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On \(\mathbf V\)-orthogonal projectors associated with a semi-norm. (English) Zbl 1332.15016

Summary: For any \(n \times p\) matrix \(\mathbf X\) and \(n \times n\) nonnegative definite matrix \(\mathbf V\), the matrix \(\mathbf X(\mathbf X^{\prime} \mathbf {VX})^{+} \mathbf X^{\prime} \mathbf V\) is called a \(\mathbf V\)-orthogonal projector with respect to the semi-norm \(\|\cdot\|_{\mathbf V}\), where \((\cdot)^{+}\) denotes the Moore-Penrose inverse of a matrix. Various new properties of the \(\mathbf V\)-orthogonal projector were derived under the condition that \(\operatorname{rank}(\mathbf {VX}) = \operatorname{rank}(\mathbf X)\), including its rank, complement, equivalent expressions, conditions for additive decomposability, equivalence conditions between two (\(\mathbf V\)-)orthogonal projectors, etc.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
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References:

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