On Moore-Penrose inverses of quasi-Kronecker structured matrices

Abstract

The Moore-Penrose inverse and generalized inverse of A + X(1)X(2)*, where A. X(1), X(2) are complex matrices are given under various assumptions. We use the result to derive the Moore-Penrose inverse and inverse for bdiag(A(k)) + uv* circle times E with p complex matrices A(k), two complex p-vectors u and v and a complex matrix E. Such block structured matrices occur in hierarchical modeling of multivariate spatial or space-time Gaussian processes. For the latter we also give expressions of the determinant and of conditional variances. (C) 2011 Elsevier Inc. All rights reserved.