Types of convergence of matrices
Olga Pryporova · 2009
This thesis is based on two papers that investigate different types of convergence of matrices. A square matrix is convergent (sometimes referred to as discrete time stable) if all its eigenvalues have modulus less than 1. The first paper investigates relations between stronger types of convergence and extends the results for real matrices to the complex case. In particular, it is proven that for complex matrices of order less than 4, diagonal convergence, DC convergence and boundary convergence are all equivalent. An example of a 4 by 4 matrix that is DC convergent but not diagonally convergent is constructed. The second paper studies potential convergence of modulus patterns. A modulus pattern Z is convergent if all complex matrices with modulus pattern Z are convergent. Also, other types of potential convergence are introduced. Some techniques are presented that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for n by n modulus patterns is given. Complete characterizations of all introduced types of potential convergence for 2 by 2 modulus patterns are also presented.