply a shear transformation.

Two different ways we can shear in

parallel to the x-axis,

and two ways in the y-axis.

It's also notable to mention a property

shear matrix have,

which the transformation preserve the

area of the object, despite changing the

shape.

Which forms a distinction with the

scalar matrix from earlier, which

preserve the shape, but changes the

area.

A three-dimensional shear is kind of

hard to describe. Personally, I'd like

to imagine there's a plane slicing

through the origin, and there's a

direction associated with this plane.

All vector are moving parallel to the

plane, while slanting towards the

direction.

Mhm, maybe that was not the best angle

to observe a shear transformation in 3D.

So, let me change the orientation of my

camera a little bit.

And for every combination of this plane

and direction,

it would have its own corresponding

shear transformation matrix.

Next, the orthogonal matrix. This one is

very important for chapter two and

three. It's a square matrix. Every

column vector is a unit vector, and

they're all orthogonal to each other.

Let's dissect the definition a little

bit.

So, what is a column vector? Basically,

if we vertically slice through the

matrix into columns, each column is a

vector. This is the reason why some

people say matrix is a collection of

vectors.

For a column vector to be a unit vector,

it means the magnitude is one. In our

case, the length of the arrow is one.

For a counterexample, the red arrows on

the screen are not unit vectors because

they're too long.

How about orthogonal?

So, the formal definition for orthogonal

is when you take the dot product between

two vectors, the number you get is zero.

And when the dot product is zero,

visually, it actually translates to the

two vectors being perpendicular to each

other.

This one is orthogonal matrix. We can

quickly verify the two column vectors

are unit and orthogonal.

What kind of transformation does it

have?

Wow.

A perfect rotation.

No scaling, no stretching, no shearing,

no reflection, but a pristine rotation.

You might say, "Come on, it's just

rotation. There's nothing special." But

it turns out finding the correct

direction to rotate is a gateway to

unlock all complexity with linear

transformation.

And it just happens the transformation

an orthogonal matrix produce is always a

rotation to some degree.

Inductively, a 3 by 3 orthogonal matrix

produces a rotation in three dimension,

and you can also rotate around the Z

axis now.

Actually, now's a really good time for

me to once again emphasize the idea that

matrix to matrix multiplication is

nothing but a composition of sequential

transformations.

Suppose I tell you orthogonal matrix A

produce a rotation around the x-axis by

60°.

And matrix B produce rotation around the

z-axis by 45°.

If I multiply those two matrices

together to get a matrix C,

what do you think C is going to do?

Well, let's see. Firstly, around X

and immediately around Z.

Since our matrix C is a composition of

the two, like you guessed it, it

encapsulates the two sequential

transformation but in just one rotation.

If you had noticed, I really tried to

punctuate the word sequential just now

because suppose we reverse the sequence

of the two rotation, namely Z first then

X.

We actually get something different than

the original composition.

Look at the position of the orange cube.

It's different.

This is actually an example to prove

matrix composition or matrix

multiplication is not commutative.

Which is saying matrix B times A is not

always equal to A times B.

Projection matrix.

Actually, before defining a projection

matrix, we need to understand what is a

subspace.

And just like all other definitions in

linear algebra, the definition for

subspace is pretty abstract. At the

moment, allow me to just provide you

with some examples. In 2D, a line

crossing through the origin is a

subspace of R2.

And this line is infinitely long.

In 3D, a plane crossing through the

origin is a subspace of R3.

I'm only showing you a small region of

the subspace here, but in reality, you

should imagine the subspace actually

spread to infinity.

For every subspace that exist, our

computer can calculate its corresponding

projection matrix, which would move

every vector outside the subspace onto

the subspace.

Let's take a look at the projection

matrix of this blue line here.

After the projection transformation,

every single dot has been compressed

onto the line.

How about the projection matrix of this

plane in 3D?

Yet another very similar vibe of

compression.

The reason why this is called projection

is because every vector always moved to

its closest point on the subspace.

For example, the vector 2 1 is currently

outside the blue line, and its closest

point would be here.

Applying the projection matrix would

move our vector 2 1 exactly over there.

And this is true for every single vector

outside the subspace.

As targets of projection matrix, they

all move towards their closest landing

spot.

And the fun fact, if some alien

civilization destroy our solar galaxy by

compressing us into a lower dimension,

this kind of transformation is similar

to a projection matrix.

The very last, but not the least, we

talk about the idea of inverse.

So far, we have seen a lot of different

matrices, and therefore many different

transformations. All those

transformations were moving our vectors

to new coordinates.

But what if I don't like a particular

transformation I had? Maybe it distorted

my image a little bit. I just want every

vector to go back where they originally

came from.

Is there one matrix that can untransform

the previous transformation for me?

Unfortunately, the short answer would be

no.

There isn't one matrix that does the

universal untransformation.

But for every matrix you're interested

in, our computer can calculate its

inverse, which is another matrix that is

capable of that particular

untransformation.

For example, the matrix which scale,

its inverse matrix unscale.

The matrix which reflects, its inverse

reflects back.

The matrix which rotates, its inverse

rotates back.

The matrix which shears,

its inverse unshears.

Or this matrix here, not sure what it

did, but its inverse matrix restores all

vector back to the original.

Notice, when you apply a matrix and then

you apply its inverse, it's like nothing

happened.

Just like the identity matrix. This is

the reason when you multiply a matrix

and its inverse, you always get the

identity matrix. When you compose the

two transformations together, you get a

transformation of no transformation.

However, some matrix cannot be inverted.

The zero matrix and projection matrix

from earlier cannot be untransformed.

It actually kind of makes sense visually

because a vector has been squashed from

a higher dimension down to a lower

dimension. There is a loss of

information.

After a long journey, we have went

through all those cool matrices. So,

what exactly is a matrix?

Does matrix intrinsically carry a visual

value?

Frankly, not at all.

Everything I did was only a very

artificial attempt to make sense of

matrices. You could say matrix

visualization is just another human

construct.

This moment, I'd like to reference a

quote from my favorite author. He was

asked whether his readers could

interpret the symbolism in his novel the

way he intended to convey. He answered,

"The books belong to the readers now,

which is a great thing because the books

are more powerful in the hands of my

readers."

I think this is true for matrix as well.

The interpretation of matrix belongs to

whoever is using it.

For a student who studies circuit,

matrix is just a tool to solve system of

equation. Personally, I still have some

PTSD from Gaussian elimination.

For probability student, matrix is a

best representation of a Markovian

process.

For data scientist, a matrix is just a

manifestation of a table, which

facilitates data analysis.

And for the deep learning folks, matrix

is just another Python function, which

takes a vector as input and returns a

vector as output.

And this list really goes on.

Perhaps,

there isn't the definition of matrix,

but only interpretations of matrices.

Then, what is so good of the visual

interpretation?

Firstly, I think it provides us with

intuition about matrix transformation on

vector.

And secondly, computer graphic is a

direct extension of this.

Thirdly, the one I like to elaborate on

personally, which is a very important

topic of matrix decomposition.

There are matrices out there whose

transformation so perplexing I cannot

easily articulate.

And there are matrices taking vector

from higher space to a lower space.

Is there a possibility we can re-express

those complicated transformation into a

sequence of simple transformation such

as rotation or scaling?

And that's where we're heading towards

next chapter two. What Professor Gilbert

Strang calls the king of all matrices.

And let you and me go spectate the

spectacular spectral decomposition.

See you there.