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Orthogonal matrices

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Mathematics for Machine Learning: Linear Algebra

Imperial College London

4.7 (3,481 ratings) | 83K Students Enrolled

Kurs 1 von 3 in Mathematics for Machine Learning Spezialisierung

In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning.

Lehrplan anzeigen

Skills You'll Learn

Eigenvalues And Eigenvectors, Basis (Linear Algebra), Transformation Matrix, Linear Algebra

Bewertungen

4.7 (3,481 ratings)

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CS

Apr 01, 2018

Amazing course, great instructors. The amount of working linear algebra knowledge you get from this single course is substantial. It has already helped solidify my learning in other ML and AI courses.

VC

Jul 11, 2018

The content of the course is very relevant, and the instructors are really fun and helpful.My only suggestion is to upload revisions for each assessment, so we can understand what we are doing wrong.

Aus der Unterrichtseinheit

Matrices make linear mappings

In Module 4, we continue our discussion of matrices; first we think about how to code up matrix multiplication and matrix operations using the Einstein Summation Convention, which is a widely used notation in more advanced linear algebra courses. Then, we look at how matrices can transform a description of a vector from one basis (set of axes) to another. This will allow us to, for example, figure out how to apply a reflection to an image and manipulate images. We'll also look at how to construct a convenient basis vector set in order to do such transformations. Then, we'll write some code to do these transformations and apply this work computationally.

Orthogonal matrices6:40

Unterrichtet von

David Dye
Professor of Metallurgy
Samuel J. Cooper
Lecturer
A. Freddie Page
Strategic Teaching Fellow

Skript

Now, it's going to be really useful if we can make

a transformation matrix whose column vectors

make up a new basis,

all of whoms't vectors are perpendicular

or what's called Orthogonal to each other.

In this video, we're going to look

at this and why this is.

But first, I want you to find

a new operation on the matrix called the transpose.

This is where we interchange

all the elements of the rows and columns of the matrix,

so I'll denote the transpose as T and I'll say that

the ijth element of the transpose of A,

is equal to the elements on the opposite diagonal A_ji.

So if I had a matrix like 1,2,3,4,

and I wanted to know its transpose,

then I will interchange the elements

that are off the diagonal.

So the one and the four stay where they are.

Because if is, I find i and j are the same;

1,1 1,1 they would stay the same, the same for 2,2.

But the element 1,2

interchange to the element 2,1 so they go like that.

So 1,2,3,4 becomes, when I transpose it 1,3,2,4.

Now, let's imagine that I have

a square matrix of dimension n by n,

it defines a transformation with

a new basis and

the columns of that transformation matrix A.

Some vectors a_1, a_2

that are the basis vectors in

the new space as we've said before

and I'm going to make

this a special matrix while I impose

the condition that these vectors

are orthogonal to each other and they have unit length.

That is, a_i dotted with a_j

is equal to zero if i isn't equal to j,

that is they're orthogonal,

and it's equal to one if i is equal to j,

that is they are of unit length.

Now, let's think about what happens if I multiply

this matrix on the left by a transpose.

So A transpose is going to be given by just flipping

the rows and columns of

all of the elements across the leading diagonal of A.

So that is, I'm going to

have a_1 is going to become a row,

a_2 is going to become the next row all the way down

to a_n because I flip

all of the elements across the leading diagonal.

So while I multiply A_T by A,

let's see what happens,

it's going to be quite magical.

So if I get A_T times A,

then I'm going to get, well,

the first element here is going

to be the first row times the first column,

and that's a_1 dotted with a_1, and that's one.

The second element here,

the first row and the second column is

a_1 dotted with a_2 and that's zero,

and a_1 dot a_3 all the way up to a_n is going to give me

a whole series of zeros where this guy remember is a_n.

Then when I do the second row on the first column,

I'm going to get another zero.

I want to do the second one and the

second column I'm going to get a one.

Second row, third column zero all the way across.

I want to do the third one,

I'm going to get the same again,

I'm going to get zero zero.

It only says it's a_3 times a_3,

and that's going to be that element all the way

across and what I'm gradually building up here,

is I'm building up the identity matrix.

So what I found is that A_t times

A gives me the identity matrix,

and what that means is,

is that A_T is a valid inverse of A.

Which is really kind of neat.

A set of unit length basis vectors

that are all perpendicular to each

other are called an orthonormal basis set,

and the matrix composed of

them is called an orthogonal matrix.

One thing also to know about

an orthogonal matrix is that

because all the basis vectors,

any of unit length,

it must scale space by a factor of one.

So the determinant of an orthogonal matrix

must be either plus or minus one.

The minus is what arises in the new basis,

if the new basis vector set flits space around.

They stay inverted, they make it left-handed.

Notice that if A_T is the inverse then I should be

able to post multiply A by A_T and get the identity.

So I can do this on the way round.

Which also means by the same logic that the rows of

the orthogonal matrix are orthonormal,

as well as the columns which is neat,

and we saw in the last video that actually

the inverse is the matrix

that does the reverse transformation.

So the transpose matrix of

an orthonormal basis vector set is itself

another orthogonal basis vector set which is really neat.

Now, remember that in the last module on

vectors we said that transforming a vector to

a new coordinate system was just taking

the projection or dot product of that vector,

so say that's the vector,

with each of the basis vectors so basis factor this one,

basis factor that one,

and so on so

long as those basis vectors

were orthogonal to each other.

If you want to pause and think about that for a moment in

the light of all we've learned about matrices,

just look and think about this for a moment.

Now in data science what we're really

saying here is that wherever possible,

we want to use an orthonormal basis vector set

when we transform our data.

That is, we want our transformation matrix

to be an orthogonal matrix.

That means the inverse is easy to compute.

It means the transmission is

reversible because it doesn't collapse space.

It means that the projection is just the dot product.

Lots of things are nice and pleasant, and easy.

If I arrange the basis vectors in right order,

then the determinant is one,

and that's an easy way to check and if they

aren't just exchange a pair of them and

actually then they will be

determinant one rather than the minus one.

So what we've done in this video is

look at the transpose and that's led us to

find out about the most convenient basis factor set

of all which is the orthonormal basis factor set

which together make the orthogonal matrix

whose inverse is its transpose. So that's really cool.

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