Lecture: Matrix Analysis

(aka the beauty of spectral analysis)

Instructor
Prof. Dr. Martin Gebert

Lecture
Tuesday and Thursday 10am-12noon, online zoom

Exercise session
Thursday 2pm-4pm, online zoom

Homework
Uni2work

Description
One of the most fundamental objects in mathematics are matrices. They appear naturally when solving systems of linear equations and turn out to be essential in physics, engineering and any kind of (applied) mathematics. In itself matrices have an interesting structure and in this class we try to investigate a number of deeper and beautiful (spectral) properties of matrices. We mostly follow the classical book Matrix Analysis by Rajendra Bhatia. Generally, we will cover in this lecture the following topics

• Quick review of linear algebra: Spectrum, Schur decomposition, singular value decomposition, polar decomposition...
• Majorization and relation between eigenvalues and singular values, e.g. Weyl's and Horn's theorem. Basic question: Are the singular values in some way bigger than the modulus of the absolute value of the eigenvalues?
• Variational principles for eigenvalues, i.e. various min-max principles.
• Unitarily invariant norms, weakly unitarily invariant norms.
• Operator monotone functions. The question we answer here is: For which functions f is f(A) < f(B) for all Hermitian matrices A,B with A < B? This will result in Löwner's theorem.
• Continuity of eigenvalues and perturbation theory for eigenvalues of (normal) matrices.
• Perturbation theory for spectal subspaces, especially Davis-Kahan theorems.
• Matrix inequalities, e.g Lieb concavity...

Most of the results covered in this class hold for compact operators or general bounded operators on infinite-dimensional Hilbert spaces as well. Therefore, having a good understanding of finite dimensional matrices is beneficial for understanding any problem on infinite dimensional space.

Prerequisites
Analysis I-III, Linear Algebra I-II, Functional Analysis I, basics of Complex Analysis.

Audience
Master Mathematics, Master Finanz- und Versicherungsmathematik, TMP master.

Exam
TBA

Literature

• R. Bhatia, Matrix Analysis
• B. Simon, Trace Ideals and application to quantum physics
• R.A. Horn, Matrix analysis