On Moore–Penrose inverses of quasi-Kronecker structured matrices

The Moore–Penrose inverse and generalized inverse of , where A, X1, X2 are complex matrices are given under various assumptions. We use the result to derive the Moore–Penrose inverse and inverse for bdiag(Ak)+uv∗⊗E with p complex matrices Ak, 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.