On V-orthogonal projectors associated with a semi-norm

摘要

For any n x p matrix X and n x n nonnegative definite matrix V, the matrix X(X'VX)(+) X'V is called a V-orthogonal projector with respect to the seminorm parallel to . parallel to v, where (.)(+) denotes the Moore-Penrose inverse of a matrix. Various new properties of the V-orthogonal projector were derived under the condition that rank(VX) = rank( X), including its rank, complement, equivalent expressions, conditions for additive decomposability, equivalence conditions between two (V-) orthogonal projectors, etc.