Description:

Theory of random matrices appeared first in 60's in the 20th century in connection with statistical physics and the theory of nucleis of atoms of heavy metals. The main interest of study is the distribution of eigenvalues of symmetric random matrices. In the 21st century the results of theory of random matrices were applied in theoretical computer science and numerics for design of random algorithms.

Contents:

1. Examples of random matrix ensembles, classes GOE and GUE, Wigner?s surmise for GOE(2), joint probability density function of spectra of GOE and its proof, Layman?s classification, Wigner?s semicircle law
2. Bernstein?s concentration inequality, Golden-Thompson inequality, Lieb?s theorem, applications of Bernstein?s inequality: sparsification of matrices, matrix multiplication, reconstruction of low-rank matrices, randomized matrix decompositions.

Recommended literature:

M.L. Mehta: Random Matrices 3rd edition, Academic Press, New York (2004)
G. Livan, M. Novaes, P. Vivo: Introduction to Random Matrices: Theory and Practice, Springer, 2018
J. Tropp: An Introduction to Matrix Concentration Inequalities, Foundations and Trends in Machine Learning, 8(1-2), 2015
M. Krbálek and P. Šeba: Statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles, J. Phys. A: Math. Theor. 33 (2000), L229

Keywords:

Wiegner?s semicircle law, GOE, joint spectral density, non-commutative Bernstein?s inequality, randomized algorithms

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar