학습>
데이터 과학>
기계 학습>
Mathematics for Machine Learning: Linear Algebra개 강좌를>
Special eigen-cases

Loading...

Special eigen-cases

Eigenvectors are particular vectors that are unrotated by a transformation matrix, and eigenvalues are the amount by which the eigenvectors are stretched. These special 'eigen-things' are very useful in linear algebra and will let us examine Google's famous PageRank algorithm for presenting web search results. Then we'll apply this in code, which will wrap up the course.

Mathematics for Machine Learning: Linear Algebra

임페리얼 칼리지 런던

Mathematics for Machine Learning 전문 분야에서 3의 강좌 1

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.

강의 계획서 보기

배우게 될 기술

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

강사:

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

스크립트

As we saw previously,

eigenvectors are those which lie along the same span both

before and after applying a linear transform to a space.

And then, eigenvalues are simply the amount that

each of those vectors has been stretched in the process.

In this video, we're going to look at

three special cases to make sure the intuition we've built so far is robust,

and then we're going to try and extend this concept into three dimensions.

The first example we're going to consider is that of a uniform scaling,

which is where we scale by the same amount in each direction.

As you will hopefully have spotted,

not only are all three of the vectors that I've highlighted eigenvectors,

but in fact, for a uniform scaling,

any vector would be an eigenvector.

In this second example,

we're going to look at rotation.

In the previous video,

we applied a small rotation,

and we found that it had no eigenvectors.

However, there is one case of

non-zero pure rotation which does have at least some eigenvectors,

and that is 180 degrees.

As you can see, the three eigenvectors are still laying on the same spans as before,

but just pointing in the opposite direction.

This means that once again,

all vectors for this transform are eigenvectors,

and they all have eigenvalues of minus one,

which means that although the eigenvectors haven't changed length,

they are all now pointing in the opposite direction.

This third case, we're going to look at a combination of

a horizontal shear and a vertical scaling,

and it's slightly less obvious than some of the previous examples.

Just like the pure shear case we saw previously,

the green horizontal vector is an eigenvector and its eigenvalue is still one.

However, despite the fact that neither of the two vectors shown are eigen,

this transformation does have two eigenvectors.

Here, I've now added the second eigenvector on to the image,

and it shows us that although the concept is fairly straightforward,

eigenvectors aren't always easy to spot.

Let's now apply the inverse transform and

watch our parallelogram go back to its original square.

But this time, with our eigenvector visible.

Hopefully, you're at least convinced that it is indeed

an eigenvector as it stays on its own span.

This problem is even tougher in three or more dimensions,

and many of the uses of eigen theory in machine learning frame

the system as being composed of hundreds of dimensions or more.

So, clearly, we're going to need

a more robust mathematical description of this concept to allow us to proceed.

Before we do, let's just take a look at one quick example in 3D.

Clearly, scaling and shear are all going to

operate in much the same in 3D as they do in 2D.

However, rotation does take on a neat new meaning.

As you can see from the image,

although both the pink and green vectors have changed direction,

the orange vector has not moved.

This means that the orange vector is an eigenvector,

but it also tells us,

as a physical interpretation,

that if we find the eigenvector of a 3D rotation,

it means we've also found the axis of rotation.

In this video, we've covered a range of special cases,

which I hope have prompted the questions in your mind about how we're

going to go about writing a formal definition of an eigen-problem.

And this is exactly what we're going to be discussing next time. See you then.

Coursera 소개

세계 최고의 대학교와 교육 기관의 최상위 강사가 가르쳐주는 강좌와 전문 강좌를 듣고 온라인 학위를 취득하세요.

Join a community of 40 million learners from around the world

Earn a skill-based course certificate to apply your knowledge

Gain confidence in your skills and further your career

Coursera는 세계 최고의 교육에 대한 보편적인 접근 기회를 제공하며, 최고의 대학 및 기관들과 제휴하여 온라인 강좌를 제공합니다.

© 2019 Coursera Inc. All rights reserved.

앱스토어에서 다운로드하기

구글 플레이에서 받기