Group inverses and Drazin inverses of bidiagonal and triangular Toeqlitz matrices

Abstract

Necessary and sufficient conditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

Some additive results on Drazin inverses

“…Now from [7], we know that the existence of the group inverses for A and a 1 , guarantee thatà # also exists. Repeating this we see that the group invertibility ofà # implies the group invertibility of a 2 .…”

“…Therefore, K has a Drazin inverse. Lastly, since K and Z are D-invertible, it again follows from [7], that W d exists, ensuring that b is D-invertible.…”

“…The smallest values of r and s are called the left and right index of a, respectively (see [7]). As shown by Drazin [2], if r and s are finite then r = s = in(a).…”

Singular Case of Generalized Fibonacci and Lucas Matrices

“…The generalized inverse for Hankel and Toeplitz matrices can be found in [1,9,14,15,13,27,29]. Hartwig and Shoaf [12] considered the group inverse and the Drazin inverse of singular bidiagonal and triangular Toeplitz matrices.…”

Schur’s theorem and the Drazin inverse

“…In a recent paper [7], Her stein and Small extended the classic result of Schur [5, p. 46] to matrices over i?-rings. These are rings for which every primitive image is artinian.…”