Matrix equations and tensor problems

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This is a proposal for research on some topics in applied mathematics. My main goal is to advance the study of nonlinear matrix equations, tensor eigenvalue problems, and tensor equations. Nonlinear matrix equations appear in many applications. They may appear in stochastic models for queueing problems arising in telecommunications, computer performance, and inventory control. They may appear in nano research, in the study of economical models, and so on. In each practical application, a special matrix satisfying the matrix equation is to be found. Great progress has been made on the design and analysis of numerical methods for solving nonlinear matrix equations from various applications, through the efforts of many researchers, myself included. However, a number of interesting (and potentially very difficult) problems remain unsolved. My goal is to solve some of these problems. They may involve further theoretical analysis of algorithms known to work well in practice, or the design of more efficient algorithms for finding the desired solution of a particular equation, or the establishment of a working theory for some matrix equations deserving further attention.  Tensors are generalizations of matrices. Tensor eigenvalue problems have applications in spectral hypergraph theory and higher order Markov chains. Tensor equations have applications in data mining and numerical partial differential equations. My research on tensor problems will be focused on nonnegative tensors and other closely related tensors. My goal is to establish a sound theory for some existing numerical methods that are useful in practice for solving tensor eigenvalue problems, and to design new and more efficient algorithms for solving tensor equations. The successful completion of my proposed research will be a significant contribution to the study of matrix equations, tensor eigenvalue problems and tensor equations. In particular, tensor analysis is a relatively new and flourishing research area. Canadian contributions to this research area is very limited so far. My further work in this area will help to improve the visibility of Canadian research in this important research area.