Weak Drazin inverses

A new type of generalized inverse is defined which is a weakened form of the Drazin inverse. These new inverses are called (d)-inverses. Basic properties of (d)-inverses are developed. It is shown that (d)-inverses are often easier to compute than Drazin inverses and can frequently be used in place of the Drazin inverse when studying systems of differential equations with singular coefficients or when studying Marcov chains.