Representations of the Drazin inverse on solution of a class singular differential equations

Abstract

In 1983, Campbell proposed a problem to find an explicit representation of the Drazin inverse for the block matrix with E and F square in terms of E and F according to the research on singular differential equations. In this article, we give the representations of the Drazin inverse for with E and F square under the condition that EF = FE and the Drazin inverse for with E square under the condition that EFG = FGE, and then give some representations under some other conditions, so as to solve the problem partially and to get some extra results.

Keywords:

• singular differential equations
• block matrix
• Drazin inverse
• index
• binomial coefficient

AMS Subject Classifications:

• 15A09
• 65F20

Acknowledgements

Supported by Natural Science Foundation of the Heilongjiang Province, No. 159110120002.