Types of convergence of matrices
Is Version Of
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
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.