Courses:

This course provides a rigorous introduction to fundamentals of random matrix theory motivated by engineering and scientific applications while emphasizing the informed use of modern numerical analysis software. Topics include Matrix Jacobians, Wishart Matrices, Wigner's Semi-Circular laws, Matrix beta ensembles, free probability and applications to engineering, science, and numerical computing. Lectures will be supplemented by reading materials and expert guest speakers, emphasizing the breadth of applications that rely on random matrix theory and the current state of the art.

Additional topics will be decided based on the interests of the students. No particular prerequisites are needed though a proficiency in linear algebra and basic probability will be assumed. A familiarity with MATLAB® will also be useful.

Course Objectives

Although applications and analysis using random matrix methods have emerged over the past decade or so, there is a gap between the mathematical theories and understanding of it by engineers. This is primarily because the theory on random matrices, developed almost concurrently by mathematicians, statisticians, and physicists has not yet been as widely used by engineers for there to be a body of literature employing consistent notation in explaining the use of such random matrix based techniques.

Our objective, in this course, is to present the random matrix theories that engineers have successfully used so far in a manner that highlights the historical and intellectual connections between the applications in mathematics, statistics, physics, and engineering. Along the way, we will

• Derive the eigenvalue density for sample covariance matrices
• Derive Wigner's semicircle law using combinatorial, free probability and resolvent based approaches
• Use MATLAB® to develop tests that assess whether a pair of random matrices is asymptotically free
• Use the Marcenko-Pastur theorem to determine the empirical distribution function for some classes of random sample covariance matrices.

Besides the measurable learning objectives described above, the students will also

• Understand the state of the art in the mathematics of finite dimensional random matrices
• Understand the fundamental mathematics and intuition for the mathematics of infinite dimensional random matrices including the tools of free probability
• Recognize the manner in which these results have been applied so far and be aware of the limitations of these techniques
• Use numerical tools such as MATLAB® to understand more difficult open questions in random matrix theory.