Syllabus – Positive Systems

Positive Systems and Nonnegative Matrices

  1. Basics of nonnegative matrices
    1. Perron theorem for positive matrices
    2. Frobenius theorem for nonnegative irreducible matrices
    3. Stochastic matrices and Markov chains
  2. Graphs and Matrices
    1. Incidence matrices
    2. Laplacians
    3. Colin de Verdiere parameters
  3. Stability and Convergence
    1. Z-matrices, M-matrices and P-matrices
    2. Stability, Diagonal stability and D-stability
    3. Paracontractivity and Projective metrices
  4. Applications and Extensions
    1. Nonnegative matrices and the internet – Google, congestion control
    2. Completely positive matrices
    3. Extensions of the Perron-Frobenius theory