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

It is well-established that, for an n x n singular k- banded complex matrix B, the submatrices of the Moore-Penrose inverse B- dagger of B located strictly below (resp. above) its k th superdiagonal (resp. kth subdiagonal) have a certain bounded rank s depending on n, k and rankB. B. In this case, B( dagger )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 k th subdiagonal) coincide with the same parts of B- dagger . When n >= 3 k, we prove that s is precisely the minimum rank of these two matrices. We also illustrate through examples that when n < 3k k those matrices may have rank less than s.