Tridiagonal generalized inverses of singular matrices possessing the triangle property

Abstract

It is known that an invertible real square matrix has the triangle property if and only if the inverse is a tridiagonal matrix. This result has an implicit importance due to the fact that nonsingular tridiagonal matrices arise in a variety of problems in pure and applied mathematics and for this reason they have been extensively studied in the literature. However, the singular case has received comparatively much lesser attention. In particular, there has been little focus on the generalized inverses of such matrices. In this paper, we provide a complete description of those singular matrices possessing the triangle property to have the tridiagonal Moore-Penrose inverse or group inverse. The converse statements are also completely answered.