Result: The Interplay of Ranks of Submatrices: The interplay of ranks of submatrices

The authors prove the following theorem. Let \(T\) be an invertible \(n\times n\) matrix over a field and let \(B\) be the submatrix of size \(m\times(n-m-p)\), say, in the upper right hand corner of \(T\) (so \(B\) lies above the \(p\)th superdiagonal of \(T\)). Let \(C\) be the submatrix of \(T^{-1}\) of size \((m+p)\times(n-m)\) in the upper right hand corner of \(T^{-1}\) (so \(C\) lies above the \(p\)th subdiagonal). Then \(\text{rank}(C)=\text{rank} (B)+p\). A short proof of this is based on a nullity theorem, originally proved by \textit{W.H. Gustafson} [Linear Algebra Appl. 57, 71-73 (1984; Zbl 0533.15002)] in a module theoretic form and later by \textit{M. Fiedler} and \textit{T. L. Markham} [Linear Algebra Appl. 74, 225-237 (1986; Zbl 0592.15002)] in the language of matrix theory. The authors discuss the history of special cases of their result, and applications to fast computations for a tridiagonal system of linear equations and numerical solution of differential and integral equations.