Meromorphic Functions and Linear Algebra
The main topic treated in the book is a study of properties of the resolvent (I-zA)-1 of a matrix A, considered as a vector-valued meromorphic function in the complex plane. The tools used in the description are naturally related to tools used for the investigation of meromorphic functions. In particular, the value distribution theory developed by Rolf Nevannlina (great-uncle of the author) is very useful for the considered question. The main point is that while eigenvalues of a matrix may move considerably under a small perturbation by a low rank matrix, the growth of the resolvent, considered as a matrix-valued meromorphic function, is much more stable. The book starts with a summary of basic value distribution theory, which is needed later. Its generalization for matrix-valued meromorphic functions is based on the notion of the total logarithmic size of a matrix. It makes it possible to study the behavior of the growth of the resolvent under low rank perturbations. The book also contains applications to rational approximations, the Kreiss matrix theorem, power boundedness and convergence of the Krylov solvers. The last chapter compares defects in the value distribution theory and defective eigenvalues of matrices. The book will certainly be valuable to mathematicians interested in numerical linear algebra but it also offers attractive information for a general mathematical audience.