Goldsmiths’ 2014 Mathematics

Monday 21st July

Eigenvalues and Eigenvectors

Dr Robert Johnson

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This is a subject that students find very difficult to get motivated with.

http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

http://www.youtube.com/watch?v=3E63wnzYHYc

http://www.youtube.com/watch?v=G4N8vJpf7hM

https://www.youtube.com/watch?v=IdsV0RaC9jM

http://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors

http://anothermathgeek.hubpages.com/hub/What-the-Heck-are-Eigenvalues-and-Eigenvectors

An eigenvalue is a number that is derived from a square matrix. A square matrix is itself just a collection of n rows of n numbers. An eigenvector of a square matrix A is a non-zero vector v that, when the matrix multiplies v, yields a constant multiple of v, the latter multiplier being commonly denoted by l That is:

A v = l v

(Because this equation uses post-multiplication by v, it describes a right eigenvector.)

The number l is called the eigenvalue of A corresponding to v.

Eigenvectors and eigenvalues are used in linear algebra

Eigenvectors make understanding linear transformations easy. They are the “axes” (directions) along which a linear transformation acts simply by “stretching/compressing” and/or “flipping”; eigenvalues give you the factors by which this compression occurs.

What are eigenvectors and eigenvalues for?

1) Mathematical motivation

image                                                                                                                                                                                                                                       is a linear map if it satisfies

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for all a, b, e, R, u and u

Matrices for a linear map

If ¦ is linear then

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Suppose

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Then

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Iterate the function

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Multiply the matrix by itself is computationally hard. One case is easy if M is a diagonal

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Concrete problems

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The method is used to model physics systems such as creatures in an ecosystem, individuals in a society and a political network.

How can we identify the most important part of the map? How do we define the importance of a vertex?

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Good things have many links. Links from important vertices are good.

Give each vertex a score

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A vertex shares its importance equally among those it links to.

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There is no connection between the two. You can’t go from 1 to 4

OR

A different random walk. At each step with probability P follow graph. (1 – P) and jump to new vertex and produce a new matrix

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