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.

Two companion algorithms are developed for constructing Pade fractions along an off-diagonal path of the Padé table for a function ${{ - A(z)} / {B(z)}}$, where $A(z)$ and $B(z)$ are formal power series over a field. One of the algorithms computes the first n Padé fractions along the off-diagonal in time $O(n^2 )$. When $A(z)$ and $B(z)$ are finite power series (i.e., polynomials), it is shown that the algorithm is equivalent to Euclid’s extended algorithm for computing greatest common divisors. The other algorithm, a generalization of the first, proceeds along the off-diagonal in quadratic steps, and is of complexity $O(n\log ^2 n)$. When $A(z)$ and $B(z)$ are polynomials, the second algorithm becomes a fast Euclid’s extended algorithm for computing greatest common divisors. The algorithm is of the same complexity as other fast greatest common divisor methods, but its iterative nature provides a practical advantage during implementation. The algorithms may also be used for computing Padé fractions along an anti-diagonal path of the Padé table. The fast algorithm is of the same complexity as other fast algorithms for anti-diagonal computations. However, it has the advantage of being able to determine easily any specific Padé fraction along the anti-diagonal.