Abstract
For any n × p matrix X and n × n nonnegative definite matrix V, the matrix X(X′V X)+ X′V is called a V-orthogonal projector with respect to the semi-norm \(\| \cdot \|_{\bf V}\) , where (·)+ denotes the Moore-Penrose inverse of a matrix. Various new properties of the V-orthogonal projector were derived under the condition that rank(V X) = rank(X), including its rank, complement, equivalent expressions, conditions for additive decomposability, equivalence conditions between two (V-)orthogonal projectors, etc.
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Tian, Y., Takane, Y. On V-orthogonal projectors associated with a semi-norm. Ann Inst Stat Math 61, 517–530 (2009). https://doi.org/10.1007/s10463-007-0150-4
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DOI: https://doi.org/10.1007/s10463-007-0150-4