Generalized inverses, symbolic computation and operator algebras

The proposed project comprises the fields of Generalized inverses, Operator theory, and Symbolic computation. The theory of generalized inverses has seen a substantial growth over the past few decades. It is a subject of great theoretical interest and finds applications in a large number of various areas including Statistics, Numerical analysis, Differential equations, Markov chains, population models, Cryptography, and Control theory, which is confirmed by the amount of published papers in this topic. Despite the popularity, there are still many open problems in this area, solving which would be important, not only theoretically, but also in resolving other specific problems. Namely, this project concerns some open problems in the theory of generalized inverses focusing on its applications. One important problem in this theory is the so-called “reverse order law“ (ROL), which has been studied intensively for various kinds of generalized inverses mostly in the matrix setting. This problem was initially proposed by Greville in 1960’s, who find necessary and sufficient conditions for (AB)+=B+A+ to hold. Following his result, there have been many variations on the subject: the case of more than two matrices, different types of generalized inverses and various settings (operator algebras, C*-algebras, rings with involution etc.). The ROL problem is still an active research field. The ROL for more general structures in the case of more than two elements will be the first topic of our joint research. The second main subject of our research will be symbolic computation of generalized inverses of rational and polynomial matrices and its applications. This is a very important problem with its main applications in automatic control theory. Especially, the computation of generalized inverses of univariate and multivariate rational matrices will be considered.