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Perron-Frobenius Theory of Nonnegative Matrices with Applications


Tuesday, 18 February 2020 to Tuesday, 9 June 2020
Total hours: 40
Hours of lectures: 40

Examination procedure

  • oral exam


The course is largely self-contained and should be accessible to anyone with a good knowledge of linear algebra. Hence, it will be accessible also to II and III year students of the Corso Ordinario. 



 The course will cover the fundamental results in the Perron-Frobenius theory concerning the spectral properties of matrices with nonnegative elements, its connections with graph theory, and selected applications to Markov chains, discrete dynamical systems, network science, data science, and numerical analysis. 

 The course is organized as follows:

1. Review of basic facts from linear algebra.

2. Positive matrices and nonnegative matrices. 

3. Graphs.  Irreducibility.  Primitive matrices. Imprimitivity.

4. The Perron-Frobenius Theorem.

5. Main consequences of the Perron-Frobenius Theorem.

6. Related matrix classes: essentially nonnegative matrices, M-matrices, monotone matrices.

7. Stochastic and doubly stochastic matrices. 

8. Applications to the theory of finite Markov chains.

9. Applications to Network science and Data Science. 

10. Computational aspects.


Educational Goal: 

 The students will familiarize themselves with an important mathematical theory rich in applications in several fields of science and technology. Furthermore, they will learn some aspects of combinatorics (graph theory), probability (Markov chains), and Data Science (network science, information retrieval, etc). 



Bibliographical references

 Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 1999.

R. Horn and C. Johnson, Matrix Analysis. Second Edition, Cambridge University Press, 2013.

R. Bapat and T. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, 1997.

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, SIAM, 1994. 

E. Estrada and P. Knight, A First Course in Network Theory, Oxofrd University Press, 2015.

G. Strang, Linear Algebra and Learning from Data, Wellesley Cambridge Press, 20019.

Additional references will be mentioned in the lectures.