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 in&de 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 Pad&Hermite and simultaneous Pad6 approximants. The notions of matrix Pad&Hermite and matrix simultaneous Pad6 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 Pad&Hermite and matrix simultaneous Pad6 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 (n,, n,, , n,) this algorithm is shown to compute the inverse components with a complexity of O(k . (n, + ... + n,>‘> 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. *Partially supported by NSERC operating grant 6194 LINEAR ALGEBRA AND ITS APPLICATIONS 177: 7-48 (1992) 0 Elsevier Science Publishing Co., Inc., 1992 7 655 Avenue of the Americas, New York, NY 10010 0024-3795/92/$5.00 8 GEORGE LABAHN