Ranks and inertias of Hermitian block Toeplitz matrices Original Research Article

The rules classifying the connections of the ranks of successive Hermitian Toeplitz matrices were presented in Iohvidovʹs book (Hankel and Toeplitz Matrices and Forms). This paper generalizes those rules to new rules for successive Hermitian block Toeplitz matrices Tm and Tm+1. Given a Hermitian block Toeplitz matrix, Tm, we present the symmetric lattice pentagon of all possible inertias for a successor, Tm+1. These results are based on our work with Nir Cohen [SIAM J. Matrix Anal. Appl. 19 (1998)]. Freund and Huckle showed that Hermitian block Toeplitz matrices which admit “singular” extensions, rank Tm+1=rank Tm coincide with those which admit a factorization with Krylov matrices. This paper extends their list of sufficient conditions.