Numerical methods for problems involving the Drazin inverse

The objective was to try to develop a useful numerical algorithm for the Drazin inverse and to analyze the numerical aspects of the applications of the Drazin inverse relating to the study of homogeneous Markov chains and systems of linear differential equations with singular coefficient matrices. It is felt that all objectives were accomplished with a measurable degree of success

Continuity of the core-EP inverse and its applications

In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition. The continuity of the core inverse can be derived as a particular case. Secondly, we study perturbation bounds for the core-EP inverse under prescribed conditions. Perturbation bounds for the core inverse can be derived as a particular case. Also, as corollaries, the sufficient (and necessary) conditions for the continuity of the core-EP inverse are obtained. Thirdly, a numerical example is illustrated to compare derived upper bounds. Finally, an application to semistable matrices is provided.This research is supported by the National Natural Science Foundation of China (No. 11771076), partially supported by FCT-'Fundacao para a Ciencia e a Tecnologia', within the project UID-MAT-00013/2013

Some Results for the Drazin Inverses of the Sum of Two Matrices and Some Block Matrices

We give a formula of (P+Q)D under the conditions P2Q+QPQ=0, P3Q=0, and PQPQ=0. Then, we apply it to give some expressions for the Drazin inverse of block matrix M=(ABCD) (A and D are square matrices) under some conditions, generalizing some recent results in the literature. Finally, numerical examples are given to illustrate our results

An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index

This paper deals with singular systems of index k ≥ 1. Our main goal is to find a state-feedback such that the closed-loop system satis- fies the regularity condition and it is nonnegative and stable. In order to do that, the core-nilpotent decomposition of a square matrix is applied to the singular matrix of the system. Moreover, if the Drazin projector of this matrix is nonnegative then the previous decomposition allows us to write the core-part of the matrix in a specific block form. In addition, an algorithm to study this kind of systems via a state-feedback is designed.This paper has been partially supported by Ministry of Education of Spain [grant number DGI MTM2010-18228].Herrero Debón, A.; Francisco J. Ramírez; Thome, N. (2014). An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index. Linear and Multilinear Algebra. 1-11. https://doi.org/10.1080/03081087.2014.904559S11