Tutorial 2 - Examples in Eigenvalue Decomposition

Assignment 1

Using the file dl-matrixmarket.jl from the package MatrixMarket.jl (copy the file to your notebook), download two randomly chosen matrices (make sure that both matrices are real and at least one is real symmetric).

For each matrix:

inspect the properties of the matrix (size, symmetry, condition number, sparsity, structure, ...),
plot the matrix using the command spy() from the package Gadfly.jl,
compute the eigenvalue decomposition with appropriate methods, and
assess the accuracy of the decomposition.

Hints

In Windows, you may need to prepend the http:// to the address in the download() command.
To plot the matrix A, use the following commands:
```
myplot=spy(A)
draw(PNG(12cm,12cm),myplot)
```
To see only the structure, use
```
myplot=spy(map(Int64,A.!=0.0))
```
For larger matrices, plotting takes a while.

Assignment 2

Choose three matrices from the package MatrixDepot.jl:

one from the class "eigen",
one which is sparse and symmetric, and
one which is ill-conditioned and symmetric,

and analyse them as described in the Assignment 1.

Assignment 3$^*$

Learn about Google's Page Rank algorithm and explain it as an eigenvalue problem. Implement the algorithm and analyse Stanford web graph.