Public defence in the field of Mathematics and Statistics, M.Sc. (Tech) Lauri Nyman

Title of the thesis: Eigenstructure of polynomial and rational matrices: perturbation theory and nearness problems

Thesis defender: Lauri Nyman
Opponent: Professor Françoise Tisseur, University of Manchester, UK
Custos: Associate Professor Vanni Noferini, Aalto University School of Science

The Millennium Bridge in London, when it first opened to the public, wobbled alarmingly and had to be closed. Clearly, the effects of vibrations to the structure were not fully understood when the bridge was built.

Understanding the natural frequencies of a structure requires solving a polynomial eigenvalue problem. This problem is prevalent in a wide range of engineering applications, not just in the study of natural frequencies. Another concrete example, where this problem plays an important role, is the design of an automatic flight control system for airplanes.

This thesis examines the properties of the polynomial eigenvalue problem, and its generalization, the rational eigenvalue problem. When solving these problems in practice, the computed solution can sometimes be very inaccurate. If these inaccuracies are not well understood when building a structure, the end result is likely to collapse. However, it is difficult to know the exact amount of inaccuracy in the computed solution.

The main result of this thesis makes it easier to estimate how inaccurate the computed solution can be. This helps in designing better and more reliable bridges, flight control systems, and other engineering systems. 

Key words: polynomial matrix, rational matrix, eigenstructure, perturbation theory, nearness problem

Thesis available for public display 10 days prior to the defence at Aaltodoc.