Facts and computations that will be useful in learning and interpreting Markov models

• What are eigenvalues/eigenvectors?
• How do you compute eigenvalues and eigenvectors?
• How do you interpret eigenspectrum of a transition matrix?
• How do you factor matrices?
  • http://en.wikipedia.org/wiki/Matrix_decomposition
• What are properties of invertible matrices?
  • http://en.wikipedia.org/wiki/Invertible_matrix
• What is the Moore-Penrose pseudo-inverse, how do you compute it, and how can we use it in MSM theory?
  • http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse
• How can we exploit matrix sparsity?
  • http://en.wikipedia.org/wiki/Sparse_matrix
• What are some neat properties of diagonal matrices?

• How do you find a matrix $U$ s.t. $U^T V$ is invertible?
  • Singular value decomposition
    • Step-by-step!

• Rank of a matrix?
  • Number of linearly independent columns (equal to the number of linearly independent rows) -- revealed by SVD, or more cheaply, LU decomposition

• Probability theory:
  • Tail inequality bounds?
  • Matrix perturbation theory?

Tutorial on Spectral Clustering:

• http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/Luxburg07_tutorial_4488%5B0%5D.pdf