On the rank structure of the Moore-Penrose inverse of singular k-banded matrices

Abstract

It is well-established that, for an $n\times n$ singular k-banded complex matrix B, the submatrices of the Moore-Penrose inverse $B^{†}$ of B located strictly below (resp. above) its kth superdiagonal (resp. kth subdiagonal) have a certain bounded rank s depending on n, k and rankB. In this case, $B^{†}$ is said to satisfy a semiseparability condition. In this paper our focus is on singular strictly k-banded complex matrices B, and we show that the Moore-Penrose inverse of such a matrix satisfies a stronger condition, called generator representability. This means that there exist two matrices of rank at most s whose parts strictly below the kth diagonal (resp. above the kth subdiagonal) coincide with the same parts of $B^{†}$. When $n\ge 3k$, we prove that s is precisely the minimum rank of these two matrices. We also illustrate through examples that when $n<3k$ those matrices may have rank less than s.