Singular matrices whose Moore-Penrose inverse is tridiagonal

Abstract

A variety of characterizations of nonsingular matrices whose inverse is tridiagonal (irreducible or not) have been widely investigated in the literature. One well-known such characterization is stated in terms of semiseparable matrices. In this paper, we consider singular matrices A and give necessary and sufficient conditions for the Moore-Penrose inverse of A to be tridiagonal. Our approach is based on bordering techniques, as given by Bapat and Zheng (2003). In addition, we obtain necessary conditions on A analogous to the semiseparability conditions in the nonsingular case, though in the singular case they are not sufficient, as illustrated with examples. We apply our results to give an explicit description of all the 3×3 real singular matrices and 3×3 Hermitian matrices whose Moore-Penrose inverse is irreducible and tridiagonal.