The Inverses of Block Hankel and Block Toeplitz Matrices
George Labahn, Dong-Koo Choi, Stan Cabay · SIAM Journal on Computing · 1990
A set of new formulae for the inverse of a block Hankel (or block Toeplitz) matrix is given. The formulae are expressed in terms of certain matrix Padé forms, which approximate a matrix power series associated with the block Hankel matrix. By using Frobenius-type identities between certain matrix Padé forms, the inversion formulae are shown to generalize the formulae of Gohberg–Heinig and, in the scalar case, the formulae of Gohberg–Semencul and Gohberg–Krupnik. The new formulae have the significant advantage of requiring only that the block Hankel matrix itself be nonsingular. The other formulae require, in addition, that certain submatrices be nonsingular. Since effective algorithms for computing the required matrix Padé forms are available, the formulae are practical. Indeed, some of the algorithms allow for the efficient calculation of the inverse not only of the given block Hankel matrix, but also of any nonsingular block principal minor.