Monday 21st July
Eigenvalues and Eigenvectors
Dr Robert Johnson
This is a subject that students find very difficult to get motivated with.
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
for all a, b, e, R, u and u
Matrices for a linear map
If ¦ is linear then
Iterate the function
Multiply the matrix by itself is computationally hard. One case is easy if M is a diagonal
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?
Good things have many links. Links from important vertices are good.
Give each vertex a score
A vertex shares its importance equally among those it links to.
There is no connection between the two. You can’t go from 1 to 4
A different random walk. At each step with probability P follow graph. (1 – P) and jump to new vertex and produce a new matrix