Low grade matrices and matrix fraction representations

The lower grade (lgrade) of an n×n matrix A is the largest rank of any subdiagonal block of a symmetric partition of A. A number of algebraic results on lgrade are given. When A has lgrade d, it can be approximately decomposed as A=U+V, where U is an upper triangular matrix and V has rank d. If A satisfies GA=N with G and N having lower bandwidths d_G and d_N, then the decomposition is exact: A=U+V, where U is an upper triangular matrix with lower bandwidth equal to d_N−d_G and V has low rank (generically d_G). This result generalizes the well-known representations of A when A=G⁻¹ and G is banded. A generalization of the Givens rotation product decomposition of unitary Hessenberg matrices is given and its structure is analyzed. These “consecutive subblock products” are used to construct a representation of an lgrade-d matrix A of the form GA=N with G and N having lower bandwidth d. G can be chosen to be lower triangular or unitary.