Matrix Differential Equations a Continuous Realization Process for Linear Algebra Problems

Many mathematical problems such as existence questions are studied by using an appropriate realization process either iteratively or continuously In this article di erential equation techniques are used as a special continuous realization process for linear algebra problems The matrix di erential equations are cast in fairly general frameworks of which special cases have been found to be closely related to important numerical algorithms The main thrust is to study the dynamics of various isospectral ows This approach has potential applications ranging from new development of numerical algorithms to theoretical solution of open problems Various aspects of the recent development and application in this direction are reviewed in this article Introduction Continuous realization methods are based on the idea of connecting two abstract problems through a mathematical bridge Usually one of the abstract problems is a make up whose solu tion is trivial while the other is the real problem whose solution is di cult to nd The bridge if it exists is regarded as a continuous path in the problem space Following the path means deforming the underlying abstract problem mathematically It is hoped that by following the path the obvious solution will systematically be deformed into the solution that we are seeking for In applying a continuous realization method two basic tasks should be carried out rst since they are most accountable for the success One needs to establish a mathematical theory that can ensure the existence of bridge con necting the two abstract problems One needs to develop a numerical algorithm that can e ectively follow the path The bridge usually takes the form as an integral curve of an ordinary di erential equation describing how the problem data are transformed from the simple system to the more complicated system The numerical algorithm thus should be an e cient ODE solver Depending upon how the bridge is constructed continuous realization methods appear in di erent forms One of the best known continuous realization methods in the literature perhaps is the so called homotopy method The philosophy behind the homotopy method is quite straightforward We use the homotopy method to demonstrate the idea of continuation as follows Suppose the original problem is to solve a nonlinear equation