pinv_of_dcap: The Moore-Penrose Inverse of the Duplication Matrix

In jeksterslab/linearAlgebra: Linear Algebra with R

Description

The Moore-Penrose inverse of the duplication matrix.

Usage

pinv_of_dcap(k)

Arguments

k	Positive integer. Dimension of the k by k matrix.

Details

The Moore-Penrose inverse of the duplication matrix \mathbf{D}_{k} is the \frac{1}{2} k ≤ft( k + 1 \right) \times k^2 matrix given by

\mathbf{D}_{k}^{+} = ≤ft( \mathbf{D}_{k}^{\prime} \mathbf{D}_{k} \right)^{-1} \mathbf{D}_{k}^{\prime}

where

\mathbf{D}_{k}^{+} \mathrm{vec} ≤ft( A \right) = \mathrm{vech} ≤ft( \mathbf{A} \right) \quad ≤ft( \mathbf{A} = \mathbf{A}^{\prime} \right)

\mathrm{vec} ≤ft( \cdot \right) is the vectorization of a matrix, and \mathrm{vech} ≤ft( \cdot \right) is the half-vectorization of a matrix.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Symmetric Functions: dcap(), mcap_cor(), mcap_diag(), mcap_sym(), sym_of_vechs(), sym_of_vech()

Examples

pinv_of_dcap(3)

jeksterslab/linearAlgebra documentation built on Dec. 20, 2021, 10:10 p.m.