Low grade matrices and matrix fraction representations (Q1347941)

The authors develop algebraic properties involving the lgrade of matrices. They give applications to signal processing including results for triangular input normal pairs [see \textit{A. Mullhaupt} and \textit{K. S. Riedel}, IEEE Trans. Signal Process. 45, 2616-2619 (1997); IEEE Trans. Autom. Control 46, No. 12, 2018-2022 (2001)]. The lgrade, i.e., the lower grade, of an \(n\times n\) matrix \(A\) is the largest rank of any subdiagonal block of a symmetric partition of \(A.\) The authors give two matrix fraction decompositions of low lgrade matrices: \(M=L^{-1}H\) and \(M=Q^{-1}H,\) where \(L\) and \(H\) have small lower band width and \(Q\) is unitary. Also, let \(M\) be a matrix with \(\text{lgrade}(M)=d.\) Then \(M\) can be approximated as the sum \(M=U+V,\) where \(U\) is upper triangular and \(V\) has rank \(d.\)