INVERSION COMPONENTS OF BLOCK HANKEL-LIKE MATRICES

The inversion problem for square matrices having the structure of a block Hankel-like matrix is studied. Examples of such matrices include Hankel striped, Hankel layered, and vector Hankel matrices. It is shown that the components that both determine nonsingularity and construct the inverse of such matrices are closely related to certain matrix polynomials. These matrix polynomials are multidimensional generalizations of Pade-Hermite and simultaneous Pade approximants. The notions of matrix Pade-Hermite and matrix simultaneous Pade systems are also introduced. These are shown to provide a second set of inverse components for block Hankel-like matrices. A recurrence relation is presented that allows for efficient computation of matrix Pade-Hermite and matrix simultaneous Pade systems. As a result it is shown that the inverse components can be computed via either the matrix Euclidean algorithm or a matrix Berlekamp-Massey algorithm applied to an associated matrix power series. An alternative algorithm based on this recurrence relation is also presented. For a block Hankel-like matrix of type (n0, n1,..., n(k)) this algorithm is shown to compute the inverse components with a complexity of O(k . (n(0) + ... + n(k))2) block matrix operations, although this can be higher in some pathological cases. This is the same complexity as with existing algorithms. This algorithm has the significant advantage, however, that no extra conditions are required on the input matrix. Other block algorithms require that certain submatrices be nonsingular. Similar results hold in the case of block Toeplitz-like matrices.