A short note on the nullity theorem

In this paper we take a close look at the nullity theorem as formulated by Markham and Fiedler in 1986. The theorem is a valuable tool in the computations with structured rank matrices, it connects ranks of subblocks of an invertible matrix A with ranks of other subblocks in his inverse A. A little earlier, Barrett and Feinsilver, 1981, proved a theorem very close to the nullity theorem, but restricted to semiseparable and tridiagonal matrices, which are each others inverses. We will adapt the ideas of Barrett and Feinsilver to come to a new, alternative proof of the nullity theorem, based on determinantal formulas. In a second part of the paper, we extend the nullity theorem to make it suitable for two types of decompositions, namely the LU and the QR decomposition. These theorems relate the ranks of subblocks of the factors L, U and Q to the ranks of subblocks of the factored matrix. It is shown, that a combination of the nullity theorem and his extended versions is suitable to predict in an easy manner the structure of decompositions and/or of inverses of structured rank matrices, e.g., higher order band, higher order semiseparable, Hessenberg, and many other types of matrices. As examples, to show the power of the nullity theorem and the related theorems, we apply them to semiseparable and related matrices.